From 7fe0a8d9fd540c953f17d9cff606c7483b14b3e7 Mon Sep 17 00:00:00 2001 From: magabrielaa <90418921+magabrielaa@users.noreply.github.com> Date: Tue, 18 Jul 2023 14:45:08 -0500 Subject: [PATCH] Adding impact evaluation files --- README.md | 14 + ps-1/PS1.Rmd | 429 + ps-1/PS1.pdf | Bin 0 -> 254935 bytes ps-1/ps1_data_22.csv | 10001 +++++++++++++++++++ ps-2/PS2.Rmd | 963 ++ ps-2/PS2.pdf | Bin 0 -> 310213 bytes ps-2/ps2_data.csv | 10001 +++++++++++++++++++ ps-2/ps2_data.dta | Bin 0 -> 565326 bytes ps-3/PS3.Rmd | 423 + ps-3/PS3.pdf | Bin 0 -> 327452 bytes ps-3/ps3_data.csv | 4001 ++++++++ ps-4/PS4.Rmd | 484 + ps-4/PS4.pdf | Bin 0 -> 512312 bytes ps-4/ps_4_data.csv | 22001 +++++++++++++++++++++++++++++++++++++++++ ps-5/data_2022.csv | 5001 ++++++++++ ps-5/ps5.Rmd | 853 ++ ps-5/ps5.pdf | Bin 0 -> 2657539 bytes 17 files changed, 54171 insertions(+) create mode 100644 README.md create mode 100644 ps-1/PS1.Rmd create mode 100644 ps-1/PS1.pdf create mode 100644 ps-1/ps1_data_22.csv create mode 100644 ps-2/PS2.Rmd create mode 100644 ps-2/PS2.pdf create mode 100644 ps-2/ps2_data.csv create mode 100644 ps-2/ps2_data.dta create mode 100644 ps-3/PS3.Rmd create mode 100644 ps-3/PS3.pdf create mode 100644 ps-3/ps3_data.csv create mode 100644 ps-4/PS4.Rmd create mode 100644 ps-4/PS4.pdf create mode 100644 ps-4/ps_4_data.csv create mode 100644 ps-5/data_2022.csv create mode 100644 ps-5/ps5.Rmd create mode 100644 ps-5/ps5.pdf diff --git a/README.md b/README.md new file mode 100644 index 0000000..0a868d2 --- /dev/null +++ b/README.md @@ -0,0 +1,14 @@ +# Impact Evaluation + +This repository contains code and analysis in R, employing different econometrics techniques for impact evaluation examples with a policy focus. Each sub-repository includes the data as .csv files for replication. + +I have also included the knitted PDF files for easier readability of the formulas. + +Some of the methods implemented include: + +- Linear regression +- RCT +- Matching +- Difference-in-Differences +- Instrumental Variable (IV) +- Regression Discontinuity (RD) \ No newline at end of file diff --git a/ps-1/PS1.Rmd b/ps-1/PS1.Rmd new file mode 100644 index 0000000..727595c --- /dev/null +++ b/ps-1/PS1.Rmd @@ -0,0 +1,429 @@ +--- +title: "PPHA 34600 Program Evaluation" +author: "Problem Set 1" +date: "4/14/2022" +output: pdf_document +--- + +```{r setup, include=FALSE} +# Clean the environment, set up the working directory: +rm(list=ls()) +setwd("~/Desktop/Spring 2022/Program Evaluation/PS1") + +# tell R not to use scientific notation +options(scipen = 999) + +# Load libraries +library(tidyverse) +library(ggplot2) +library(kableExtra) +library(broom) + +#Load data +keller <- read_csv("ps1_data_22.csv") +``` + +### (1) KELLER would like to know about the payment impacts of their disconnections program. They say they’re interested in measuring the impact of their disconnections, but don’t exactly know what that means. Use the potential outcomes framework to describe the impact of treatment (defined as “disconnecting a household’s electricity”) for household i on electricity payments (measured in rupees) formally (in math) and in words.### + +Using the potential outcomes framework, we would theoretically measure the +impact of disconnecting a household's electricity by taking the difference of the +potential outcome on electricity payments for a given household under +treatment(disconnecting its electricity) minus the outcome for that same +household not being treated(not having its electricity disconnected). + +Mathematically, this would be represented as follows: + +$$\tau_{household\space i} = Y_{household\space i}(D = 1) - Y_{household\space i}(D = 0)$$ +$$\tau_{household\space i} = Y_{household\space i}(1) - Y_{household\space i}(0)$$ +Where: +$D_i=1$ for treated households and $D_i = 0$ for untreated households (ie. control +group). In this case treatment refers to disconnecting a household's electricity. + +$Y_i(D_i)$ is the outcome in electricity payments (measured in rupees) for +household i with treatment status $D_i$ and $\tau_i$ is the treatment effect +for an individual household i. + + +### (2) KELLER are extremely impressed. They want to know how they can go about measuring tau_i. Let them down gently, but explain to them why estimating tau_i is impossible. ### +The treatment effect for each individual household is only a theoretical +estimation. In reality, it is impossible to calculate $\tau_i$ because we need +to observe the two potential outcomes for each individual household under both +states of the world: treatment and no treatment(ie. control). That is, +observing the potential outcome for a treated household and at the same time, +what would have been its potential outcome had the household not been treated. +This is known as the fundamental problem of causal inference, since we cannot +ever observe both $Y_{household\space i}(1)$ and $Y_{household\space i}(0)$ +simultaneously for the same household. + + +### (3) KELLER are on board with the idea that they can’t estimate individual-specific treatment effects. They suggest estimating the average treatment effect instead. They are willing to give you some of their early data on payments. They have data on households who did and didn’t get disconnected, and want you to compare the average payments across the two sets of households. Describe what this is actually measuring, and provide an example of why this may differ from the average treatment effect. ### + +This early data on payments does not correspond to a process of randomized +assignment to treatment and control groups, it simply is a registry of household +payments and whether they were disconnected or not. In the absence of random +assignment, this is not the same as the ATE and instead is a naive estimator +based on **observed outcomes** instead of **potential outcomes**. + +What this is measuring is: +$$\tau^N = \overline{Y}(1) - \overline{Y}(0)$$ +Which is NOT the same as ATE: + + +$$\tau^{ATE} = E[Y_i(1)] - E[Y_i(0)]$$ +For example, if the households that were behind on their payments and paid after +their electricity was disconnected were households in a coal mine neighborhood +where workers hadn't been paid and once they were paid, they paid for their +electricity bill, then what caused them to settle their bill was receiving their +overdue wages and not the fact that their electricity was cut-off. This would be +a case of selection bias, where the households that pay their bills are systematically +different from the households that don't pay their electricity bills for reasons +**other** than treatment. In this example, we would be incorrectly overestimating +the effect of disconnecting a household's electricity on their payments when there +are economic and labor factors behind that are responsible for that causality. + + +### (4) KELLER have realized the error of their ways. Their CEO tells you, “Okay, we understand that our data won’t let us estimate the average treatment effect. But can’t we estimate the average treatment effect on the treated?” First formally (in math) define the ATT in this context, and then explain whether or not the KELLER data will allow you to estimate it. If so, describe how what you see in the data corresponds to the necessary components of the ATT. If not, explain why not, and describe what you can’t see in the data that you’d need to observe.### +In this context, the average treatment effect on the treated is: +$$\tau^{ATT} = E[\tau_i|D_i= 1] = E[Y_i(1) - Y_i(0)|D_i = 1]$$ +$$\tau^{ATT} = E[Y_i(1)|D_i = 1] - E[Y_i(0)|D_i = 1]$$ +From the data, we cannot observe the second term $E[Y_i(0)|D_i = 1]$, that is +what would have been the potential outcome (endline_payments) for households whose +electricity actually got disconnected ($D_i = 1$) had their electricity not been +disconnected $Y_i(0)$. Similar as before, the fundamental problem of causal +inference remains and we cannot actually calculate ATT from the given data. + + +### (5) KELLER forgot to tell you that they ran a randomized pilot study to estimate the effects of disconnections on payments. They’re happy to share those data with you: find it in ps1_data_22.csv. This experience has made you a little bit skeptical of KELLER’s skills, so start by checking (with a proper statistical test) that the treatment group and control group are balanced in pre-treatment payments, electricity usage, household size, and household head age. Use keller_trt as your treatment variable. Report your results. What do you find? ### +```{r} +# First we segment the data into treatment and control groups: +treatment <- filter(keller, keller_trt == 0) +control <- filter(keller, keller_trt == 1) + +# Then, we segment the treatment and control groups by each baseline characteristic +# and run a t-test to check if they are balanced: + +# (1) Pre-treatment payments (baseline_payments) +payments_t <- treatment %>% pull(baseline_payments) +payments_c <- control %>% pull(baseline_payments) +mt1 = mean(treatment$baseline_payments) +mc1 = mean(control$baseline_payments) +sdt1 = sd(treatment$baseline_payments) +sdc1 = sd(control$baseline_payments) +t_test <-t.test(payments_t, payments_c) +p1 = t_test$p.value + +# (2) Electricity usage (baseline_elec_use) +elec_t <- treatment %>% pull(baseline_elec_use) +elec_c <- control %>% pull(baseline_elec_use) +mt2 = mean(treatment$baseline_elec_use) +mc2 = mean(control$baseline_elec_use) +sdt2 = sd(treatment$baseline_elec_use) +sdc2 = sd(control$baseline_elec_use) +t_test <-t.test(elec_t, elec_c) +p2 = t_test$p.value + +# (3) Household size (baseline_hhsize) +hsize_t <- treatment %>% pull(baseline_hhsize) +hsize_c <- control %>% pull(baseline_hhsize) +mt3 = mean(treatment$baseline_hhsize) +mc3 = mean(control$baseline_hhsize) +sdt3 = sd(treatment$baseline_hhsize) +sdc3 = sd(control$baseline_hhsize) +t_test <-t.test(hsize_t, hsize_c) +p3 = t_test$p.value + +# (4) Household head age (baseline_hh_head_age) +hage_t <- treatment %>% pull(baseline_hh_head_age) +hage_c <- control %>% pull(baseline_hh_head_age) +mt4 = mean(treatment$baseline_hh_head_age) +mc4 = mean(control$baseline_hh_head_age) +sdt4 = sd(treatment$baseline_hh_head_age) +sdc4 = sd(control$baseline_hh_head_age) +t_test <-t.test(hage_t, hage_c) +p4 = t_test$p.value + +baseline = c("Payments", "Electricity Use", "Household Size", "Household Head Age") +p_value = c(p1, p2, p3, p4) +mean_treatment = c(mt1, mt2, mt3, mt4) +sd_treatment = c(sdt1, sdt2, sdt3, sdt4) +mean_control = c(mc1, mc2, mc3, mc4) +sd_control = c(sdc1, sdc2, sdc3, sdc4) +diff_means = c(mt1-mc1,mt2-mc2, mt3-mc3, mt4-mc4) +df = data.frame(baseline, mean_treatment, sd_treatment, mean_control,sd_control,diff_means, p_value) + +kbl(df, caption = "Balance Table", booktabs = T, digits = 3,col.names = + c("Baseline Characteristic", "Mean Treated", "Std Dev Treated", + "Mean Control", "Std Dev Control", "Diff Means", "p-value"))%>% kable_styling( + latex_options = c("striped","hold_position"), position = "center") +``` + +From the table, we can observe that after running a Welch Two-Sample t-test, +the p-values of all four baseline characteristics are not statistically +significantly different than 0 at any of the conventional levels of statistical +significance (1%, 5% and 10%). + +$$ H_0: mean\space treatment - mean\space control = 0$$ +$$ H_A: mean\space treatment - mean\space control \neq 0$$ + +This means that for all cases, we fail to reject the null hypothesis that the +true difference in means between the the treatment and control groups is equal +to 0, therefore the t-statistic results indicate that the treatment and control +groups are statistically balanced in all the baseline characteristics. + +It's noticeable from the table that the standard deviation for the treatment group baseline payments is much larger than the respective standard deviation for the control group (ie. 282869.182 > 74.530). This suggests that there may be an issue with the distribution of the data for this baseline characteristic within the treatment group. + +### (6) Plot a histogram of pre-treatment payments for treated farms and control households. What do you see? Re-do your balance table to reflect any necessary adjustments. What does this table tell you about whether or not KELLER’s randomization worked? What assumption do we need to make on unobserved characteristics in order to be able to estimate the causal effect of keller_trt?### +```{r histogram} + +hist(treatment$baseline_payments, + main = "Histogram of pre-treatment payments for \nTreated Households", + xlab="Baseline electricity payments", + col="darkmagenta") + +min(treatment$baseline_payments) +max(treatment$baseline_payments) + +``` +In the histogram of electricity payments for the treatment group, the graph +looks very distorted due to an outlier that is very large = $20.000.000 +```{r} +hist(control$baseline_payments, + main = "Histogram of pre-treatment payments for \nControl Households", + xlab="Baseline electricity payments", + col="blue") +``` +The histogram of electricity payments for the control group does not show any +extreme outliers like in the case of the treatment group. It shows that the data +for electricity payments at the baseline is evenly distributed. + +Since the outlier in the previous case distorts the data, it is important to +drop this observation: +```{r} +keller_filtered <- keller[keller$baseline_payments <20000000,] +treatment <- filter(keller_filtered, keller_trt == 0) +hist(treatment$baseline_payments, + main = "Adjusted Histogram of pre-treatment payments for \nTreated Households", + xlab="Baseline electricity payments", + col="darkmagenta") + +``` +After dropping the extreme outlier, we can observe that the data for baseline +payments is evenly distributed for the treatment group. We need to recalculate +the balance table with this adjustment: +```{r} +# (1) Pre-treatment payments (baseline_payments) +payments_t <- treatment %>% pull(baseline_payments) +payments_c <- control %>% pull(baseline_payments) +mt1 = mean(treatment$baseline_payments) +mc1 = mean(control$baseline_payments) +sdt1 = sd(treatment$baseline_payments) +sdc1 = sd(control$baseline_payments) +t_test <-t.test(payments_t, payments_c) +p1 = t_test$p.value + +# (2) Electricity usage (baseline_elec_use) +elec_t <- treatment %>% pull(baseline_elec_use) +elec_c <- control %>% pull(baseline_elec_use) +mt2 = mean(treatment$baseline_elec_use) +mc2 = mean(control$baseline_elec_use) +sdt2 = sd(treatment$baseline_elec_use) +sdc2 = sd(control$baseline_elec_use) +t_test <-t.test(elec_t, elec_c) +p2 = t_test$p.value + +# (3) Household size (baseline_hhsize) +hsize_t <- treatment %>% pull(baseline_hhsize) +hsize_c <- control %>% pull(baseline_hhsize) +mt3 = mean(treatment$baseline_hhsize) +mc3 = mean(control$baseline_hhsize) +sdt3 = sd(treatment$baseline_hhsize) +sdc3 = sd(control$baseline_hhsize) +t_test <-t.test(hsize_t, hsize_c) +p3 = t_test$p.value + +# (4) Household head age (baseline_hh_head_age) +hage_t <- treatment %>% pull(baseline_hh_head_age) +hage_c <- control %>% pull(baseline_hh_head_age) +mt4 = mean(treatment$baseline_hh_head_age) +mc4 = mean(control$baseline_hh_head_age) +sdt4 = sd(treatment$baseline_hh_head_age) +sdc4 = sd(control$baseline_hh_head_age) +t_test <-t.test(hage_t, hage_c) +p4 = t_test$p.value + +baseline = c("Payments", "Electricity Use", "Household Size", "Household Head Age") +p_value = c(p1, p2, p3, p4) +mean_treatment = c(mt1, mt2, mt3, mt4) +sd_treatment = c(sdt1, sdt2, sdt3, sdt4) +mean_control = c(mc1, mc2, mc3, mc4) +sd_control = c(sdc1, sdc2, sdc3, sdc4) +diff_means = c(mt1-mc1,mt2-mc2, mt3-mc3, mt4-mc4) +df = data.frame(baseline, mean_treatment, sd_treatment, mean_control,sd_control,diff_means, p_value) + +kbl(df, caption = "Adjusted Balance Table", booktabs = T, digits = 3,col.names = + c("Baseline Characteristic", "Mean Treated", "Std Dev Treated", + "Mean Control", "Std Dev Control", "Diff Means","p-value"))%>% kable_styling( + latex_options = c("striped","hold_position"), position = "center") +``` +From the adjusted balance table, we can observe that the treatment and control +groups remain balanced in all four baseline characteristics at all conventional +levels of statistical significance. The p-values increased in size in every +case, showing that the degree of balance is higher than previously anticipated. +With the outlier removed, the difference in means for baseline payments is not +as drastic as before and the standard deviation is 74 both for the treatment +and control groups. + +This tells us that KELLER'S randomization worked and that treatment and control +groups are statistically similar, on key observables, at the baseline. In order +to estimate the causal effect of keller_trt, we need to assume: +$$E[\varepsilon_i|D_i] = 0$$ + +Which means that the expectation of the error term, conditional on treatment, +is zero. This is to say that there is no selection bias because we observe +everything that is correlated with treatment and affects the outcome $Y_i$ and +those observable characteristics are balanced at the baseline. +With this assumption in place, if we find a treatment effect, we can attribute it +solely to the treatment and not to other unobservable characteristics that might +be correlated with the treatment and the output. + + +### (7) Assuming that keller_trt is indeed randomly assigned, describe how to use it to estimate the average treatment effect, and then do so. Please describe your estimate: what is the interpretation of your coefficient (be clear about your units)? Is your result statistically significant? Is the effect you find large or small, relative to the mean in the control group?### +Assuming that keller_trt is randomly assigned, we can use it to estimate ATE as +follows: +$$\tau^{ATE} =\overline{Y_i}(D = 1) - \overline{Y_i}(D = 0)$$ + +This difference in means is essentially what a regression does: +$$Y_i = \beta + \tau D_i+ \varepsilon_i$$ +Where $\tau$ captures the ATE coefficient. Since the treatment and control groups are +balanced (statistically equal) at the baseline, we can run this regression to recover the ATE. + +```{r} +reg <- lm(endline_payments ~ keller_trt, data = keller_filtered) +summary(reg) +``` +From the regression, the difference in endline payments between a household that +receives treatment (gets electricity disconnected) and a household that doesn't +(ie. control), is estimated to be approximately -4.679,1 rupees. The coefficient +is statistically significant at all conventional levels of significance since the +p-value of 0.000 < 0.05 and 0.01 and 0.1. + +Relative to the mean endline payments for the control group (given by the intercept), which +is equivalent to 48.673,3 rupees, the estimated ATE is economically significant since the percentage +change in endline payments for a treated household is -9.61%, which is a considerable amount. + +```{r} +(((48673.3 -4679.1) - 48673.3)/48673.3)*100 +``` + + + +### (8) KELLER is convinced that the reason their disconnections are effective is because they are getting households to use less electricity. They want you to estimate the effects of the disconnections, but controlling for the endline amount of power consumed. Is this a good idea? Why or why not? Run this regression and describe your estimates. How do they differ from your results in (7)? What about controlling for baseline electricity consumption? Run this regression and describe your estimates. How do they differ from your results in (7)? How do the two estimates differ? What is driving any differences between them?### +It is a bad idea to include the endline amount of power consumed in the regression +because this post-treatment variable could potentially be correlated with our outcome +of interest (electricity payments) and therefore, it could likely have been impacted +by the treatment. That is, a decrease in the amount of power consumed could be a direct +effect of the households that were treated by getting their electricity cut off. + + +```{r} +reg2 <- lm(endline_payments ~ keller_trt + endline_elec_use, data = keller_filtered) +summary(reg2) +``` + +By running this regression, we obtain that, ceteris paribus, our ATE estimate is not statistically +significant at any of the conventional levels of significance. (p-value: 0.441 > 0.05). +On the contrary, the coefficient on endline electricity use is highly statistically +significant, because as expected, this variable was directly affected by the treatment. + +This regression provides misleading results that the intervention had no statistically +significant ATE due to bias introduced by including a post-treatment outcome control. + +```{r} +reg3 <- lm(endline_payments ~ keller_trt + baseline_elec_use, data = keller_filtered) +summary(reg3) +``` +When we control for baseline electricity consumption, we find that holding everything +else constant, the difference in endline payments between a household that +receives treatment (gets electricity disconnected) and a household that doesn't +(ie. control), is estimated to be approximately -6.043,18 rupees. The coefficient +is highly statistically significant at all conventional levels of significance. + +Compared to the regression in (7), we find that the size of the coefficient increases from -4.679,1 rupees in (7) to -6.043,18 rupees, a difference of -1.0722.28 rupees. In both cases the coefficient on ATE is statistically significant at all conventional levels of significance. In terms of economic significance, +the estimated percentage change in endline payments between a treated and control household is +equivalent to -12.42%, a considerable increase from -9.61% in (7). + +```{r} +(((48673.3 -6043.18) - 48673.3)/48673.3)*100 +``` + +Also, the standard errors decrease, which is what drives the differences in the size of the coefficients. For example, the standard error for the ATE coefficient in (7) was 985.2 compared to 60.57 in this regression, +which shows more precision in our estimate. + +Since baseline electricity consumption was balanced at the baseline, it is not required to include it in order to recover the ATE, however, the fact that the size of the ATE estimate increases in magnitude, suggests that baseline electricity consumption might be correlated with the outcome and the error term such that excluding it, lead to omitted variable bias and therefore led us to underestimate the size of the ATE. + + +### (9) One of the KELLER RAs (the real workforce!) informs you that not everybody who was supposed to be disconnected -- (keller_trt = 1) actually got disconnected. She tells you that the actual treatment indicator is keller_trt_yes. (Since disconnections are expensive, KELLER assures you that nobody in the control group got disconnected). In light of this new information, what did you actually estimate in question (7)? How does this differ from what you thought you were estimating?### + +Since not every household that was assigned to treatment was treated, this tells us that there +was non-compliance in the RCT, either from outside the experiment (eg. because KELLER implementers +failed to disconnect all the households it had intented to) or from inside the experiment (eg. households that +reconnect themselves illegally). + +In light of this new information, what I actually estimated in (7) was the Intent to Treat (ITT), that is, +the impact of offering the program. Mathematically: + +$$ITT = \bar{Y_i}(R=1) - \bar{Y_i}(R=0)$$ +Where $R = assignment \space status$ and $D = actual \space treatment \space indicator$ +This is **different** from the ATE for the treated: +$$ATE =\overline{Y_i}(D = 1) - \overline{Y_i}(D = 0)$$ +Instead of recovering the ATE of the treatment (disconnecting households' electricity), +we are recovering the ATE of the assignment of treatment, known as *ITT*. + +### (10) KELLER aren’t actually interested in the effect of assignment to treatment - they want to know about the actual effects of their disconnections. Describe (in math, and then in words) what you can estimate using the two treatment variables we observe, keller_trt and keller_trt_yes. Estimate this object (you can ignore standard errors just for this once). Interpret your findings. How does this compare to what you estimated in (7)?### +Using the two treatment variables we observe, we can estimate the Local Average Treatment +Effect (LATE), which is the effect of the treatment for the compliers. We can classify four +types of households: + +* **Never takers**: Never treated, regardless of ($R$) status, $D=0$ +* **Always takers**: Always treated, regardless of ($R$) status, $D=1$ +* **Compliers**: Treated in the treatment group, untreated in the control group. If ($R = 1$), then ($D = 1$). If ($R = 0$), then ($D = 0$) +* **Defiers**: Untreated in the treatment group, treated in the control group.If ($R = 1$), then ($D = 0$). If ($R = 0$), then ($D = 1$) + +**Assumptions:** + +* Typically there are no defiers in the sample +* Never takers and always takets are equally distributed in the treatment and control groups + +With these assumption, LATE is essentially a weighted average of the ITT with respect to the share of compliers. + +Mathematically, LATE is estimated as follows: + +$$ \tau^{LATE} = \frac{\overline{Y}(R_i = 1) - \overline{Y}(R_i=0)}{\hat{\pi} ^{c}}$$ +$$ \tau^{LATE} = \frac{\hat{\tau} ^{ITT}}{\hat{\pi} ^{c}}$$ +**To estimate LATE, we need to follow three steps:** + +1. Regress $Y_i$ on $R_i$ to recover $\tau^{ITT}$ +2. Regress $D_i$ on $R_i$ to recover $\pi ^{c}$ +3. Calculate $\tau^{LATE}$ +```{r} +# Step (1): +itt <- lm(endline_payments ~ keller_trt, data = keller_filtered) +summary(itt) + +# Step (2): +pi_c <- lm(keller_trt_yes ~ keller_trt, data = keller_filtered) +summary(pi_c) + +# Step (3): +late = itt$coefficients[2] / pi_c$coefficients[2] +late +``` +$$ \tau^{LATE} = \frac{-4,679.066} {0.7032593 }$$ +$$ \tau^{LATE} = -6,653.4 $$ +As a result, we obtain that the estimated LATE is -6,653.4, which means that on average, electricity disconnections are associated with a decrease of -6,653.4 rupees in electricity payments for complying households. Compared to (7), this is larger than the estimated ITT of -4,669.6, which tells us that the effect of disconnecting +electricity on endline payments is higher for complying households than an average household assigned to treatment. + + + + + diff --git a/ps-1/PS1.pdf b/ps-1/PS1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..7dbc305ba8a5653c4a5d6ab0a3202edc9923912d GIT binary patch literal 254935 zcmeFYWpL(7wytSrW`;5|Gcz;0>@qVmGcz+YbD5c$nVFff>|1+x_lchA6VZEO?#w+u z`up)kN+Br~Z!V?Qdh!*Cys#(@11$>_$;`sw3KRlBPd>8Cj+d-Ha^4@eF55cV>kGt7 zhh}3ne9o6%YPhYppSP4u5>D(YLYu5=UoEe|)R|U4RgID{wOYhg6w)Ufoytd~G@31J z-~m(U-R@H8cOvANWw%e1Pf> zZ$Zl0j~Awe6%H5UV&L z@U8on!H|qPR>Z;N+yF1fUL@|mg^f)@`{h`au&lk~yh-pX)YByUMtI?h^f3QyQ2M5% z)}MIfttC(_NM_d~W1O3bL9PnZv3uR@$$ryl^<#YF1~QqW$6bzEH)-nd(EgAy4n3=R z=-#%k)mtf=oaq*u z^-K3bydD%L_*%u%1qxi$vyv|F;#Gbuc5Fo?*ZTO_@EA#d8U-YrRWFc3XEQM6)NgSr zF9|R3nw^(@@-{>vvn_pV)wCFkOuZ=tFQq!p`jbguo((6OjJkGP;VzLOx3wVQO+3&;Zz{7#WeDcx7w+g 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environment, set up the working directory: +rm(list=ls()) +setwd("~/Desktop/Spring 2022/Program Evaluation/PS2") + +# tell R not to use scientific notation +options(scipen = 999) + +# Load libraries +library(tidyverse) +library(ggplot2) +library(kableExtra) +library(broom) +library(MatchIt) +library(dplyr) +``` + +### (1) HARRIS are interested in answering the following question: What was the effect of FIONA on profits for the average farmer? To make sure everybody is on the same page, explain to them what the ideal experiment would be for answering this question. Describe the dataset that you’d like to have to carry out this ideal experiment, and use math, words, and the potential outcomes framework to explain what you would estimate and how you would do so. Make sure to be clear about the unit of analysis (ie, what is “i” here?). ### + +Using the potential outcomes framework, we would ideally like to measure the impact of the rainfall-index insurance on the average farmer's profits. + +In the ideal experiment, we would be able to randomize farmers into treatment and control groups where treated farmers take up the rainfall insurance, while farmers in the control group do not have insurance. To prevent spillovers from happening, in the ideal experiment, randomization would be conducted at the district level so as to minimize the possibility that farmers that are not offered the insurance find out about the program from their neighbors and get access to the insurance (ie. treatment). In the ideal experiment, there would not be issues of non-compliance either, such that everyone assigned to treatment actually takes up the insurance and vice versa with the control group. + +To run this ideal experiment, I would like to have a dataset of farmers by districts with their baseline profits and other baseline observable characteristics so that I could randomize access to the rainfall-index insurance at the district level. I would then construct a balance table by conducting t-tests to make sure the treatment and control groups are balanced in all baseline characteristics and subsequently run the experiment. + +Therefore, we could answer HARRIS' question of the effect of the rainfall-index insurance on the profits of the **average farmer** by estimating the **Average Treatment Effect (ATE):** + + +$$\tau^{ATE} = E[Y_{i}(1)] - E[Y_{i}(0)]$$ +Where: + +* $Y$ is the outcome variable of interest, 2016 profits. +* $i$ is each individual farmer + +Under randomization, the **expected** profits for farmers that take up the insurance (ie. $Profits_{farmer}(1)$) is equivalent to the **mean** profits of farmers that take the insurance, and likewise for farmers in the control group. As such, we could estimate ATE by taking the difference in means between both groups: + +$$\tau^{ATE} =\overline{Y_{i}}(D = 1) - \overline{Y_{i}}(D = 0)$$ + +Where: + +* $D$ refers to treatment status: $D = 1$ for farmers that took up the rainfall insurance and $D = 0$ for farmers that did not take it. +* $Y$ is the outcome variable of interest, 2016 profits. +* $i$ is each individual farmer + +Even though I would be randomizing at the district level, the unit of analysis i would be farmer, since HARRIS is interested in the profits of the _average_ **farmer**. + + +### (2) HARRIS like what you’re suggesting, but think it’s answering the wrong question. They aren’t going to be able to get every single farmer to participate. They’d instead like to know: What was the effect of FIONA on profits among farmers who took up insurance? Describe in math and words, using the potential outcomes framework, what they’d like to estimate. Explain how this differs from what you described in (1), and describe what component of this estimand you will be fundamentally unable to observe.### + +Since HARRIS will not be able to get every single farmer to participate in their insurance scheme, this means that HARRIS anticipates an issue with perfect compliance (were they to run an RCT), in which farmers assigned to receive the rainfall insurance will not actually opt in. In this case, we effectively could not estimate ATE. + +Instead, what HARRIS is interested in measuring is the the **Average Treatment Effect on the Treated (ATT)** or in other words, the ATE only among those farmers that did take up the insurance. Mathematically, this can be expressed as: + +$$\tau^{ATT} = E[\tau_i|D_i= 1] = E[Y_i(1) - Y_i(0)|D_i = 1]$$ +$$\tau^{ATT} = E[Y_i(1)|D_i = 1] - E[Y_i(0)|D_i = 1]$$ +Where: + +* $D$ refers to treatment status: $D = 1$ for farmers that took up the insurance and $D = 0$ for farmers that did not take it. +* $Y$ is the outcome variable of interest, 2016 profits. +* $i = farmer$. + +The ATT, therefore is calculated by taking the difference between **(1)** the potential profits for farmers that actually took up the insurance ($D_i = 1$) had they been offered the insurance ($Y_i(1)$) and, **(2)**, the potential profits for farmers that actually took up the insurance ($D_i = 1$) _had they not been offered the insurance_ ($Y_i(0)$). + +This is different from **ATE** because, instead of measuring the average effect for both the treatment and control groups, the **ATT** only focuses on the average effect of the treated group (those that take up the insurance). + +The issue here is the **fundamental problem of causal inference** because we are fundamentally unable to observe the second term $E[Y_i(0)|D_i = 1]$. It is impossible to know what the potential profits of farmers that took up insurance would have been if they never got access to it in the first place. As such, we cannot calculate the ATT. + + +### (3) HARRIS are on board with your explanation. Because FIONA already exists in the real world, they can’t run an RCT to study it. However, they do know that not all farmers were offered insurance through FIONA. It turns out that FIONA only impacted certain districts. Non-FIONA districts were not offered any insurance products. Explain what you would recover if you simply compared FIONA farms to non-FIONA farms on average. Describe three concrete examples of why this might be problematic.### + +In the absence of randomization, if we simply compared FIONA farms to non-FIONA farms, we would be calculating the **naive estimator:** + +$$\tau^N = \overline{Y}(1) - \overline{Y}(0)$$ +Where: + +* $N = farms$ since the unit of analysis is FIONA and non-FIONA farms. +* $Y$ is the outcome variable of interest, 2016 profits. + +This naive estimator is simply based on **observed outcomes** and it can be very problematic since the treated and untreated farms are very likely to be systematically different in many characteristics, both observable and unobservable, in ways that affect farmer's selection into treatment (opting for insurance or not). As such, if we use this estimator, we would have **selection bias** and would erroneously attribute the impact on profits to FIONA, when it can well be the result of many other characteristics. + +To give three concrete examples: + +**1.** It may be the case that due to geographic locations, some districts had a very good rainfall year while other districts didn't. If, for example, the farms that are offered insurance happened to be those farms with a good rainfall year, then those farms do not receive any tangible benefit from opting for insurance against bad rainfall in the year. + +Since they had good rainfall that year, they would have more income to invest in fertilizer to further increase their profits. In this case, the increase in profits would come from good rainfall and not because the FIONA scheme was actually effective. + +**2.** Alternatively, it could be the case that in some farms, farmers are culturally more risk adverse. If most of the farmers that are offered the FIONA insurance happened to be risk-adverse, then it is unlikely they will invest in riskier up-front inputs, even if they have access to insurance. Here, we might erroneously conclude that FIONA doesn't work at all, when it might actually have an effect on non risk-adverse farmers. + +**3.** If, for instance, farms that were offered treatment also happened to be located in districts that had an economic boost during the year, then the increase in profits may be the result from that boost rather than from the FIONA scheme, and it may be that doing well economically is a deciding factor on feeling the need to opt for insurance or not. + + +### (4) HARRIS hears your concerns, but still wants an estimate of the impacts of FIONA. Given that you’re unable to implement your ideal experiment, and you are worried about simple comparisons of FIONA-aided farmers and those without insurance, you’ll need to do something a little more sophisticated. Luckily for you and for HARRIS, India makes data on farmers available to the public, in the form of ps2_data.csv. Read the data into R and, as always, make sure everything makes sense. Document and fix any errors.### + +```{r data, include=FALSE} +#Load data +fiona <- read_csv("ps2_data.csv") +``` + +```{r} +#First I check the unique values for discrete variables, to check for +# consistency in data types. +unique(fiona$farmer_birth_year) +unique(fiona$crop) +unique(fiona$district) +unique(fiona$fiona_farmer) +unique(fiona$fertilizer_use) + +# Replace written birth years to numbers in the farmer_birth_year variable +fiona$farmer_birth_year[fiona$farmer_birth_year == + "nineteen seventy-three"]<- 1973 +fiona$farmer_birth_year[fiona$farmer_birth_year == + "nineteen seventy-two"] <- 1972 + +# Convert birth year from string type to integer type +typeof(fiona$farmer_birth_year) +fiona$farmer_birth_year = strtoi(fiona$farmer_birth_year, base=0L) +typeof(fiona$farmer_birth_year) + +``` +When looking at the data, there are some observations that need to be cleaned up for consistency. For example, in farmer_birth_year there are observations that are written in words (ie. "nineteen seventy-three") instead of numbers (ie. 1973). I replaced these observations with numbers and converted the variable from string type to integer. + +Otherwise, all other observations are of the same data type. There are no variations in spelling in the case of string variables (ie. district and crop) and luckily, there are no missing values either. + +Then, I plot histograms of the outcome variable of interest, profits, at baseline and endline to detect if there are any visible outliers in the data. +```{r} +hist(fiona$profits_2005, + main = "Histogram of 2005 profits", + xlab="Baseline profits", + col="orange") + +hist(fiona$profits_2016, + main = "Histogram of 2016 profits", + xlab="Endline profits", + col="purple") +``` + + +While the baseline 2005 profits follows a normal distribution, compared to a right skewed distribution in the case of endline 2016 profits, there are no visible outliers in the data that need to be dropped. + +### Use the variables contained in the dataset to describe, using math and words, two (related) potential approaches to estimating the effect of FIONA on profits. Make sure to be clear about your unit of analysis, and be explicit about how these designs apply to FIONA (ie, describe things in terms of “profits,” not just “outcome”). Hint: HARRIS wants you to describe two selection-on-observables designs.### + +Given the variables available in the dataset and the absence of randomization, we can estimate ATE under Selection on Observables (SOO) designs. + +**1. REGRESSION ADJUSTMENT:** +In this approach, we adjust the regular ATE regression by including controls in the form of additional covariates $X_i$'s. + +$$ Y_i = \alpha + \tau D_i + \gamma X_i + v_i$$ + +Where: + +* $D_i$ refers to treatment status $Di = 1$ for farms that took up FIONA insurance and $D_i = 0$ for those that didn't. +* $X_i$ represents control variables, such as: crop dummies, farmer_birth_year, etc. +* $Y_i$ is the outcome variable, 2016 profits +* $i = farmer$ + +Since there was no randomization in the FIONA scheme, $E[\varepsilon \mid D_i] \neq 0$. Instead, we have **conditional independence** $E[\varepsilon \mid D_i, X_i] = 0$, where the expected error term _conditional_ on taking up the insurance AND covariates (eg. crop dummies, farmer_birth_year) is 0. This means that, assuming this assumption holds as well as the functional form above, we need to include additional covariates in our regression so as to recover $\tau ^ {ATE}$. + + +**2. MATCHING:** +In this second approach, we match untreated farmers with treated farmers with identical covariates (eg. crop dummies, farmer_birth_year). Then, the difference in 2016 profits between treated and untreated farmers is the average treatment estimate $\hat {\tau}$. + +This requires dividing the data into _cells_ by the different covariates (eg. farmer_birth_year, crops). For each cell, we need to calculate the mean profits for the treated $\overline Y_T$ and the mean profits of the untreated $\overline Y_U$, and then take the differences between both $\overline Y_T - \overline Y_U$ for each covariate. Finally, $\hat {\tau}^{ATE}$ is a weighted average of these differences. We can also calculate ATT and ATN by using weights for the treated and untreated, respectively. + + +### (5) Produce a balance table which displays the differences between FIONA and non-FIONA farmers on observable characteristics. Interpret this table. Does this table make you feel better or worse about your concerns in (3)?### + +```{r} +# Convert crop and district to numerical categorical values +fiona$crop_cat = as.numeric(as.factor(fiona$crop)) +fiona$district_cat = as.numeric(as.factor(fiona$district)) + +# Convert crop variable to dummies for each crop +fiona$rice = ifelse(fiona$crop == 'RICE', 1, 0) +fiona$lentils = ifelse(fiona$crop == 'LENTILS', 1, 0) +fiona$wheat = ifelse(fiona$crop == 'WHEAT', 1, 0) +fiona$cotton = ifelse(fiona$crop == 'COTTON', 1, 0) + +# Convert district variable to dummies for each district +fiona$karur = ifelse(fiona$district == 'KARUR', 1, 0) +fiona$tenkasi = ifelse(fiona$district == 'TENKASI', 1, 0) +fiona$pudukkottai = ifelse(fiona$district == 'PUDUKKOTTAI', 1, 0) +fiona$thanjavur = ifelse(fiona$district == 'THANJAVUR', 1, 0) +fiona$dindigul = ifelse(fiona$district == 'DINDIGUL', 1, 0) +fiona$madurai = ifelse(fiona$district == 'MADURAI', 1, 0) + +``` + +```{r} +# Separate the dataset into treated and untreated groups: +treated <- filter(fiona, fiona_farmer == 1) +untreated <- filter(fiona, fiona_farmer == 0) + +# Segment treated and untreated by each available baseline characteristic and +# run a t-test to check if they are balanced: + +# (1) Pre-treatment profits (profits_2005) +profits_t <- treated %>% pull(profits_2005) +profits_u <- untreated %>% pull(profits_2005) +mt1 = mean(treated$profits_2005) +mu1 = mean(untreated$profits_2005) +sdt1 = sd(treated$profits_2005) +sdu1 = sd(untreated$profits_2005) +nt1 = nrow(treated) +nu1 = nrow(untreated) +t_test <-t.test(profits_t, profits_u) +p1 = t_test$p.value + +# (2) Farmer birth year (farmer_birth_year) +fby_t <- treated %>% pull(farmer_birth_year) +fby_u <- untreated %>% pull(farmer_birth_year) +mt2 = mean(treated$farmer_birth_year) +mu2 = mean(untreated$farmer_birth_year) +sdt2 = sd(treated$farmer_birth_year) +sdu2 = sd(untreated$farmer_birth_year) +nt2 = nrow(treated) +nu2 = nrow(untreated) +t_test <-t.test(fby_t, fby_u) +p2 = t_test$p.value + +# (3) All crops(crop_cat) +all_crops_t <- treated %>% pull(crop_cat) +all_crops_u <- untreated %>% pull(crop_cat) +mt3 = mean(treated$crop_cat) +mu3 = mean(untreated$crop_cat) +sdt3 = sd(treated$crop_cat) +sdu3 = sd(untreated$crop_cat) +nt3 = nrow(treated) +nu3 = nrow(untreated) +t_test <-t.test(all_crops_t, all_crops_u) +p3 = t_test$p.value + +# (4) Rice +rice_t <- treated %>% pull(rice) +rice_u <- untreated %>% pull(rice) +mt4 = mean(treated$rice) +mu4 = mean(untreated$rice) +sdt4 = sd(treated$rice) +sdu4 = sd(untreated$rice) +nt4 = nrow(subset(treated, rice=="1")) +nu4 = nrow(subset(untreated, rice=="1")) +t_test <-t.test(rice_t, rice_u) +p4 = t_test$p.value + +# (5) Lentils +lentils_t <- treated %>% pull(lentils) +lentils_u <- untreated %>% pull(lentils) +mt5 = mean(treated$lentils) +mu5 = mean(untreated$lentils) +sdt5 = sd(treated$lentils) +sdu5 = sd(untreated$lentils) +nt5 = nrow(subset(treated, lentils=="1")) +nu5 = nrow(subset(untreated, lentils=="1")) +t_test <-t.test(lentils_t, lentils_u) +p5 = t_test$p.value + +# (6) Wheat +wheat_t <- treated %>% pull(wheat) +wheat_u <- untreated %>% pull(wheat) +mt6 = mean(treated$wheat) +mu6 = mean(untreated$wheat) +sdt6 = sd(treated$wheat) +sdu6 = sd(untreated$wheat) +nt6 = nrow(subset(treated, wheat=="1")) +nu6 = nrow(subset(untreated, wheat=="1")) +t_test <-t.test(wheat_t, wheat_u) +p6 = t_test$p.value + +# (7) Cotton +cotton_t <- treated %>% pull(cotton) +cotton_u <- untreated %>% pull(cotton) +mt7 = mean(treated$cotton) +mu7 = mean(untreated$cotton) +sdt7 = sd(treated$cotton) +sdu7 = sd(untreated$cotton) +nt7 = nrow(subset(treated, cotton=="1")) +nu7 = nrow(subset(untreated, cotton=="1")) +t_test <-t.test(cotton_t, cotton_u) +p7 = t_test$p.value + +# (8) All districts (district_cat) +all_dist_t <- treated %>% pull(district_cat) +all_dist_u <- untreated %>% pull(district_cat) +mt8 = mean(treated$district_cat) +mu8 = mean(untreated$district_cat) +sdt8 = sd(treated$district_cat) +sdu8 = sd(untreated$district_cat) +nt8 = nrow(treated) +nu8 = nrow(untreated) +t_test <-t.test(all_dist_t, all_dist_u) +p8 = t_test$p.value + +# (9) Karur (karur) +karur_t <- treated %>% pull(karur) +karur_u <- untreated %>% pull(karur) +mt9 = mean(treated$karur) +mu9 = mean(untreated$karur) +sdt9 = sd(treated$karur) +sdu9 = sd(untreated$karur) +nt9 = nrow(subset(treated, karur=="1")) +nu9 = nrow(subset(untreated, karur=="1")) +t_test <-t.test(karur_t, karur_u) +p9 = t_test$p.value + +# (10) Tenkasi (tenkasi) +tenkasi_t <- treated %>% pull(tenkasi) +tenkasi_u <- untreated %>% pull(tenkasi) +mt10 = mean(treated$tenkasi) +mu10 = mean(untreated$tenkasi) +sdt10 = sd(treated$tenkasi) +sdu10 = sd(untreated$tenkasi) +nt10 = nrow(subset(treated, tenkasi=="1")) +nu10 = nrow(subset(untreated, tenkasi=="1")) +t_test <-t.test(tenkasi_t, tenkasi_u) +p10 = t_test$p.value + +# (11) Pudukkottai (pudukkottai) +pudukkottai_t <- treated %>% pull(pudukkottai) +pudukkottai_u <- untreated %>% pull(pudukkottai) +mt11 = mean(treated$pudukkottai) +mu11 = mean(untreated$pudukkottai) +sdt11 = sd(treated$pudukkottai) +sdu11 = sd(untreated$pudukkottai) +nt11 = nrow(subset(treated, pudukkottai=="1")) +nu11 = nrow(subset(untreated, pudukkottai=="1")) +t_test <-t.test(pudukkottai_t, pudukkottai_u) +p11 = t_test$p.value + +# (12) Thanjavur (thanjavur) +thanjavur_t <- treated %>% pull(thanjavur) +thanjavur_u <- untreated %>% pull(thanjavur) +mt12 = mean(treated$thanjavur) +mu12 = mean(untreated$thanjavur) +sdt12 = sd(treated$thanjavur) +sdu12 = sd(untreated$thanjavur) +nt12 = nrow(subset(treated, thanjavur=="1")) +nu12 = nrow(subset(untreated, thanjavur=="1")) +p12 = NaN + +# (13) Dindigul (dindigul) +dindigul_t <- treated %>% pull(dindigul) +dindigul_u <- untreated %>% pull(dindigul) +mt13 = mean(treated$dindigul) +mu13 = mean(untreated$dindigul) +sdt13 = sd(treated$dindigul) +sdu13 = sd(untreated$dindigul) +nt13 = nrow(subset(treated, dindigul=="1")) +nu13 = nrow(subset(untreated, dindigul=="1")) +t_test <-t.test(dindigul_t, dindigul_u) +p13 = t_test$p.value + +# (13) Madurai (madurai) +madurai_t <- treated %>% pull(madurai) +madurai_u <- untreated %>% pull(madurai) +mt14 = mean(treated$madurai) +mu14 = mean(untreated$madurai) +sdt14 = sd(treated$madurai) +sdu14 = sd(untreated$madurai) +nt14 = nrow(subset(treated, madurai=="1")) +nu14 = nrow(subset(untreated, madurai=="1")) +t_test <-t.test(madurai_t, madurai_u) +p14 = t_test$p.value + + +# Create data frame to store computed values +baseline = c("Profits 2005", "Farmer Birth Year", "Crop", "Rice", "Lentils", + "Wheat", "Cotton", "District", "Karur", "Tenkasi", "Pudukkottai", + "Thanjavur", "Dindigul", "Madurai") +mean_treated = c(mt1, mt2, mt3, mt4, mt5, mt6, mt7, mt8, mt9, mt10, mt11, + mt12, mt13, mt14) +sd_treated= c(sdt1, sdt2, sdt3, sdt4, sdt5, sdt6, sdt7, sdt8, sdt9, sdt10, + sdt11, sdt12, sdt13, sdt14) +n_treated = c(nt1, nt2, nt3, nt4, nt5, nt6, nt7, nt8, nt9, nt10, nt11, nt12, + nt13, nt14 ) +mean_untreated = c(mu1, mu2, mu3, mu4, mu5, mu6, mu7, mu8, mu9, mu10, mu11, + mu12, mu13, mu14) +sd_untreated = c(sdu1, sdu2, sdu3, sdu4, sdu5, sdu6, sdu7, sdu8, sdu9, sdu10, + sdu11, sdu12, sdu13, sdu14) +n_untreated = c(nu1, nu2, nu3, nu4, nu5, nu6, nu7, nu8, nu9, nu10, nu11, nu12, + nu13, nu14) +diff_means = c(mt1-mu1, mt2-mu2, mt3-mu3, mt4-mu4, mt5-mu5, mt6-mu6, mt7-mu7, + mt8-mu8, mt9-mu9, mt10-mu10, mt11-mu11, mt12 - mu12, mt13-mu13, + mt14-mu14) +p_value = c(p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11, p12, p13, p14) +df = data.frame(baseline, mean_treated, mean_untreated, diff_means, sd_treated, + sd_untreated, n_treated, n_untreated, p_value) +``` + +\newpage + +The available observables at baseline in this dataset are the following: + +1. Profits 2005 + +2. Farmer birth year + +3. Crops + + Rice + + Lentils + + Wheat + + Cotton + +4. District + + Karur + + Tenkasi + + Pudukkottai + + Thanjavur + + Dindigul + + Madurai + +In the case of crops and district, it is necessary to convert them into numerical categorical variables in order to run formal t-tests and check for balance (eg. cotton, lentils, rice and wheat are converted to 1, 2, 3, 4). I additionally convert them to dummy variables with the purpose of checking for balance for each type of crop and district (eg. 1 if rice, 0 otherwise). + +As a result, I obtained the following balance table: +```{r} +# Create Balance Table +kbl(df, caption = "Balance Table", booktabs = T, digits = 3,col.names = + c("Variable", "Mean Treated","Mean Untreated", "Diff Means", "SD Treated", + "SD Untreated", "N Treated", "N Untreated", + "p-value"))%>% kable_styling(latex_options = "scale_down") + +``` + + +The balance table reports the means, standard deviations, difference in means and the p-value from running a Welch Two-Sample t-test, which tests for the following hypotheses: + +$$ H_0: mean\ \text{treatment} - mean\ \text{control} = 0$$ +$$ H_A: mean\ \text{treatment} - mean\ \text{control} \neq 0$$ + +There are several observations that can be made from the balance table: + +* The **baseline profits** are **balanced** between treated and untreated groups. Since the p-value = 0.589 > 0.05, we fail to reject the null that the true difference in means is 0. The difference in means is relatively small, -11.835 between treated and untreated. We can observe the standard deviation in profits is very high in both groups, particularly for the untreated group: 1036.465. + +* The **farmer birth year** is also **balanced** between both groups. The p-value = 0.406 > 0.05, such that we fail to reject the null at the 5% level. The difference in means is very small at 0.112. + +* If we analyze the **crops** variable, in general terms, we find that it is **unbalanced**. The p-value is small at 0.024 < 0.05, such that we reject the null in favor of the alternative at the 5% significance level. The difference in means, however, is very small, at 0.042. To understand better, we can check for balance for each type of crop: + + + If we specifically look at **rice**, the p-value 0.059 > 0.05 such that we fail to reject the null, therefore the + variable is statistically **balanced**. However, we can see that the number of rice crops in the treated group, + 3000, is more than double that of the treated group, 947. + + + As to **lentils**, it is completely **balanced** since the p-value is at it's highest possible value of 1 > + 0.05, such that we we fail to reject the null. There are 2250 lentil crops in the treated group compared to 750 in + the untreated group and the difference in means is 0. + + + For **wheat**, this characteristic is also completely **balanced** with a p-value of 1 > 0.05. The statistics are + exactly the same as those for lentils. + + + For **cotton**, the characteristic is completely **unbalanced**, where the p-value of 0 < 0.05, such that we + reject the null in favor of the alternative. This is visible as there are 0 cotton crops in the treatment group, compared to 53 in the untreated group, so there is clearly no balance at all. It is because of cotton that the + variable crop as a whole is unbalanced, even though the difference in means for crop is very, very small. + +* Looking at **district**, this variable is completely **unbalanced**, where the p-value of 0 < 0.05. This is to be expected since the FIONA program only impacted certain districts. By design, each district is entirely offered treatment or not, so we can expected every district to be unbalanced: + + + All of the districts **Karur, Tenkasi, Pudukkottai, Dindigul and Madurai** are completely **unbalanced**. In the + case of **Thanjavur**, it is not possible for R to run the t-test because all the data is constant. However, for + the same reason as the 5 other districts, it is also unbalanced given that all observations in the district + correspond to the treated group. + +To better understand the disparity in the sample size for the treated and the untreated, I made the following table to summarize the offer of insurance, which was done at the district level: + +```{r} +ndt1 = nrow(subset(treated, district_cat=="1")) +ndu1 = nrow(subset(untreated, district_cat=="1")) +ndt2 = nrow(subset(treated, district_cat=="2")) +ndu2 = nrow(subset(untreated, district_cat=="2")) +ndt3 = nrow(subset(treated, district_cat=="3")) +ndu3 = nrow(subset(untreated, district_cat=="3")) +ndt4 = nrow(subset(treated, district_cat=="4")) +ndu4 = nrow(subset(untreated, district_cat=="4")) +ndt5 = nrow(subset(treated, district_cat=="5")) +ndu5 = nrow(subset(untreated, district_cat=="5")) +ndt6 = nrow(subset(treated, district_cat=="6")) +ndu6 = nrow(subset(untreated, district_cat=="6")) +tot_ndt = 1500 * 5 + +district = c("Dindigul", "Karur", "Madurai", "Pudukkottai", "Tenkasi", + "Thanjavur", "Total") +n_treated = c(ndt1, ndt2, ndt3, ndt4, ndt5, ndt6, tot_ndt) +n_untreated = c(ndu1, ndu2, ndu3, ndu4, ndu5, ndu6, ndu6) +df2 = data.frame(district, n_treated, n_untreated) + +kbl(df2, caption = "Treated and Untreated Observations by District", + booktabs = T, digits = 3,col.names = + c("District", "N Treated", "N Untreated"))%>% kable_styling( + latex_options = c("striped","hold_position"), position = "center") +``` + +As we can observe, 5 out of 6 districts reported in the dataset were not offered the insurance, with 1500 farmers each. On the other hand, the district of Thanjavur was the only one to be offered the insurance, with a total of 2500 farmers. In total, 7500 farmers were not offered the insurance compared to only 2500 that were (ie. onlt 25% of the sample was offered the insurance). + +In terms of the numbers of farmers offered insurance, there is a lot of unbalance, which makes it challenging to compare these two groups when we have relatively little observations for those farmers that were offered insurance. + +From the histograms in part (3), I got a positive sense that the data does not have any drastic outliers and that there are no missing values. However, this analysis makes me feel worse, because most of the baseline variables (accounting for dummies) are unbalanced. The district variable will not be very useful since entire districts received treatment or did not. Within the crop variable, 3 out 4 crops are balanced, with the exception of cotton, which makes it difficult to analyze the effect of the insurance scheme for this type of crop, if there is no counterfactual to compare it to. + + +### (6) HARRIS are interested in your approach in (4), but would like to know a bit more about how much they should believe your proposal. Describe the assumptions required for these designs to be valid in math and in words. To the extent possible, assess the validity of these assumptions using the provided data. + +When it comes to SOO designs, two main assumptions are required to hold: + + +**1. Conditional independence: ** conditional on observables, treatment assignment is independent of potential outcomes, which means that all differences between $D_{farmer} = 1$ and $D_{farmer} = 0$ are assumed to be _as good as random_. This is a **very strong assumption** and can be mathematically expressed as: + +$$ (Y_i(1), Y_i(0)) \perp D_i \mid X_i$$ + +Where: + +* $D_i$ refers to treatment status $Di = 1$ for farms that took up FIONA insurance and $D_i = 0$ for those that didn't. +* $Y_i$ is the outcome variable, 2016 profits +* $X_i$ represents control variables, such as: crop dummies, farmer_birth_year, etc. +* $i = farmer$ + +This means that once we control for $X_i$'s (eg. baseline profits, farmer birth year, crop), opting for insurance $D_i$ is _as good as random_ and we have essentially eliminated selection bias. + + +**2. Common support - ** The probability that $D_i = 1$ for all levels of $X_i$ is between 0 and 1: +$$ 0 < Pr(D_i = 1 \mid X_i = x) < 1$$ +Where: + +* $D_i$ refers to treatment status $Di = 1$ for farms that took up FIONA insurance and $D_i = 0$ for those that didn't. +* $X_i$ represents control variables, such as: crop dummies, farmer_birth_year, etc. +* $i = farmer$ + +This means that for each covariate, there are _both treated and untreated units_ (ie. farmers that took up the insurance and farmers that did not). + +Additional to the conditional independence and the common support assumptions, there are two more assumptions that need to hold when employing the regression adjustment approach: + +**Regression adjustment additional assumptions:** + +* The mean covariates for the treated are **close** to the mean covariates for the untreated, ie: $\overline X_T$ close to $\overline X_U$. If the difference between the covariates is large, $\mid \overline X_T - \overline X_U \mid$, then the $\hat {\tau}$ estimate will be biased. + +* The true relationship is assumed to be of the following **functional form**: +$$Y_{i}= \alpha + \tau D_{i} + \gamma X_{i}$$ + +Where: + +* $Y_i$ is the outcome variable, 2016 profits +* $D_i$ refers to treatment status $Di = 1$ for farms that took up FIONA insurance and $D_i = 0$ for those that didn't. +* $X_i$ represents control variables, such as: crop dummies, farmer_birth_year, etc. +* $i = farmer$ + +In the case of **Matching**, since treated and untreated units are matched _exactly_, it's a more flexible estimator in which the functional form does not matter and only the first two SOO assumptions described above are required. + + +**ASSESSING THE VALIDITY OF THESE ASUMPTIONS:** + +* **(1) Conditional Independence Assumption** + + + It is not possible to test this assumption. However, it is very, very unlikely for this assumption to hold as we + would essentially have to account for all possible covariates $X_i$'s that are correlated with taking up insurance. + This is never actually possible because, for one, we can never account for unobservables, and second, due to + limitations in the data we can only control for _some_ covariates. In this dataset, we have few covariates: + profits_2005, farmer birth year, crop and district. There are many more covariates that could be inducing selection + bias for which we have no data. For example: size of farmer household, education or training in agriculture, + additional jobs/income, savings, etc. + +* **(2) Common Support Assumption** + + From the balance table, we found that there were 0 farmers that produce cotton and took up the insurance. All 53 + farmers that produce cotton were not treated. Likewise, by design, offer of insurance was done at the district + level, so for each district, all farmers are entirely treated or entirely untreated. + +```{r} +crop = c("Rice", "Lentils", "Wheat", "Cotton") +n_treated = c(nt4, nt5, nt6, nt7) +n_untreated = c(nu4, nu5, nu6, nu7) +df3 = data.frame(crop, n_treated, n_untreated) + +kbl(df3, caption = "Treated and Untreated Observations by Crop", + booktabs = T, digits = 3,col.names = + c("Crop", "N Treated", "N Untreated"))%>% kable_styling( + latex_options = c("striped","hold_position"), position = "center") + +kbl(df2, caption = "Treated and Untreated Observations by District", + booktabs = T, digits = 3,col.names = + c("District", "N Treated", "N Untreated"))%>% kable_styling( + latex_options = c("striped","hold_position"), position = "center") +``` + + + + Therefore, the common support assumption **doesn't hold for cotton and district**, since there are not treated + units and untreated for _every level_ of X. If we look at the numbers, there are a lot more untreated farmers + (7500) than treated farmers (2500). We can expect a lot of farmers to be unmatched due to lack of overlap. + + + Also, in the case of **farmer birth year**, it will be difficult to find exact matches without running into the + **curse of dimensionality** since there are a total of 65 different farmer birth years. As we can see below, there + are many birth years for which there are very few farmers (eg. 7 where there is only 1 farmer), so that it will not + be possible to find an exact match at every single level. + + + Thus, the common support assumption **does not hold** for birth year either. + +```{r} +fiona %>% + group_by(farmer_birth_year) %>%count() %>% arrange(n) +``` + + With the exact matching estimator, it will **not be possible** to find exact matches for 2005 profits because it + is a continuous variable and we will run into the **curse of dimensionality**. + + + In conclusion, the following table summarizes that the common support assumption **only holds** for three + covariates: **rice, wheat and lentils.** +```{r} +var = c("Profits 2005", "Farmer Birth Year", "Rice", "Lentils", + "Wheat", "Cotton", "Karur", "Tenkasi", "Pudukkottai", + "Thanjavur", "Dindigul", "Madurai") +holds = c("No", "No", "Yes", "Yes", "Yes", "No", "No", "No", "No", "No", "No", + "No") +df4 = data.frame(var, holds) +kbl(df4, caption = "Common Support Assumption by Variable", + booktabs = T, digits = 3,col.names = + c("Variable", "Assumption Holds"))%>% kable_styling( + latex_options = c("striped","hold_position"), position = "center") +``` + +* **(3) Closeness Assumption ** + + + For the $\overline X_T$ close to $\overline X_U$ regression adjustment assumption, we can refer back to the + balance table in (5). The covariates that are statistically balanced are: 2005 profits, farmer birth year, rice, + lentils and wheat. For these 5 variables, the difference in means between treated and untreated is very small. + + +### Discuss whether you think you will be able to obtain a credible estimate of the answer to the questions described in (1) and (2) based on the data, and use concrete examples to explain why or why not.### + +Since this is not an RCT where $E[\varepsilon_i|D_i] = 0$ holds, we cannot establish causality in any of these relationships. I predict that it will not be possible to obtain a very credible estimate with neither SOO design, in particular because of the conditional independence assumption being violated. As previously explained, we are missing key covariates that very likely are correlated with the treatment variable, such as the number of dependents or children. Farmers that have more dependents to sustain economically, might be more willing to enroll in insurance to avoid losing profits. Likewise, if there are long-term diseases in the family that require stable profits. + +Additionally, there is the issue of unobservables such as farmers being risk adverse or not, or having distrust of insurance schemes. SOO designs are unable to control for unobservables and cannot provide reliable estimates in such situations, so it's likely that the estimates will not perform significantly better than the naive estimator. + +### (7) Use a regression-based approach to estimate the effect of FIONA on farmer profits. Describe which variables you chose to include in your regression, and explain why you chose these. Did you leave any variables out? If yes, explain why. Interpret your results. What are the strengths and weaknesses of this approach? How do your results differ from what you find if you instead use the naive estimator? ### + +The data we have available from FIONA represents farmers that self-selected into treatment (opting-in for the rain insurance). For this reason, I chose to include as many covariates available in the dataset for which **both the common support and closeness assumption holds**. The goal is to minimize the risk of violating the conditional independence assumption by excluding covariates that are potentially correlated with farmer selection into the insurance scheme, given that those covariates satisfy the required SOO assumptions. + +These are the variables I included: + +* **fiona_farmer** since this is out treatment variable, this is the coefficient that will estimate our ATE +* **lentils**, **rice** and **wheat**, because some types of crop may be more rainfall-dependent, making it a variable that farmers consider when self-selecting into treatment. I. + +I excluded all of the following variables: + +* **farmer_birth_year**, while it satisfy the closeness assumption, it does not satisfy common support +* **cotton** because it does not satisfy common support nor closeness asumption +* **all district dummies** because they don't satisfy common support nor the closeness assumption +* **fertilizer use**, this is a post-treatment variable that should _never be included_ as doing so would introduce bias into the model. +* **profits_2005**, satisfies closeness but does not satisfy common support. Also, 2005 profits are likely very different to profits in 2014 (if anything due to inflation) when the study actually started. It is not relevant to include this variable because it's unlikely that the profits of a farmer in 2005 is correlated with his/her decision to opt into insurance 9 years later. + + + +```{r} +reg <- lm(profits_2016 ~ fiona_farmer + lentils + rice + wheat, data = fiona) +summary(reg) +``` +From the regression, we obtain the following results: + +* Holding everything else constant, the difference in endline 2016 profits between a farmer that opted for FIONA insurance and a farmer that didn't, is estimated to be approximately 2375.95 Indian rupees. This ATE estimate is highly statistically significant at all conventional levels of significance, such that we reject the null in favor of the alternative that ATE is significantly different from 0. + +* Ceteris paribus, rice, lentils and wheat are statistically significant at the 5% level. Rice is not statistically significant at the 5% level. + +* The coefficient on the three crops is positive, which means that growing each of these crops, all else constant, is associated with an increase in profits, in particular wheat. + + + +**Running the naive estimator** + +If, instead we run the naive estimator, we obtain the following results: + +```{r} +# Naive estimator +naive <- lm(profits_2016 ~ fiona_farmer, data = fiona) +summary(naive) +``` +* Ceteris paribus, the difference in endline 2016 profits between a farmer that opted for FIONA insurance and a farmer that didn't, is estimated to be approximately 2369.56 Indian rupees. This ATE estimate is highly statistically significant at all conventional levels of significance. This is a minimal difference of -6.39 rupees from the estimate obtained in the adjusted regression. + +* The $R^2$ of the naive estimator model is of a relative size at 0.38, meaning a lot of the change in 2016 profits is not being captured by the model. + +* In the absence of randomization, we cannot determine causality from this model and the estimated ATE coefficient is not reliable, as the likelihood of selection bias is very high (ie. farmers that self-selected into the insurance are very likely to be systematically different from those who did not) + + +**ASSESSMENT OF THE ADJUSTED REGRESSION APPROACH:** + +**1. Strengths** + +* Compared to the naive estimator, the adjusted regression is barely different. The $R^2$ of 0.40 is slightly higher than the 0.38 obtained in the naive estimator, which means that adding the covariates allowed us to control for a little more of the variation in endline profits. + + +**2. Weaknesses** + +* It is very, very likely that the conditional independence assumption does not hold in this case. We can infer this from the $R^2$, because there is much in the change in 2016 profits that is unexplained by the model. This is without a doubt because the selection bias is not eliminated simply by adding a few crop controls. + +* Although we added controls to the regression, they are either not enough or not the right controls. For this approach to perform better, we would need to include more, if not all, the covariates that are strongly correlated with the selection into the FIONA insurance. + +* We also cannot add more covariates available in the dataset like 2005 profits, farmer birth year, cotton and district variables because these do not fulfill the required SOO assumptions. Given the high selection bias, however, it is unlikely that including those controls would change the result by much. + +* It is very possible that there are unobservables that drive selection into the insurance, for example, farmers that are more risk-adverse might be more willing to choose insurance and see it is a benefit. The adjusted regression approach cannot account for this issue and is therefore **biased**. + +* Since this is not an RCT where $E[\varepsilon_i|D_i] = 0$ holds, and the conditional independence assumption very likely is violated, we cannot establish causality in any of these relationships. + +* In all, the adjusted regression performs the same as the naive estimator. + + +### (8) Use an exact matching approach to estimate the effect of FIONA on farmer profits. What variables should you include in the matching procedure? Begin by estimating the answer to the question in (1). Then, estimate the answer to the question in (2). Are these meaningfully different? Would you have expected these results to be the same? Why or why not? What are the strengths and weaknesses of this approach? How do your results differ from what you find if you instead use the naive estimator? From what you found in (8)? Did you run into the Curse of Dimensionality with this analysis? If yes, describe how it affected your approach. If not, describe how the Curse could have generated problems in this setting.### + +There are 4 steps to calculate Exact Matching: + +**1. Divide the data into cells uniquely identified by the covariates.** + +I am going to use the three crop dummies for which there are observations both in the treated and untreated groups: rice, lentil and wheat. + +All other variables that do not satisfy the common support assumption are excluded (ie.for district there are no observations in both groups and profits_2005 is a continuous variable which would lead to the curse of dimensionality to automatically kick in with no exact matches found and the method automatically failing ). + +```{r} +# STEP 1 - divide data into cells +fiona_match <- matchit(fiona_farmer ~ lentils + rice + wheat, method = "exact", + data = fiona) +df_match <- match.data(fiona_match) +nrow(df_match) +length(unique(df_match$subclass)) +``` + +From this first step, we obtain **9447 matched observations**. This is a very high number of matches considering the total number of observations in the data is 10000. In total, we have **3 cells** since only 3 crop dummies are being utilized for exact matching. + +```{r} +# To check if matching worked, we do a balance table. + +# Separate the dataset into treated and untreated groups: +treated <- filter(df_match, fiona_farmer == 1) +untreated <- filter(df_match, fiona_farmer == 0) + +# Run t-tests and calculate means: + +# (1) Rice +rice_t <- treated %>% pull(rice) +rice_u <- untreated %>% pull(rice) +mt1 = mean(treated$rice) +mu1 = mean(untreated$rice) +t_test <-t.test(rice_t, rice_u) +p1 = t_test$p.value + +# (3) Lentils +lentils_t <- treated %>% pull(lentils) +lentils_u <- untreated %>% pull(lentils) +mt2 = mean(treated$lentils) +mu2 = mean(untreated$lentils) +t_test <-t.test(lentils_t, lentils_u) +p2 = t_test$p.value + +# (4) Wheat +wheat_t <- treated %>% pull(wheat) +wheat_u <- untreated %>% pull(wheat) +mt3 = mean(treated$wheat) +mu3 = mean(untreated$wheat) +t_test <-t.test(wheat_t, wheat_u) +p3 = t_test$p.value + +# Create data frame to store computed values +baseline = c("Rice", "Lentils", "Wheat") +mean_treated = c(mt1, mt2, mt3) +mean_untreated = c(mu1, mu2, mu3) +p_value = c(p1, p2, p3) +diff_means = c(mt1-mu1, mt2-mu2, mt3-mu3) +df = data.frame(baseline, mean_treated, mean_untreated, diff_means, p_value) + +# Post-Matching Balance Table +kbl(df, caption = "Post-Matching Balance Table", booktabs = T, digits = 3,col.names = + c("Variable", "Mean Treated", + "Mean Untreated", "Diff Means", + "p-value"))%>% kable_styling(latex_options = + c("stripped", "hold_position"), + position = "center") +``` +From this new balance table, post-matching, we can observe that matching was successful because all p-values > 0.05, we fail to reject the null that the true difference in means is 0. This tells us that all three covariates selected for matching are balanced. + + +**2. Next, calculate $\overline Y_T$ and $\overline Y_U$ _for each cell_. ** + + +```{r} +# STEP 2 + +# Filter the matched observations by treated and untreated +matched_treated <- filter(df_match, fiona_farmer == 1) +matched_untreated <- filter(df_match, fiona_farmer == 0) + +# Group by cell (subclass) and calculate the mean of outcome variable +# (ie.profits_2016) +means_treated <- matched_treated %>% + group_by(subclass) %>% + dplyr::summarize(Mean = mean(profits_2016)) + +means_untreated <- matched_untreated %>% + group_by(subclass) %>% + dplyr::summarize(Mean = mean(profits_2016)) + +# Convert lists to a dataframe with the means of treated and untreated +means <- cbind(data.frame(means_treated), data.frame(means_untreated)) +colnames(means) <- c('cell_t', 'treated_mean','cell_u', 'untreated_mean') +means <- means[ -c(3) ] + +head(means) + +``` + + +**3. Calculate the difference in means $\overline Y_T - \overline Y_U$ for each cell.** + +```{r} +#Step 3, calculate the difference in means for each cell +means$diff_means <- means$treated_mean - means$untreated_mean +head(means) +``` + + +**4. Estimate $\hat {\tau}$ as a weighted average.** + +* First I calculate this for (1), which is the $\tau ^{ATE}$: + + +```{r} +# STEP 4 (1) WEIGHTED ATE + +# Store the sample size. For ATE, weights are based on ALL observations from +# the full sample: +n = nrow(df_match) + +# Calculate weights by cell (subclass) +cell_n <- df_match %>% + group_by(subclass) %>%count() + +# Convert to dataframe +cell_size_df <- cbind(data.frame(cell_n)) +colnames(cell_size_df) <- c('cell_t', 'cell_n') + +# Merge into a single dataframe +matching <-cbind(means, cell_size_df) + +# Drop duplicate identifier columns used for merging dataframes +matching <- matching[ -c(5) ] + +# Calculate weights by dividing cell_size/n +matching $weights <- matching $cell_n / n + +# Calculate weights * difference in means +matching $w_x_diffm <- matching $diff_means * matching $weights + +# Get top observations of the dataframe +head(matching) + +# Sum to get ATE estimate +ate <- sum(matching $w_x_diffm) +ate +``` + + +As a result of this process, we get that our estimated $\tau ^{ATE} = 2366.59$ rupees. + +* Next, I do the weights calculation for (2), which is $\tau ^{ATT}$: + +```{r} +# STEP 4 (2) WEIGHTED ATT + +# Store the treatment sample: +n_treat = nrow(treated) + +# Calculate treated counts by cell (subclass) +n_treated<- matched_treated %>% + group_by(subclass) %>%count() + +# Convert to dataframe +cell_treated_df <- cbind(data.frame(n_treated)) +colnames(cell_treated_df ) <- c('cell_t', 'cell_treated_n') + +# Merge into a single dataframe +matching <-cbind(matching, cell_treated_df) + +# Drop duplicate identifier columns used for merging dataframes +matching <- matching[ -c(1, 8) ] + +# Calculate treated weights by dividing cell_treated_n/sample n_treat +matching $weights_treated <- matching $cell_treated_n/n_treat + +# Calculate treated weights * difference in means +matching $wtreat_x_diffm <- matching $diff_means * matching$weights_treated + +# Get top observations of the dataframe +head(matching) + +# Sum to get ATT estimate +att <- sum(matching $wtreat_x_diffm) +att +``` +For our (2) estimate, we obtain that $\tau ^{ATT} = 2380.43$ rupees. + + +**COMPARISON BETWEEN (1) ATE AND (2) ATT WITH THE MATCHING ESTIMATOR** + + +* My $\tau ^{ATE}$ and $\tau ^{ATT}$ are very close: ATE = 2366.59 rupees vs ATT = 2380.43 rupees. + +* This tells us that the estimated effect of FIONA on endline 2016 profits for the average farmer, _given the matched covariates and assuming the SOO assumptions hold_, is approximately 2366.59 rupees, compared to 2380.43 rupees for the average farmer among _farmers that took up the insurance_. This represents only a 0.58% percentage difference between the ATE and ATT estimates, so they are **not meaningfuly different**. + +* I would not have expected these results to be the so close because when there is selection bias $\tau^{ATE} \neq \tau^{ATT}$. Since there was no randomization in this case, it is very likely that farmers that opt for insurance are systematically different that those who do not, **ie. heterogenous effects.** As such, I was expecting that the average treatment effect for the those who took up FIONA insurance would be even higher than the ATE. This result does not mean that there is no selection bias but instead, that we are unable to capture it with the exact matching estimator. + +```{r} +# Percentage change +((att - ate)/ate) *100 +``` + +**ASSESSMENT OF THE EXACT MATCHING APPROACH:** + +**1. Strengths** + +* Based on the selected covariates, most of the observations had identical matches (ie. 9947). This creates observably identical comparison between the treated and untreated groups. + +* The balance table post matching showed that the exact matching procedure was successful in achieving balance in the three included covariates. + +* This approach does not make any functional form assumptions and is therefore more flexible than the adjusted regression. + + +**2. Weaknesses** + +* This approach doesn't work for continuous variables like baseline 2005 profits. This makes it impossible to find exact matches. When attempting to include it for testing purposes, I ran into **the Curse of Dimensionality** because no matches could be found. In this scenario, an alternative approach might be needed like matching by k-nearest neighbors. On one hand, we want to include more covariates to sustain the conditional independence assumption but, we the more covariates, the lower the likelihood of finding exact matches. + +* Since cotton is completely unbalanced, it's not possible to match on cotton, which limits our analysis. Same with the districts, these are not useful variables because they don't satisfy the common support assumption and as such, it is not sensible to include them in the matching procedure. + +* It still doesn't account for selection on unobservables, which an approach like an RCT is able to. + + +**EXACT MATCHING VS. NAIVE ESTIMATOR** + +When I calculated the naive estimator in (7), I obtained an estimated ATE of 2369.56 rupees, compared to an ATE of 2366.59 rupees with the exact matching estimator, **the difference is minimal** (2.97). The fact that the matching estimated ATE is so close to the naive estimator points to the unreliability of SOO designs, given the high selection bias. + + +### (9) Based on your results in (8), explain to HARRIS whether or not they should implement a FIONA-like program in Bangladesh. Be sure to tell them the reasoning behind your recommendation. + +Based on my results in 8, I do not feel confident enough to tell HARRIS whether they should implement a FIONA-like program in Bangladesh. + +* Due to the lack of an RCT in the FIONA data, I employed an exact matching approach as an alternative. However, only 3 covariates satisfied the common support assumption, limiting the scope of the technique. + +* Second, the ATE estimate obtained with exact matching is essentially the same to the one obtained with the naive estimator. If we analyze the context, it is __very, very likely__ that there is **selection bias** in the treatment variable (opting into the FIONA insurance). As such, farmers that opt for the scheme are statistically different from those who do not. The fact that the estimated ATE is so close to the naive estimator, shows how unreliable these results are. + +* Third, the required SOO assumptions do not hold entirely. + + + The **conditional independence assumption** is unlikely to hold because the covariates we have available in the + dataset are not the only ones that are correlated with the treatment variable, so we cannot safely assume we're + in a _"as good as random"_ situation. + + + I only included those covariates for which the **common support assumption** holds, by excluding cotton, the + district dummies and 2005 profits. However, this means we are excluding precisely those problematic variables for + which we would like to have counterfactuals. With the way FIONA was designed, offering + insurance only in certain districts, this is simply not possible. + +* The ATE and ATT matching estimates are very, very close to the naive estimator, showing this method fails to perform any better. + +* An additional issue is that only 2005 profits are available for the baseline, when the study actually started in 2014. Profits in 2005 are unlikely to influence self-selection into treatment 9 years later. It would have been useful to have baseline profits in 2011, for instance and then conduct matching with KNN techniques or bandwidths. With what I estimated in (8), we are unable to compare farmers that had the exact same income before the intervention. + +Given these reasons, I cannot determine causality nor be determinant in my conclusions. Instead, I would recommend to HARRIS that we run an RCT, even at a smaller scale through a pilot, to test if the rainfall-index insurance is effective, before doing a full-scale implementation. 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+library(rpart.plot) +``` + +## I. Section 03: Keiser, David A. and Joseph S. Shapiro 2018. “Consequences of the Clean Water Act and the demand for water quality".## + +### I.A: Describe the research question in this paper in words. Explain, in words and math, the ideal experiment one might want to use to answer this question. Explain, in words and math, the naïve estimator. Provide three concrete examples of why the naïve estimator is unlikely to provide an unbiased parameter estimate in this setting, and explain whether each would bias your estimate upwards or downwards relative to the truth.### + +**(1) The research question** + +The research paper poses several research questions: + +1. What are the effects of the Clean Water act in water pollution concentration? +2. **Did the Federal grants for municipal wastewater plant treatment have an effect in water pollution?** +3. What is the benefit-cost ratio of the Clean Water Act? + +Given that the authors state, _"Because this article exploits the timing and location of grants to identify the effect of the Clean Water Act's grants program"_ (pg. 357) and the cost-effectiveness analysis is also based on grant-allocation, I consider the 2nd research question as the main question that motivates the paper. Since $650 billion of the $1 trillion invested by the Clean Water Act was destined to municipal grants for water treatment, the authors seek to measure whether these grants had an effect in water pollution concentration. + +**(2) The ideal experiment** + +In the ideal experiment, we would run a **Randomized Controlled Trial (RCT)**, where municipalities are _randomly assigned_ into receiving federal grants for wastewater plant treatments. Those municipalities that receive grants would be the treatment group, which we would compare against those municipaliies that do not receive grants (ie. the control group). + +Using the **potential outcomes framework**, we can expect the potential water pollution for a grant-recipient municipality to be the same level of water pollution for a control municipality that did not receive a grant, had that grant-recipient municipality never received a grant. + +If an RCT is properly randomized (which can be checked with a balance table pre-treatment), we effectively solve the _selection bias problem_, since treated and control municipalities would be statistically similar in observable and unobservable characteristics. This means that the following **identifying assumption** for an RCT holds: + +$$E[\varepsilon_i|D_i] = 0$$ + +Where: + +* $D$ refers to treatment status: $D = 1$ for grant-recipient municipalities and $D = 0$ for municipalities that do not receive grants +* $i$ is each individual municipality + +This means that the expectation of the error term, conditional on receiving a grant, is zero. In other words, if we find a treatment effect, we can attribute it solely to the grant for water treatment and not to other unobservable characteristics that might be correlated with the grant allocation and water pollution. + +With this _ideal RCT_, we can estimate the **Average Treatment Effect (ATE)** of the _average municipality grant_ on water pollution: + + +$$\tau^{ATE} = E[Y_{i}(1)] - E[Y_{i}(0)]$$ +Where: + +* $Y$ is the outcome variable of interest, water pollution +* $i$ is each individual municipality. + +Under randomization, the **expected** water pollution for municipalities that are recipients of grants (ie. $Y_{i}(1)$) is equivalent to the **mean** water pollution for grant-receiving municipalities, and likewise for municipalities in the control group. We would then estimate ATE by taking the difference in means between both groups: + +$$\tau^{ATE} =\overline{Y_{i}}(D = 1) - \overline{Y_{i}}(D = 0)$$ + +Where: + +* $D$ refers to treatment status: $D = 1$ for grant-recipient municipalities and $D = 0$ for municipalities that do not receive grants +* $Y$ is water pollution, measured by the quantity of pollutants in the water +* $i$ is each individual municipality + +In the ideal experiment, there would be **perfect compliance** such that only municipalities randomly assigned to receive a grant actually receive it and vice versa. There would also be **no spillover effects** such that only the waters in municipalities that receive grants benefit from the wastewater treatment plants built with those grants and waters in control municipalities do not have access to those treatment plants. + +**(3) The naive estimator** + +In the absence of randomization, the naive estimator consists of a simple difference in means between **observed outcomes** instead of **potential outcomes**: + +$$\tau^N = \overline{Y}(1) - \overline{Y}(0)$$ +Where: + +* $Y$ is the outcome variable, ie. water pollution + +This naive estimator is very likely to be **biased** because of selection into treatment. Three examples of why it might be problematic: + +1. It could be the case there is a corruption issue within municipalities that receive the grants. For example, if municipality politicians redirect the funds designated for water treatment plants for personal gain or other projects in their political interest, then we could have a **downward bias** assuming that these grants don't work in reducing water pollution when in fact, the funds where never destined to address the issue. + +2. Some municipalities may have a concentration of water-polluting factories while other municipalities do not. If coincidentally, those municipalities with _very limited_ water-polluting factories are the ones that receive the grants (due to environmentally-focused politicians that lobby for the grants or respond to climate-conscious citizens that hold them accountable), then we would have an **upward bias** of the effect of the grants, incorrectly concluding that they are more effective than they actually are because we are observing municipalities whose waters were unpolluted to begin with. + +3. There could be governance problem or a delay in between grants being allocated and grants actually being received by the municipalities. If it takes too long to receive the grants or they money comes in fractions over time, it would take longer than 2-10 years to construct water treatment plants. Then, we would have a **downward bias** where we conclude the grants are not effective in reducing water pollution when in reality, those grants are not received entirely, take too long or the operation costs of transferring them reduce the quantity received. + + +### I.B: Copy down the authors’ main regression specification (and be sure to list which equation number this is in the paper). Explain, in words and math, what treatment parameter the authors are recovering with their main specification. Does this approach recover the population average treatment effect? If yes, why? If no, why not? What assumptions are required for this regression to recover the causal effect of interest? Do you think these assumptions are likely to be satisfied in this context? Why or why not? Include references to evidence presented in the paper to support your conclusion.### + +For the second research question, which I consider to be the main question of the paper, ie. _did the Clean Water Act grants for municipal wastewater plant treatment have an effect in water pollution?_, the authors employ a **Triple difference-in-differences DDD** design with the following regression (page 367, regression #3): + +$$ Q_{pdy} = \gamma G_{py}d_d + X'_{pdy} \beta + \eta _{pd} + \eta _{py} + \eta _{dwy} + \epsilon _{pdy}$$ +Where: + +* $Q_{pdy}$ is water pollution, which varies by treatment plan, year and whether the water is upstream or downstream +* $G_{py}$ is the cumulative number of grants than each plant received by year +* $\gamma _{}$ is the **main coefficient of interest**, and represents the mean effect of each cumulative grant on downstream water pollution +* $X'_{pdy}$ are covariates that are controlled for, such as temperature and precipitation +* $\eta _{pd}$ are **plant x downstream** fixed effects +* $\eta _{py}$ are **plant x year** fixed effects +* $\eta _{dwy}$ are **plant x downstream x year** fixed effects +* $p$ is each individual treatment plant +* $y$ is each year +* $d$ is downstream + +With this regression specification, the authors include time (year y) and unit (treatment plant p) fixed effects as part of the standard DD estimator, and add a third difference where they additionally compare between downstream (d) and upstream areas. In this case, upstream areas _"act as a counterfactual for downstream areas"_ (pg. 353). The focus is on areas 25 miles downstream of a treatment plant because downstreamis the direction that pollution travels in a water source. + +We can also rewrite this regression as follows: + +$$ +\begin{split} +\tau ^{DDD} = \beta _0 + \beta _1 CumGrant_{pi} + \beta _2 Post_y + \beta _3 Downstream_{pj} + +\eta _1 (CumGrant_{pi} \times Post_y) + \\ \eta _2 (Post_y \times Downstream_{pj}) + \eta 3 (CumGrant_{pi} \times Downstream_{pj}) + \\ +\tau (CumGrant_{pi} \times Post_y \times Downstream_{pj}) + \varepsilon_{ijy} +\end{split} +$$ +Where: + +* $CumGrant$ refers to the cumulative number of grants per year by plant, because the authors measure grants as investment in durable capital (instead of the dollar value of each grant) +* $_{pi}$ is plant i and $_{pj}$ is plant j +* $_y$ is time measured in years +* $\tau$ is the triple DDD estimate +_(Equation was adapted from slide 13 of Lecture 13 and the regression in the paper.)_ + +With their main specification, the authors are recovering the **Average Treatment Effect** of the average cumulative grant on downstream water pollution: + +$$\tau^{DDD} = E[Y_{pyd}(1)] - E[Y_{pyd}(0)]$$ + +Where: + +* $Y$ is downstream water pollution +* $p$ is each individual plant +* $y$ is each year +* $d$ is 1 if dowwnstream, 0 otherwise + +The DDD estimator effectively recovers the population ATE if the following identifying assumption is met: + +$\rightarrow$ Conditional on changes over time, within treatment plants and within downstream areas, grant-recipient and non grant-recipient plants would have been trending similarly before, PRE, grants being allocated. This is the **similar difference in trends over time** identifying assumption, which is a weaker assumption than the parallel trends assumption for the standard DD estimator because a more narrow comparison is being made. + +One reason why this identifying assumption may not hold is due to **coincident treatments**, which is mentioned by the authors. They refer to air pollution regulations that happened at the same time as the Clean Water Act, where _"less acid rain may have led to less acidic rivers and lakes"_. (pg 352) Another coincident phenomenon is climate change, which is inversely correlated with a oxygen saturation deficits. + +While we can never truly observe the counterfactual, that is, how grant-recipient plants would have trended had they not received grants, the researchers include **event study graphs** in evidence that grant-recipient and non grant-recipient plants were trending similarly before grants were allocated. In equation (4) on page 369, they specify the following event-study specification: + +$$ +\begin{split} +Q_{pdy} = \sum^{\tau = 25} _{\tau = -10} \gamma_{\tau}1[G_{p,y-\tau} = 1]d_d + X'_{pdy} \beta + +\eta_{pd} + \eta_{py} + \eta_{dwy} + \epsilon_{pdy} +\end{split} +$$ +Where: + +* $\tau$ refers to the event study periods, from -10 years before grants were allocated, to 25 years since the grants were allocated +* $1[G_{p,y-\tau} = 1]$ is a dummy that equals 1 for each of the years before/since the grants were allocated and 0 otherwise +* All other variables are the same as described above + +Based on the corresponding event-study graph on page 377, we can observe that in the years before the grants were allocated, water pollution (measured as dissolved oxygen deficit and share of not fishable waters) in downstream and upstream areas were on similar trends. According to the authors, _"In years before a grant, the coefficients are statistically indistinguishable from zero, have modest magnitude, and have no clear trend."_(pg. 377) This suggests that statistically, plants in downstream and upstream areas were on similar trends in the years **before** grants were allocated, which provides evidence in favor that the identifying assumption holds and that the authors are recovering the ATE from their triple DDD estimator. We cannot definitively proof that the identifying assumption but the provided event-study graph supports that the assumption holds. + +Additionally, to recover a precise ATE, it is necessary for the research design to have **clustered standard errors** before claiming statistical significance. The authors pose they cluster standard errors by watershed (which are water treatment plants that receive NPDES permits, pg 356). That is, SE's are clustered at the unit level, which in this case are the plants and since the analysis is at the national level, we can expect there to be more than 45 clusters. In all, the evidence in the paper supports the identifying assumption. + + +### I.C: Describe the main results of the paper. Include a discussion of (at a minimum) one table and one figure, in which you interpret the estimated coefficients and describe their magnitudes. What is the main policy take-away of the paper? ### + +Based upon the three different research questions, these are the main results of the paper: + +**1.** The authors find a substantial decline in water pollution concentration from 1962-2001 but, this decline was smaller over time (marginal returns to scale). From 1972-2001, the share of waters safe for fishing increased by _12 percentage points_. However, the authors also note that there has been a decrease in air pollution over the same time period (ie. **coincident treatments**), which explains less acidic pH levels in lakes and rivers (and therefore the decline in water pollution may not be solely attributed to the the Clean Water Act). (pgs 349, 352) + +**2.** The Clean Water Act's grants destined for municipal wastewater treatment plans caused part of the decline in water pollution concentration. The authors employ a _triple difference-in-differences DDD design_ comparing water pollution 1) pre and post grant allocation, 2) upstream vs downstream of wastewater plants, and 3) across plants. The authors find that on average, each grant reduces the probability that a downstream area violates the regulation's standards by a _0.5 percentage point._ This effect is found within 25 miles downstream of treatment plants and are estimated to last for 30 years. The authors also find that municipal had near complete pass-through (that is, it wasn't misplaced). $1 of a federal grant led to $1 of spending on municipal sewers. (pgs 349, 352) + +**Table II. interpretation, pg. 378** + ++ Holding all else constant, on average, each cumulative grant is estimated to: + + Decrease _dissolved oxygen deficit_ by -0.68 percentage points, statistically significant at all conventional levels of significance. + + Decrease the _share of not fishable waters_ by -0.7 percentage points, statistically significant at all conventional levels of significance + + Decrease in the _share of not swimmable waters_ by -0.4 percentage points, statistically significant at the 10% level. + + Decrease in _biochemical oxygen demand (BOD)_ by -0.10 milligrams of oxygen per liter, statistically significant at the 5% level. + + Decrease of -204.06 _fecal coliforms_ per 100 milliliters of water, statistically significant at the 5% level. + + Decrease of -0.5 by milligrams of _total suspended solids TSS_ per liter of water (mg/L), statistically insignificant. + + In general, the DDD estimates show a decrease in the different types of pollutants and share of non-swimmable and non-fishable waters, statistically significant for 6 of the 7 measures. The magnitude of the coefficients are argued by the researchers to be sizable in terms of water pollution decline, especially considering that $650 billion were destined to municipal grants. If, each average cumulative grant causes a decrease of -0.0.68 pp in dissolved oxygen deficit, this is a quite large effect given the amount of investment destined to grants. Additionally, all 7 coefficients have a negative sign, as one would expect, showing each average cumulative grant has a negative relationship with water pollution. + +**3.** The estimated effect of the Clean Water Act's grants on housing values was less than the cost of the grants themselves. To estimate the benefits, the authors ask residents within a 25-mile radius of affected rivers, how much they valued these grants. This is essentially estimating the demand curve to understand the value or utility derived from residents from these grants. The authors find that on average, the benefit over home values was 25% the cost of the grant. These results are mostly homogenous across areas and types of grants. The researches find that the average grant was $31 million and the estimated effect on housing value was $7 million, which represents a benefit of 22.5% of the grant's cost. (pgs 349, 352) + +**Graph IV. interpretation, pg. 385** + ++ The graph shows the event-study results, both for homes very close to the treatment plants (within 0.25 miles) and for homes further away (25 miles of the downstream river). ++ On panel A, for **homes close to treatment plants**, we can observe that in the years before grants were allocated, home values were in a down-sloping trend. After grant allocation, there is an immediate steep increase in home values, however, the trend in home values varies up and down over the following years. This panel barely provides evidence in support of grant allocation increasing house values for households near treatment plants. ++ On panel B, for **homes further from treatment plants**, there is no clear indication that grants are associated with an increase in house values. There is up and down variation that could be due to many reasons other than the grants. ++ It looks as though residents near the plants have a slight benefit, as measured by a modest increase in home values, but this could well be the result of other factors. The authors mention that treatment plant construction as a result of the grants, could also lead to increased taxes, which would reduce home values (pg. 389). They also pose that it's possible that the price function may have shifted, leading to willingness to pay (WTP) estimates being underestimated. (pg. 389) While there are different explanations for the directions of possible correlations, the event study graphs do not support an effect in home values. + +**Main policy take-away** + +This paper provides evidence that the Clean Water Act did lead to a decline of water pollution concentration. However, the benefit-cost ratio is small because of the huge costs (estimated at $1 trillion invested in 2014 dollars) and benefits that do not seem as large as are expected. + +As a main policy take-away, the Clean Water Act may not be the policy instrument best-suited nor the most cost-effective to decrease water pollution at the desired levels for several reasons: + +1. The Clean Water Act is effectively a type of **subsidy**, by allocating municipal grants and subsidizing pollution control equipment. Economic theory shows that market-incentives are often more cost-effective. As an example, the authors compare the cap-and-trade system behind the Clean Air Act.(pg. 391) + +2. The scope of the policy was limited to only certain sources of pollution. It only addressed industrial pollution or "point sources" and excluded urban and agricultural pollution, or "nonpoint sources." (pg. 356) Agriculture is a major water pollution source, according to the authors. In contrast, the Clear Air Act included all major sources of pollution. + +3. Most of the benefit from unpolluted water is recreational trips and aesthetics, which people can substitute easily by traveling to clean rivers instead of polluted rivers. (pg. 391) + +4. In terms of health, people drink treated and filtered water, even if it was previously polluted, so they are less likely to view polluted water sources as an issue. This is different with breathing polluted air which has direct health implications. (pg. 392) + +Half of these issues are specific to the way the Clean Water Act was design (subsidy design and the scope of pollution sources) and the other half are related to the inherent differences between water and air that change how people value these unpolluted resources. As the most important **take-away**, the authors note that _"allowing the stringency of pollution regulation to vary over space has potential to increase social welfare"_. This means that the net benefits of a policy like the Clean Water Act vary by location, population density and how popular water recreation is, so this policy should be adapted to those areas in order to maximize net benefits. (pg. 392) + + +## II. A local activist group, the Universal Committee to Heighten Interest in Cars Against Greenhouse Outlays (UCHICAGO) promotes the adoption of electric vehicles in Chicago. They would like you to help them design a pilot program to demonstrate the impacts of a new electric vehicle (EV) subsidy they are trying to roll out at scale.## + +### II.A: UCHICAGO hypothesizes that providing homeowners with subsidies for electric vehicles will increase miles driven in EVs. Using the potential outcomes framework, describe the impact of treatment - a 20% EV subsidy – on electric vehicle adoption, household electricity use, and electric vehicle driving at the household level. EV adoption is a binary variable; household electricity use is measured in kWh, and electric vehicle driving is measured in eVMT. Explain to UCHICAGO what the ideal experiment would be for answering this question. Describe the dataset you’d like to have to carry out this ideal experiment and use math, words, and the potential outcomes framework to explain what you would estimate and how you would do so. Make sure to be clear about the unit of analysis (ie, what is i here?) ### + +Using the **potential outcomes framework**, we are interested in measuring the change in potential outcomes (ie. electric vehicle adoption, household electricity use, and electric vehicle driving) caused by a 20% EV subsidy (ie. treatment). Here, potential outcomes refer to _all possible outcomes_ we could have observed with the subsidy and in the absence of the subsidy. + +Hypothetically, we could measure this as: + +$$\tau_{household\space i} = Y_{household\space i}(D = 1) - Y_{household\space i}(D = 0)$$ +$$\tau_{household\space i} = Y_{household\space i}(1) - Y_{household\space i}(0)$$ + +Where: + +* $D_i=1$ for households that receive the subsidy and $D_i = 0$ for households that do not +*$Y_i(D_i)$ are the different _potential_ outcomes of interest (ie. electric vehicle adoption, household electricity use, and electric vehicle driving) for each household i with treatment status $D_i$ +* $\tau_{household\space i}$ is the treatment effect for an individual household i. + +This treatment effect for each **individual household is impossible to estimate** because we would need to observe two potential outcomes for each individual household under both states of the world: as a recipient of the EV subsidy and as a non-recipient of the EV subsidy _at the same time_. This is the **fundamental problem of causal inference** because we cannot ever observe both $Y_{household\space i}(1)$ and $Y_{household\space i}(0)$ simultaneously for the same household. This means, we cannot actually estimate the treatment effect of the subsidy at the _individual household level._ + +**Ideal experiment** + +In the ideal experiment, we would randomize homeowners who are recipients of the 20% EV subsidy (ie. treatment group) and homeowners that do not receive the subsidy (ie. control group). There would be **perfect compliance** in the ideal experiments, such that only those homeowners assigned to receive the treatment actually receive it, and viceversa for the control group. + +We could answer UCHICAGO's question by instead estimating the effect of the **average 20% EV subsidy** on each of the potential outcomes, by estimating the **Average Treatment Effect (ATE)**. Since there are three outcomes of interest, we would calculate three separate ATE's of the following form: + +$$\tau^{ATE} = E[Y_{i}(1)] - E[Y_{i}(0)]$$ + +Where: + +* $Y$ is the outcome variable of interest, 1) electric vehicle adoption as a binary variable, 2) household electricity use measured in kWh, and 3) electric vehicle driving measured in eVMT +* $i$ is each individual homeowner + +Under randomization, the **expected** potential outcome for homeowners that receive the EV subsidy (ie. $Y_{i}(1)$) is equal to the **mean** outcome of subsidy-recipient homeowner, and likewise for homeowners in the control group. Then, we can estimate the three different ATE's by taking the difference in means between both groups: + +$$\tau^{ATE} =\overline{Y_{i}}(D = 1) - \overline{Y_{i}}(D = 0)$$ + +Where: + +* $D$ refers to treatment status: $D = 1$ for homeowners that receive the subsidy and $D = 0$ for those who don't +* $Y$ is each of the 3 outcome variables, previously described +* $i$ is each individual homeowner + +The dataset I would use to estimate the ATE through an RCT, would include a **cross-sectional data** of multiple homeowners and their baseline characteristics, which would allow me to check the randomization process was effective through a balance table. That is, the dataset should include a number of observable characteristics that could potentially be correlated with the different outcomes, such as the homeowner's wealth, whether they already own an EV, average driving, etc. I would also need the dataset to include measures of three outcome variables of interest: binary variable on EV adoption, household electricity use in kWh and electric vehicle driving in eVMT. + + +### II.B: UCHICAGO have secured funding to run a randomized trial to test their hypothesis. However, their funder is worried about implementing an individually-randomized design. In particular, they are concerned that control group individuals may be less likely to purchase an EV if all of the cars on the market are bought by the treatment group. Is this a problem for an RCT that is randomized at the individual level? If yes, explain why, and describe what this would do to your treatment effects relative to the truth. If no, explain why not. Assume for the remainder of Question II that everyone offered a subsidy by UCHICAGO purchases an EV.### + +If effectively, control group individuals are less likely to buy an EV because _all of the cars on the market are bought by the treatment group_, this is would be a problem. This is because our treatment is an EV subsidy with the goal of measuring how that subsidy impacts electric vehicle adoption, household electricity use, and electric vehicle driving. This means that we want to compare the behavior towards EV adoption and driving between individuals that received and that did not receive the subsidy. + +If all cars on the market are bought by the treatment group, we have inadvertently created a **spillover**, an artificial situation where the control group cannot act as they normally would have. In the absence of this situation, some individuals in the control group may opt to buy an EV at full price (without receiving the subsidy). However, since there are no EV's available for them to buy, they will be unable to acquire one. We would therefore be recover an **upwards biased ATE** because we would observe that the whole control group did not acquire an EV, not because they did not want to but because there were no vehicles available as a by-product of the research design. + +This is a violation of the **Stable Unit Treatment Value Assumption (SUTVA)**: +$$ D_i \neq D'_i \rightarrow Y_i(D) \neq Y_i(D')$$ + +Where treatment status of individuals receiving the 20% EV subsidy ($D_i$), affects potential EV adoption for individuals that did not receive the subsidy ($Y_i(D')$). This is the case of **spillovers**. Since the EV subsidy impacts the availability of EV's for non subsidy recipients: + +$$(D_i = 0) \downarrow $$ + +When we estimate ATE by taking the difference in means, we are **overestimating the effect of the subsidy**: +$$\tau^{ATE} = \overline Y(1) - \overline Y(0) \downarrow$$ +$$\tau^{ATE} \uparrow$$ +Where: + +*$Y$ is one of three outcomes variables (EV adoption, household electricity and EV driving) + +As such, we would incrrectly conclude that the 20% EV subsidy is more effective than it actually is because _only subsidy-recipient individuals bought an EV._ Randomization makes treated and control groups statistically similar on observable and unobservable charactersitics so that the only difference in outcomes between them can be attributed to the treatment (ie. the EV subsidy). However, this problem is not solved by randomization because we are creating a situation that restricts the control group's choices in ways that would not be restricted in the absence of the experiment. + +If this design were to be implemented no matter what, then we could treat the estimated ATE as a upper bound (ie. we know it is overestimated so it can never be higher than our estimate). Alternatively, the RCT could be designed to avoid this spillover effect or to actually measure it (if it were in UCHICAGO's interest). + +### II.C: The founder is adamant that they will not support an individually-randomized design. They would like UCHICAGO to instead provide subsidies at the neighborhood level. Will this address the concern they raised in IIA? If yes, explain how. If no, explain why this will be no better than the individually randomized design.### + +If instead, subsidies are provided at the neighborhood level, this would indeed solve the issue raised in (II.B) as **SUTVA will not be violated**. We would no longer have unintended **spillover effects** from subsidy-recipient individuals buying all available EV's from the stores and leaving none to buy for the control group because the whole neighborhood gets a subsidy and we assume they all buy an EV. + +We are effectively randomizing entire neighborhoods into receiving the subsidy and entire neighborhoods to not receive the subsidy (control group), which gets rid of the spillover effects. Control neighborhoods who are not offered the subsidy are free to act as they normally would, so we can expect some individuals to buy EV's and others who won't, according to their WTP for an EV at full-price. Since we are now assuming that _"everyone offered a subsidy by UCHICAGO purchases an EV"_, we would be comparing the impact of the subsidy on the three outcomes in neighborhoods that are randomized into receiving the subsidy vs. those randomized not to receive it. + + +### II.D: After listening to the founder’s feedback, UCHICAGO has decided that they are actually interested in studying how providing EV subsidies to some people in a neighborhood impacts EV adoption, home electricity use, and EV driving for people who don’t get the subsidy. The founder is excited about this too, and is willing to devote substantial funding to the project. Given a large budget, describe an RCT design that will allow you to measure treatment effects on both households who get a subsidy from UCHICAGO and households who don’t. Make sure to describe any necessary steps, and clearly lay out any treatment arms. A tree diagram may be helpful. Use words and math to explain what treatment parameters you can estimate with this design. Be sure to be clear about the comparisons you are making. Write down a regression equation that you would use to estimate these treatment parameters, and describe how you would interpret any coefficients you recover. Finally, describe how you would use your estimates to recommend to UCHICAGO whether they should scale their subsidy program or not.### + +To measure treatment effects on both households who get a subsidy and households who do not (control group), I would propose running the following **Stratified Design**: + +![Stratified RCT Design, Tree Diagram]("~/Desktop/Spring 2022/Program Evaluation/Final/img.png") + + +For the previous question (II. C), I would have conducted a cluster-randomized RCT, where the subsidy was randomized at the neighborhood level in order to correct for spillovers. However, since UCHICAGO is specifically interested in "how providing EV subsidies to _some people in a neighborhood_ impacts EV adoption, home electricity use, and EV driving for _people who don’t get the subsidy"_, this requires comparing the outcomes of treated and untreated household in the _same neighborhoods_. + +As such, I propose a stratified design that **randomizes the subsidy within neighborhoods**. The steps included in the diagram are the following: + +1. Collect household-level data on baseline characteristics for each of the three desired outcomes: EV adoption, household electricity use and EV driving PRE subsidies being allocated. +2. Randomize _within clusters_ (ie. neighborhoods), such that some households will receive the subsidy (treatment group) and others will not (control group). Here, a balance table should be done by running t-tests to make sure that the clusteres are statistically balanced. As per UCHICAGO's specifications, the treatment itself does not change (ie. 20% EV subsidy) but there are different treatment arms for each of three outcomes of interest: ++ EV adoption ++ Household electricity usage ++ EV driving +4. Collect end-line POST measurements at the household-level and conduct statistical analysis. + +This design ensures balance between the neighborhoods and increases statistical power. With this design, we can recover the **ATE** of the average household that received the subsidy on EV adoption, household electricity use and EV driving: + +$$\tau^{ATE} = E[Y_{ni}(1)] - E[Y_{ni}(0)]$$ +Where: + +* $Y$ is each of the three outcome variables of interest, 1) EV adoption, 2) household electricity use, and 3) EV driving +* $i$ is each individual household +* $n$ is each neighborhood or strata + +Under stratified randomization, in **expectation** the potential outcomes for households that receive the subsidy is equivalent to the average outcomes for households that received it. This expectation holds the other way around for households that do not receive the subsidy. Then, to estimate ATE, we need only take the difference in means between both groups: + +$$\tau^{ATE} =\overline{Y_{ni}}(D = 1) - \overline{Y_{ni}}(D = 0)$$ +Where: + +* $D$ refers to treatment status: $D = 1$ for subsidy-recipient households within a neighborhood $D = 0$ for households that do not receive subsidies within a given neighborhood. +* $Y$ is the outcome variable of interest: EV adoption, household electricity, use and EV driving +* $i$ is each individual household +* $n$ is each neighborhood or strata + +In order for the recovered ATE to be **unbiased**, we require as an identifying assumption that the potential outcome of households within a neighborhood strata are unaffected by other households within the same neighborhood, such that there is no **cross-neighborhood/cluster interference**. + +Another way to estimate this difference-in-means to recover ATE, is to run a regression: + +$$ Y_i = \alpha + \tau Subsidy_{i} + (\gamma Subsidy_i \times X_n) + \delta X_i + \varepsilon _i$$ +Where: + +* $Y_i$ is the outcome variable of interest: EV adoption, household electricity, use and EV driving +* $\tau$ is the ATE of the subsidy on the average household i +* $\gamma Subsidy \times X_n$ is an interaction term that accounts for heterogenous effects by neighborhood strata. X covariates that may be correlated with the outcome variables, vary by neighborhood n, whereas the treatment status into receiving subsidy varies by household i +* $X_i$ refer to covariates that vary by household i +* $i$ is each individual household +* $n$ is each neighborhood/strata + +If the stratified design is randomized, it is not necessary to include $X_i$ covariates in the regression because we will still recover the ATE. Doing so will soak up noise and improve the ATE estimates. + +I would interpret the ATE coefficient as follows: the average difference in household electricity use/EV driving between households that receive the 20% EV subsidy and households that do not, holding everything else constant, is predicted to be an (x coefficient value) increase/decrease (depending on a positive/negative sign) in kWh of household electricity consumed/ electrical driving eVMT. In the case of EV adoption, since the dependent and independent variables are dummies, it would be interpreted as the x% probability that an EV is adopted. I would note whether the coefficients are statistically significant, the sign of the coefficient and whether it is economically significant. + +Assuming the identifying assumption holds and that the recovered ATE estimate is unbiased, I would recommend to UCHICAGO to scale their subsidy program if we obtain statistically significant estimates of a sizable magnitude for most (if not each) of the three outcome variables: EV adoption, household electricity use and EV driving. To determine whether the magnitude of the ATE estimates is significant or not, I would have to discuss with UCHICAGO experts based on their past experiences of how much is economically significant, or alternatively, research this in the literature. + + +## III: Around the world, women are under-represented in politics. A bunch of people have hypothesized that having more women in positions of political power could improve well-being. A policy advocacy group, Powerful, Respected, and Organized Government Requires that All Men Elect Valuable Accomplished Ladies (PROGRAMEVAL), is interested in demonstrating the impact of women in government on local public goods provision in India. They’ve asked you for help with their analysis.## + +### III.A: PROGRAMEVAL would like you to compare the average number of public goods (roads, schools, public buildings, etcetera) in towns with female-headed governments as compared with towns that have male-headed governments. Describe this comparison in math and words. Under what conditions would this comparison estimate the causal effect of female leaders on public goods provision? Provide two concrete examples of reasons why this comparison may be problematic.### + +If we simply compare the average number of public goods in towns with female-headed governments vs. male-headed governments, we would be obtaining the **naive estimator**: + +$$\tau^N = \overline{Y}(1) - \overline{Y}(0)$$ + +Where: + +* $Y(1)$ is the average number of public goods for _female-headed_ governments +* $Y(0)$ is the average number of public goods for _male-headed_ governments + +This naive estimator is based soley on **observed public goods**, instead of the **potential public goods** for female-headed governments had they been male-headed governments instead. It is very, very unlikely that this comparison would ever estimate the effect of female leaders on the provision of public goods. For this to be the case, we would have to completely **rule out any selection bias on both observable and unobservable characteristics**, meaning that all differences between female-led and male-led governments are "as good as random". + +In order to estimate the naive estimator, we could run a regression of the following form: + +$$ Y_i = \alpha + \tau D_i + \varepsilon_i$$ +Where: + +* $D_i$ refers to treatment status $Di = 1$ for female-led towns and $D_i = 0$ for male-headed towns +* $Y_i$ is the outcome variable, average number of public goods provided (roads, schools, public buildings, etc.) +* $i$ is each individual town + + +This assumption is _highly unlikely_ to hold since we can expected female-headed governments to be systematically different from male-headed governments in many ways other than in public good provision. For example, female-headed governments are likely to be more liberal, more developed and with more educated citizens, on average. + +Two examples why the naive estimator would be problematic: + +1. If female-headed governments also happen to be governments with better governance and a better public goods provision, then the naive estimator would lead us astray with a **upwards bias** over-estimation that female-led governments are better at providing public goods than they actually are, because we are only analyzing systematically more efficient governments. + +2. Alternatively, if male-headed governments happen to be governments where public goods provision is much cheaper, like certain lower-income developing countries, then we might have a **downward bias** were we underestimate the true effect of female-led governments on public goods provision because we are comparing them to governments where the increased provision of public goods is a result of their low-cost and not because of governments being male-led. + +### III.B: PROGRAMEVAL gets it - this is not the best comparison. However, they have data on a bunch of other town characteristics: per-capita income, number of residents, year of incorporation, average population age, and share of gross city product devoted to manufacturing. Describe, using math and words, a comparison between female- and male-headed towns which leverages these administrative data. Under what conditions would this comparison estimate the causal effect of female leadership on public goods provision? Provide two concrete examples of reasons why this comparison may be problematic (different from what you described above).### + +Here we have a **cross-sectional** dataset for multiple towns in the same time period, that includes several observable characteristics. + +Assuming we can only rely on this data and that an RCT is not possible, we can estimate ATE with a **Regression Adjustment as part of a Selection on Observables (SOO)** design where we essentially add controls. + +To leverage this administrative data, we adjust the regular ATE regression by including controls in the form of additional covariates $X_i$'s. + +$$ Y_i = \alpha + \tau D_i + \gamma X_i + v_i$$ + +Where: + +* $D_i$ refers to treatment status $Di = 1$ for female-led towns and $D_i = 0$ for male-headed towns +* $X_i$ represents a series of control variables available to us in the administrative data, including: + + per-capita income + + number of residents + + year of incorporation + + average population age + + share of gross city product devoted to manufacturing +* $Y_i$ is the average number of public goods provided (roads, schools, public buildings, etc.) +* $v_i$ is the error term, which captures anything else correlated with public good provision that is not being controlled for in the regression +* $i$ is each individual town + +Since we don't have random assignment into female-led governments, the error term conditional on treatment status is non-zero, ie. $E[\varepsilon \mid D_i] \neq 0$. Therefore, in order to recover an unbiased ATE from this adjusted regression, we need the following **four identifying assumptions** to hold: + +1. **Conditional independence: ** conditional on the controlled observables (eg.per-capita income, average population age, etc.), treatment assignment into a female-headed government is independent of potential public goods provision. in other words, all differences between $D_{town} = 1$ and $D_{town} = 0$ are assumed to be _as good as random_. This is a **very strong assumption** and can be mathematically expressed as: + +$$ (Y_i(1), Y_i(0)) \perp D_i \mid X_i$$ + +Where: + +* $D_i$ refers to treatment status $Di = 1$ for female-led towns and $D_i = 0$ for male-headed towns +* $Y_i$ is the average number of public goods provided +* $X_i$ represents control variables, such as: per-capita income, number of residents, year of incorporation, etc. +* $i$ is each individual town + +This assumption indicates that once we control for these covariates $X_i$'s (eg.per-capita income, average population age, etc.), selection into a female-led town $D_i$ is _as good as random_ and we have essentially **eliminated selection bias.** + + +2. **Common support - ** The probability that $D_i = 1$ for all levels of $X_i$ is between 0 and 1: +$$ 0 < Pr(D_i = 1 \mid X_i = x) < 1$$ +Where: + +* $D_i$ refers to treatment status $Di = 1$ for female-led towns and $D_i = 0$ for male-headed towns +* $X_i$ represents control variables, such as: per-capita income, number of residents, year of incorporation, etc. +* $i$ is each individual town + +This means that for each covariate, there must be _both treated and untreated units_ in the administrative (ie. female and male-headed towns, respectively.). + +3. **Close mean covariates between treated and untreated -** The mean covariates for female-headed towns must be **close** to the mean covariates for male-headed towns, ie: $\overline X_T$ close to $\overline X_U$. If the difference between the covariates is large, $\mid \overline X_T - \overline X_U \mid$, then the $\hat {\tau}$ estimate will be biased. + +4. **Functional form adherence -** The true relationship is assumed to be of the following **functional form**: + +$$Y_{i}= \alpha + \tau D_{i} + \gamma X_{i}$$ +Where: + +* $Y_i$ is the average number of public goods provided +* $D_i$ is the treatment status $Di = 1$ for female-led towns and $D_i = 0$ for male-headed towns +* $X_i$ represents control variables, such as: per-capita income, number of residents, year of incorporation, etc. +* $i$ is each individual town + +NOTE: _(The write-up for these assumptions were adapted from my own work in Problem Set 2)_ + +**Examples of why this SOO Regression Adjustment may be problematic ** + +1. The **conditional independence** assumption is a very, very **strong assumption** and it is fundamentally untestable. It is _highly unlikely_ that this assumption would hold because it requires essentially controlling for every single observable that might be potentially correlated with the provision of public goods. For instance, some observables that are likely correlated with public goods provision in any given town (and for which we do not have administrative data), include: town's government budget, the town's debt level, corruption, etc. A town with a higher budget allocation would have more funds that can be destined to public goods provision, likewise an indebted town is unlikely able to provide public goods if most of the budget is oriented to pay-off its debt. If a given town has a high degree of corruption, which can be measure in accounting terms, then public goods provision will be reduced. Essentially, out estimates are affected by **omitted variable bias** because we cannot account for every variable that is correlated with public goods provision. + +2. Another example is that we might not have **common support** for all covariates in our administrative data. It's possible that there are no observations for female-led governments (since there are significantly fewer in quantity) for all covariates (per-capita income, share of gross city product devoted to manufacturing, etc). If this is the case, we will recover a **biased** ATE estimate because we will be unable to compare female-led vs. male-led towns in all observables. It could well be that the covariates on which we are controlling both groups are precisely those covariates for which female-led towns have a lower performance relative to male-led towns, leading us to underestimate the true effect of having a female-headed government. Since we don't have the full picture, we are making biased comparisons with part of the picture. + +### III.C: PROGRAMEVAL understands your concerns, but has some in-house machine learning experts. They tell you that they can use this same administrative data to solve your issues. Do you agree? Why or why not? Be specific.### + +In principle, machine learning methods cannot solve my concerns of using an Adjusted SOO Regression. Causal inference is focused on estimating the _causal effect of female-led towns in the provision of public goods_, that is, finding $\tau ^{ATE}$. The focus of program evaluation is recovering **unbiased estimates**. + +On the other hand, Machine Learning is concerned with **prediction**, that is, finding the best guess for a given outcome $\hat{Y}$. For example, PROGRAMEVAL's experts might be able to predict what the _average number of public goods_ is in a given town, given features/covariates $X_i$ such as per-capita income or the number of residents in a town. The experts may employ different ML methods to find the function $f(X)$ that best predicts the outcome $Y$ by minimizing bias without sacrificing variance too much, but ultimately their goal is prediction $\hat{Y}$, when I need the causal effects $\tau ^{ATE}$. + +ML experts, however, could help to figure out which covariates actually matter in driving public goods provision. The root of the problem is we don't know if we are controlling for the right stuff. So, ML can help us by following four steps: + +1. Predict public good provision $\hat{Y}$ as a function of the covariates $X$ (ie. per-capita income, number of residents, etc.) + +2. Predict treatment status, ie. female-headed town $D$ as a function of the covariates $X$ + +3. Calculate the outcome and treatment status residuals: +$$Y^R = Y - \hat Y$$ +$$D^R = D - \hat D$$ +4. Estimate the treatment effects $\tau$ by running the following regression: +$$Y_i^R = \alpha + \tau D^R + \varepsilon_i$$ +Where: + +* $Y_i^R$ are the public good provision residuals +* $D^R$ are the treatment status residuals, ie. female-led towns + +Of these four steps, (1) and (2) can be done with any ML methods, ideally some that get the best bias-variance trade-off without overfitting the training data. + + +$\longrightarrow$ Nonetheless, for this approach to work, **SOO assumptions need to hold** and my main concern in part (III.B) is precisely that the Adjusted Regression identifying assumptions are _highly unlikely to hold_. Unfortunately, ML experts will _not be able to help me solve these issues_, even with the most sophisticated techniques available. + + +### III.D: PROGRAMEVAL forgot to tell you that, in India, certain local government positions are “reserved” for women -- meaning only women can run for office to fill these seats (this is, again, a Real Thing!). They inform you that towns are selected to have reserved seats based on their political party. In Uttar Pradesh, an Indian state, all towns are put on a list, ordered by the share of women in the population. Each election cycle, the top 500 towns on the list are required to reserve the leadership positions for women (though, be warned -- official rules aren’t always perfectly followed). PROGRAMEVAL asks you whether you can use Uttar Pradesh as a test case. Describe, in math and words, the research design you would use to leverage this new information. Be sure to include a regression equation. Under what conditions would this approach estimate the causal effect of female leadership on public goods provision? For whom is this causal effect identified? ### + +Yes, Uttar Pradesh can be used as a test case. I would employ a **Regression Discontinuity Design RD** to leverage this new information. The benefit of an RD design is that it allows us to mimic random assignment into female-led governments by comparing the towns that just made it into the top 500 towns, based on the lowest share of women in the town's population, with towns that just barely didn't make it to the cut-off list of top 500. This is "as good as random" because the cutoff point of top 500 towns is an arbitrary policy cut-off point. + +Since we are warned that _"offical rules aren't always perfectly followed"_, this means there is **imperfect compliance** across the cut-off point. That is, not all towns that made it to the top 500 list reserve leadership positions for women and viceversa. Therefore, our design will be a **Fuzzy RD**, that focuses on the **probability** of having leadership positions reserved for women. Mathematically, this can be expressed as: + +$$Pr(D_i = 1 |X_i \geq c) - Pr(D_i = 1 | X_i < c) = k$$ +Where: + +* $D_i$ is treatment status, $Di = 1$ for leadership positions reserved for women and $D_i = 0$ for no leadership position reserved for women +* $X_i$ is the running variable, which is the list rank +* $i$ is each individual town +* $c$ is the cut-off point, ie. rank 500 on the list +* $k$ is the difference in probabilities, that can take any value between 0 and 1. If there was perfect compliance: k = 1 + +This means that the probability of being treated (ie. leadership positions reserved for women) is higher if the town is above the cutoff point (ie. in the top 500 list with the lowest share of women), and likewise, the probability of leadership positions being reserved for women is lower if the town did not make it to the top 500 list. + +**(1) Regression Equations ** + +Since treatment status $D_i$ (ie. leadership seats reserved for women) does not change by 100% at the cutoff, crossing the cutoff essentially changes the _probability of treatment_. To account for this incomplete change in treatment status, and in order to estimate treatment effects $\tau$ for a Fuzzy RD, we instrument on being below the cutoff point $\longrightarrow Z_i = 1[X_i \leq c]$. As with any **instrumental variable approach**, we can estimate treatment effects $\tau$ with a **2 Stage Least Squares (2SLS approach)**, which requires the following two regressions: + +1. **First Stage -** This captures the effect of moving from above the threshold $X_i > c$ to below the threshold $X_i \leq c$ on treatment status $D_i$ (ie. leadership seats reserved for women). + +The First Stage regression allows for quantifying non-compliance and can be expressed as: + +$$ D_i = \alpha + \gamma 1[X_i \leq c] + v_i$$ +For: $$\begin{split} c-h \leq X_i \leq c + h \end{split}$$ Which is the bandwidth at which we're close to the cutoff point c. This is because we purposefully compare units around the cutoff which we assume to be statistically similar. + +Where: + +* $Y_i$ is the outcome, ie. number of public goods provided +* $\gamma$ estimates the change in probability of leadership seats reserved for women from crossing the threshold c +* $1[X_i \leq c]$ is a dummy variable that equals 1 if the town is below the threshold (ie. list rank 500 and below) +* $X_i$ is the running variable, ie. list rank +* $c$ is the cut-off point, position 500 on the list + + +2. **Reduced Form -** Which refers to the effect of going from above the threshold $X_i > c$ to below the threshold $X_i \leq c$ on the number of public goods provided (outcome $Y$). _Note: I reversed the specification from the slides because in this scenario, treatment happens BELOW the cutoff point instead of above_. The reduced form regression would be: + +$$ +\begin{split} +Y_i = \alpha + \theta 1[X_i c$ for at least some towns. In other words, we need that crossing the threshold c actually changes treatment status for some towns. + +$$ E[D_i \mid X_i \leq c] \neq E[D_i \mid X_i > c]$$ +Here, our instrument is an indicator function for being below the threshold $Z_i = 1[X_i \leq c]$ that takes the value of 1 if below the threshold and 0 otherwise. + +NOTE _I use below the threshold instead of above because in this example, towns get seats of leadership reserved for women if their share of women is in rank 500 or BELOW_ + +**Independence:** +The potential outcomes $Y_i$ and potential treatment status $D_i$ need to be independent of treatment assignment +(ie. being below the threshold $1[X_i \leq c]$). + +$$ Y_i(D_i,1[X_i \leq c]), D_i(X_i \leq c), D_i(X_i > c) \perp 1[X_i \leq c]$$ +**Exclusion Restriction:** +The only way in which crossing the threshold affects the number of public goods provided $Y_i$ is through its effect on treatment status ie. leadership seats reserved for women $D_i$. That is to say, the instrument is _as good as randomly assigned_. + +\begin{equation} +Z_i \rightarrow D_i \rightarrow Y_i +\end{equation} + +And is not directly related to public goods provision $Y_i$: +\begin{equation} +Z_i \not \rightarrow Y_i +\end{equation} + +This occurs when the instrument $Z_i = 1[X_i \leq c]$ is uncorrelated with the error term $\varepsilon_i$. + +$$Cov(Z_i,\varepsilon_i) = 0$$ +Since the policy cutoff point at 500 is arbitrary, we can expect this assumption to hold. + +**Monotonicity:** +Towns below the threshold have higher treatment status than towns that are above the threshold for all the towns. There cannot be some towns that are less likely to have seats reserved for women as they move _below the cutoff point_. + +$$ \mid D_i(X_i \leq c) - D_i(X_i > c) \mid \geq 0$$ +That is, moving across the cutoff point need not move _every single town's_ treatment status (ie. having leadership seats reserved for women), but _all affected towns_ need to move in the **same direction** (whether positive or negative). + +**(3) For whom the causal effect is identified** + +Provided that the 5 identifying assumptions hold, the treatment effect recovered is a **Local Average Treatment Effect (LATE)** specifically for _complier_ towns right below and right above the cutoff that do not move across the threshold. +$$ \tau^{FRD} = \tau^{ATE}$$ + +This $\tau^{LATE}$ can be different from the ATE or LATE at other cutoff points (for example if the cutoff was designated at the top 1000 list of towns with the lowest share of women). + + +### III.F: PROGRAMEVAL likes this idea, and is willing to share data with you to try this out. Use the dataset contained in final_exam_2022.csv. What empirical tests would you like to perform, prior to attempting to estimate the effect of female leadership on public goods provision, to provide evidence in support of the identifying assumption(s)? Perform at least two tests (hint: these should be simple graphical exercises). What do they tell you about the validity of the identifying assumption(s) in this case?### + +```{r data, include=FALSE} +#Load data +data <- read_csv("data_2022.csv") +``` + +Before attempting to employ a **Fuzzy RD design**, we need to run empirical tests of the identifying assumptions, in graphical form. + +1. **Density of the running variable** +Towns could manipulate whether they receive treatment. For example, in a town where the patriarchy dominates, politicians may lobby for their town not to have leadership positions reserved for women, even if the town is in the top 500 list with the lowest share of women in the population. Likewise, more progressive towns might manipulate the system to get leadership positions reserved for women, even if they didn't make it to the top 500 list. With this graphical test, we can visualize if there is smooth density across the cutoff point. If it is smooth, then the cutoff is essentially random. + +```{r} +# Plot density of running variable, ie. share of women +hist(data$list_rank, main = "Density of Running Variable + ie. List Rank", breaks = 15, freq = FALSE, + xlab = "List Rank", col = "purple") +abline(v = 500, lwd=5.0, col = "red") +``` +From the graph, we can see that the density of the running variable (ie. the list rank) is uniform across the sample. In particular, it is exactly smooth below and above the cutoff point of 500. This provides evidence in favor of the threshold point being _essentially random_. + +2. **Outcome across the running variable** +This test allows us to see the potential for an RD design because it will show whether there is a discontinuous jump in public goods provision between female-led and male-led towns. If there is no jump, then there is no potential for this design. + +```{r} +# Bin observations by share of women, this is the running variable +# that determines if a town gets reserved seats for women or not +data$bin = floor((data$list_rank - 500/10)) +bin_data = data %>% group_by(bin) %>% summarise(mean_outcome = +mean(public_goods_number), med_rank = median(list_rank)) + +# Plot +plot(bin_data$med_rank, bin_data$mean_outcome, main="Public Goods + Provision by Share of Women", xlab="List Rank (Running Variable)", + ylab = "Number of public goods") +abline(v=500, lwd=5.0, col = "red") +``` +At a first glance, there does not appear to be a very dramatic point to the left of the cutoff. However, if we focus on the observations to the left of the cutoff point, we can see that the lower rank in the list (ie. the lower the share of women in a town's population), the _modestly_ higher the number of public goods provided. As we move to the left, there seems to _some_ increase in public goods provision, even if not very pronounced. This supports employing an RD design for this case study.s + +3. **Continuity in Covariates** +If we observe continuity in the observables $X_i$ across the threshold, then we can attribute any discontinuity in public goods provision $Y$ between female-led and male-led towns solely to the treatment (leadership seats reserved for women). This acts as a _proxy test_ for the RD identifying assumption, since we cannot observe nor plot potential outcomes of treated towns in the absence of treatment. + +```{r} +# Use bins +covar = data %>% group_by(bin) %>% summarise(mean_income = mean(per_capita_income_rupees), + mean_incorp = mean(incorp_year), + mean_manuf = mean(manufacturing_product_share), + mean_res = mean(number_of_residents), + med_rank = median(list_rank)) + +# (1) Plot per capita income +plot(covar$med_rank, covar$mean_income, main = "Per capita income in rupees", + xlab = "List Rank (Running Variable)", + ylab = "") +abline(v = 500, lwd=5.0, col = "red") +``` +* The ***per capita income (in rupees)** looks pretty smooth just below and just above the cutoff. + +```{r} +# (2) Plot incorporation year +plot(covar$med_rank, covar$mean_incorp, main = "Incorporation Year", + xlab = "List Rank (Running Variable)", + ylab = "") +abline(v = 500, lwd=5.0, col = "red") +``` +* The **incorporation year** appears to be _mostly smooth_ just below and just above the threshold. Overall, there does not appear to be a very marked difference in continuity in this covariate. + +```{r} +# (3) Plot share of gross city product devoted to manufacturing +plot(covar$med_rank, covar$mean_manuf, main = "Share of town product + devoted to manufacturing", + xlab = "List Rank (Running Variable)", + ylab = "") +abline(v = 500, lwd=5.0, col = "red") +``` +* As to the **share of town product devoted to manufacturing**, the difference in continuity immediately to the left vs. immediately to the right of the threshold, is _not very smooth_. Just below the threshold there is a lower share of town product devoted to manufacturing, vs. the opposite above the threshold. + +```{r} +# (4) Plot number of town residents +plot(covar$med_rank, covar$mean_res, main = "Number of residents", + xlab = "List Rank (Running Variable)", + ylab = "") +abline(v = 500, lwd=5.0, col = "red") +``` +* In terms of the **number of residents**, this covariate appears to be _smooth_ just below and just above the threshold. + +With the exception of share of town product devoted to manufacturing, all other covariates are smooth right below and above the cutoff point, providing evidence in support of the identifying assumption, based on the analyzed covariates. While we cannot definitively test for the RD identifying assumption, these graphs act as a proxy test, suggesting that among the available covariates in the dataset, towns just below the cutoff point are very similar to towns just above the cutoff point. + +**Conclusion:** + +In all, the performed graphical tests suggest that the RD identifying assumption is valid for the state of Uttar Pradesh in India, since 1) the density of the running variable (ie. list rank) is completely smooth around the cutoff point, 2) there is a _relative_ visible change in outcome, where we see higher observations in the number of public goods provision as we move to the left of the threshold, and 3) out of 4 of the available covariates, 3 are smooth just below and just above the cutoff point. + +As to the IV identifying assumptions, the **independence and exclusion restriction** give us **covariate smoothness**, which we checked for in these graphs. Even though we have not explicitly tested for the IV assumptions (ie. the exclusion restriction is fundamentally untestable), these visual graphs so far suggest the identifying assumptions are valid and that as part of a natural experiment, the cutoff point is essentially random. Therefore, we can expect the exclusion restriction to hold. Overall, we can feel more comfortable about the identifying assumptions holding and proceed to implement a Fuzzy RD design. + +### III.G: Plot the relationship between a town’s position on the list and its likelihood of having a female leader. Describe what you’re plotting, using a definition from the course. Plot the relationship between the probability of having a female leader and public goods provision. Describe what you’re plotting, using a definition from the course. Informed by these plots, write down your preferred regression equation(s) for estimating the causal effect of female leadership on public goods provision. Defend your choice of bandwidth and any functional form choices you make.### + +In the data we are given, we are provided with the variable _reservation_, which is a dummy that takes a value of 1 for all treated towns (ie. all those with rank 500 and below in the list). + +We are also provided with the variable _female_leader_, which I will assume represents the probability of having a female leader. In other words, this represents our **instrument**, because the likelihood of having a female leader is contingent upon moving below the threshold: $\rightarrow Z_i = 1[X_i \leq c]$ + +As we can observe from the data, some towns in the top 500 list have a 0 likelihood of having a female leader, which reflects the non-compliance in the sample. + +1. **(1) Plot relationship between a town's position on the list and its likelihood of having a female leader** + +```{r} +# Subset data by a bandwidth h = 80, where lower bound = 420 and upper bound = +# 580 +filtered_data <- data %>% filter(list_rank %in% 420:580) +# Plot list rank against likelihood of female leader +ggplot(filtered_data, aes(x=female_leader, y=list_rank)) + +geom_point(shape=18, color="green") + geom_smooth(method=lm) + +ggtitle("Relationship between a town's position on the list and + its likelihood of having a female leader at +/- 80 from the cutoff") + + theme(plot.title = element_text (hjust = 0.5)) + xlab("Likelihood of female leader (instrument)") + + ylab("List rank") +``` +Focusing around +/- 80 bandwidth from the cutoff point of 500, this plot represents our **First Stage**, because it indicates the effect of the instrument (ie. moving from above to below the threshold $X_i > c$ to $X_i \leq c$) ON our treatment status $D_i$, which is partially determined by the town's rank in the list. + +The line of best fit, comes from a regression of our instrument (likelihood of female leader) on treatment status. We can observe a downward sloping relationship between list rank and the likelihood of a female leader, where, from list rank 500 and above, the likelihood of female leaders is essentially 0. Conversely, from list rank 500 and below, we see a higher probability of having female leaders. + +```{r} +female = nrow(subset(data, female_leader==1)) +male = nrow(subset(data, female_leader== 0)) +total = female + male +cat("The overall probability of having a female leader is", female/total) + +cat("\nThe overall probability of having a male leader is", male/total) +``` + + +1. **(2) Plot relationship between the probability of having a female leader and public goods provision** +```{r} +ggplot(filtered_data, aes(x=female_leader, y=public_goods_number)) + +geom_point(shape=18, color="orange") + geom_smooth(method=lm) + +ggtitle("Relationship between the likelihood of having a female leader + and the provision of public goods at +/- 80 from the cutoff") + + theme(plot.title = element_text (hjust = 0.5)) + xlab("Likelihood of female leader (instrument)") + + ylab("Number of public goods") + +``` +This second graph reflects the **Reduced Form**, since we are plotting our instrument (ie. probability of having a female leader by moving from above to below the threshold $X_i > c$ to $X_i \leq c$) against our outcome Y, which is the number of public goods provided. + +From the graph, we can observe a very clear positive relationship between $Z$ and $Y$, such that when there is a high likelihood of having a female leader, the number of public goods provided is higher, and conversely, when there is no probability of having a female leader, there is a low amount of public goods provision. This graph appears to support the notion that having more women in positions of political power can improve welfare, as measured in terms of public goods provision. + + +**Preferred regression equations for estimating the causal effect of female leadership on public goods provision** + +Based on these graphs, my preferred equations for estimating the causal effect of female leadership on public goods provision is essentially to estimate the First Stage and the Reduced Form in order to get the ratio between both and recover $\tau^{FRD}= \tau^{ATE}$. + +1. As to the **First Stage**, I would run the following regression equation: + +$$ D_i = \alpha + \gamma 1[X_i \leq c] + v_i$$ +Where: + +* $Y_i$ is the outcome, ie. number of public goods provided +* $\gamma$ estimates the change in probability of leadership seats reserved for women from crossing the threshold c +* $1[X_i \leq c]$ is the instrument of moving below the threshold, in this example the likelihood of a female leader +* $X_i$ is the running variable, ie. list rank +* $c$ is the cut-off point, 500 + +Limited to the following bandwidth: +$$\begin{split} 500-80 \leq X_i \leq 500 + 80 \end{split}$$ +$$\begin{split} 420 \leq \text {list rank} \leq 580 \end{split}$$ +There is a bias-variance trade-off with the choice of bandwidth, $h$. Ideally we want a small bandwidth because it can yield a more unbiased estimate (since we are limiting the analysis to towns very close to the cutoff point, which are good counterfactuals for each other). Essentially, the closest to the cutoff point, the more statistically similar are towns just above to towns just below the threshold. + +However, the closer we get to the threshold, the fewer observations we have to compare with, which increases the standard errors, decreasing precision. For this reason, I chose a bandwidth of 80 because it is close enough to the cutoff point to observe effects, without being too close to risk decreasing precision too much. + +2. Secondly, I would run the following **Reduced Form**, regression: + +$$ +\begin{split} +Y_i = \alpha + \theta 1[X_i \leq c] + \varepsilon _i +\end{split} +$$ +Where: + +* $\theta$ measures the effect of crossing the threshold c on the number of public goods provided +* The other variables are the same as above + +Again, this would be estimated for the following bandwidth: +$$\begin{split} 420 \leq \text {list rank} \leq 580 \end{split}$$ + +3. With out $\gamma$ and $\theta$ estimates, we can proceed to recover our _Fuzzy RD estimator_ by dividing the Reduced Form by the First Stage: + +$$\hat\tau^{FRD} = \frac{\hat{\theta}} {\hat{\gamma}}$$ + +### III.H: Finally, estimate the causal effect of female leadership on public goods provision. What do you find? Interpret your results. Advise PROGRAMEVAL: should they expand female leadership to all towns?### + +1. **First Stage Regression and Interpretation** +```{r} +# Run First Stage +first_stage <- lm(reservation ~ female_leader, data = filtered_data) +gamma <- first_stage$coefficients[[2]] +cat("The effect of crossing the threshold on treatment status (gamma) is",gamma) +summary(first_stage) +``` +Our First Stage estimate, $\hat{\gamma}$ shows that the estimated difference between the likelihood of having a female leader (which is determined by crossing from above $X_i > c$ to below the cutoff point $X_i \leq c$), and having a male leader, ON the likelihood of having leadership positions reserved for women) is _approximately 45%_.This coefficient is highly statistically significant at all levels of conventional significance. The sign of the coefficient is positive, as expected. That is the likelihood of female leaders increases if the town is in the top 500 list, where seats are often reserved for women (not always). + +Also, the F-statistic 2495 > 20, shows the instrument (likelihood of female leader) satisfies the first stage assumption. + + +2. **Reduced Form Regression and Interpretation** +```{r} +# Run Reduced Form +reduced_form <- lm(public_goods_number ~ female_leader, data = filtered_data) +theta <- reduced_form$coefficients[[2]] +cat("\n The effect of crossing the threshold on public goods provision (theta) + is", theta) +summary(reduced_form) +``` +This regression provides a * ${\hat{\theta}}$ estimate, which captures the effect of crossing the threshold (ie. likelihood of female leader) on the outcome (ie. public goods provision). The estimated difference between the likelihood of having a female leader (which is determined by crossing from above $X_i > c$ to below the cutoff point $X_i \leq c$), and having a male leader, ON public goods provision is approximately _8 additional public goods being provided_. + +This coefficient is highly statistically significant at all levels of conventional significance. Moreover, given that the mean provision of public goods is 4.12, this is a very, very meaningful coefficient. + +```{r} +mean_goods = mean(data$public_goods_number) +cat("The mean number of public goods provided is", mean_goods) +``` + +the instrument (randomized electricity price variation) on the outcome (groundwater consumption). As described in question (4), this is the **Reduced Form**. + +Every additional USD dollar in electricity price variation, is associated with a - 147.50 acre-feet decrease in +groundwater consumption. This ${\hat{\theta}}$ estimate is highly statistically significant at all conventional levels of significance. The sign of the estimate is logical by economic theory, since we can expect an increase in electricity prices to reduce consumption. + +2. **Recover Fuzzy RD LATE estimate** + +```{r} +# Fuzzy RD LATE +tau_late= theta/gamma +cat("\n The LATE effect of female leadership on public goods provision for + complier around the cutoff is estimated to be ", tau_late) +``` + +After estimating the First Stage and the Reduced Form, we obtain that: + + +$$\hat\tau^{FRD} = \frac{\hat{\theta}} {\hat{\gamma}}$$ +$$\hat\tau^{FRD} = \frac{\hat{8.0856}} {\hat{0.45094 }}$$ +$$\hat\tau^{FRD} = 17.93 = \tau^{LATE}$$ +Therefore, our estimated causal impact of female leadership on public goods provision for _complier towns_ around the cutoff with a bandwidth of +/- 80 is estimated to be$\tau^{FRD}$ is 17.93, which is _quite a sizable effect_. It should be highlighted that this LATE is specifically for complier towns and should the threshold change OR should the bandwidth be adjusted, we can expect this effect to change. + + +4. **Advise PROGRAMEVAL** +Based on the results obtained from the Fuzzy RD design, there is evidence to support that having female-headed government is correlated with an increase in public goods provision, and therefore in social welfare. However, I would emphasize that the finding from this analysis is _specific and particular_ to 1) complier towns around the cutoff, (that is to say that towns that didn't make it to the top 500 list did not reserve leadership positions for women and viceversa), and 2) towns within +/- 80 rank positions around the cutoff. + +Therefore, I cannot advise PROGRAMEVAL to expand female leadership to ALL towns, because that would require estimating the **Average Treatment Effect (ATE)**, which we cannot do with this research design. If that is PROGRAMEVAL's interest, we would have to employ a different method such as an RCT. However, what I can advise PROGRAMEVAL is to increase the cutoff point to at least 580 instead of 500 to benefit more towns that can stand to gain from female-led governments. To determine the ideal cutoff point would require adjusting the bandwidth in coordination with the state of Uttar Pradesh. + +## BONUS## + +### Find an example of a popular press article describing a study using causal language, when, given what you’ve learned in this quarter, this is likely not appropriate. Use a few sentences to describe the study and the main problem of the study through the lens of this course. Attach the article in PDF form to your exam submission.### + +There is a popular press article I found on several news media about how a new study "shows" that _"drinking coffee may be linked to a lower risk of death, even with a little sugar."_ + +The study, which was published in the Annals of Internal Medicine, focused on a sample of 120.000 people in the UK who have regularly been drinking either sweetened or unsweetened coffee over the past 7 years. The findings suggested that "those who drank 1.5 to 3.5 cups a day had a lower risk of death during those seven years than non-coffee drinkers, even if they added a teaspoon of real sugar — not artificial sweetener — to every cup." + +Simply comparing coffee drinkers to non-coffee drinkers is the **naive estimator**, which as we have seen is very problematic because coffee drinkers are systematically different from non-coffee drinkers in many characteristics, both observable and unobservable. It could be the case that coffee drinkers tend to partake more in physical exercise, which is very correlated with improved health outcomes. It could also be the case that people who drink 1.5 to 3.5 cups of coffee every single day for 7 years, are more wealthy on average and therefore have better access to medicine and treatment. There are multiple causes of death and it is highly unlikely that coffee consumption can directly be linked to a decreased probability of dying, solely based on a naive estimator and without a proper research design that addresses bias. + +**Source:** https://www.insider.com/drinking-coffee-daily-may-stave-off-early-death-study-suggests-2022-5 + diff --git a/ps-5/ps5.pdf b/ps-5/ps5.pdf new file mode 100644 index 0000000000000000000000000000000000000000..db6d5d40c43f92197cafb073e8e0d5006221b11e GIT binary patch literal 2657539 zcma&tW0Yi3wl?alv~AnAZQHhO+jgYwthBSzcBO4p+SaY^bGlETaqs=c_a|b;+OgKV zX8ed4dq1(s6h+19nCMxc$%c;Zj?T;P3+9GKpjinS2_1~Bp?P@;8N{t@UCo^TKHD0( znu(g3IGCCdGRT_QTew;hvNN+W5%TjxySO@=8QDR5{;18_h$kI&3cr4*$_k|S4%{+c zsYCz<4I#G5zJn8s?}v%PC?@NW+1Ml_`%Mnl8%`j^K-%Btj~;LiNK9J$-1=I1zkZVV;%V9X^1XcO{dWFd`+EG!cz!o=;lpgv99ci+j?ZOL=l3|VapBT`D)+aP zAJ;sI#T)T>f7-fQb@Ongj9XvnX1yLviEJL+@__@0J;eM`MA9?=xg%XSKQ 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/dev/null +++ b/ps-3/PS3.Rmd @@ -0,0 +1,423 @@ +--- +title: "PPHA 34600 Program Evaluation" +author: "Problem Set 3" +date: "5/12/2022" +output: + pdf_document: default + html_document: + df_print: paged +--- + +```{r setup, include=FALSE} +# Clean the environment, set up the working directory: +rm(list=ls()) +setwd("~/Desktop/Spring 2022/Program Evaluation/PS3") + +# tell R not to use scientific notation +options(scipen = 999) + +# Load libraries +library(tidyverse) +library(ggplot2) +library(kableExtra) +library(broom) +library(dplyr) +library(AER) +``` + +### (1) CALBEARS are interested in answering the following question: What is the effect of average groundwater costs between April and September (measured in dollars per acre-foot) on total groundwater consumption (measured in acre-feet) during the same time period? To make sure everybody is on the same page, explain to them what the ideal experiment would be for answering this question. Describe the dataset that you’d like to have to carry out this ideal experiment, and use math, words, and the potential outcomes framework to explain what you would estimate and how you would do so. Make sure to be clear about the unit of analysis (ie, what is “i” here?).### + +In the ideal experiment, to find the effect of _average groundwater costs_ on total groundwater consumption, we would conduct a Randomized Control Trial where we would randomize a shock in groundwater costs among farmers, for example by varying electricity prices or distributing efficient water pumps among farmers. + +Here, treatment would be the shock/change in groundwater costs and the control group would be composed of farmers whose groundwater extraction costs remained unchanged. + +Through a well-conducted randomization that results in balanced baseline characteristics, farmers in the treatment group would be statistically similar to farmers in the control group, allowing us to recover the Average Treatment Effect (ATE) of groundwater costs between April and September on consumption during the same period: + +$$\tau^{ATE} = E[Y_{i}(1)] - E[Y_{i}(0)]$$ +Where: + +* $Y$ is the outcome variable of interest, groundwater consumption (measured in acre-feet) +* $i$ is each individual farmer + +Under randomization, the **expected** groundwater consumption for farmers who are subject to a shock in groundwater costs is equivalent to the **mean** groundwater consumption for those same farmers, and viceversa for the control group. This property enables us to estimate the ATE by simply taking the difference in means between both groups: + +$$\tau^{ATE} =\overline{Y_{i}}(D = 1) - \overline{Y_{i}}(D = 0)$$ + +Where: + +* $D$ refers to treatment status: $D = 1$ for farmers subject to a shock in consumption prices and $D = 0$ for farmers who are not +* $Y$ groundwater consumption (measured in acre-feet) +* $i$ is each individual farmer + +The ideal dataset would contain **1)** baseline and endline farmer groundwater consumption (measured in acre feet), **2)** groundwater extraction costs and its cost determinants including pump efficiency, electricity prices and aquifer depth, and, **3)** other baseline characteristics needed to check for balance in the randomization process to ensure treated and control farmers are statistically similar. We would need this data at the farmer level, covering the time period between April and September. For the baseline characteristics, we would need the data to be collected before the experiment starts, for example in March. + +### (2) CALBEARS are on board with your explanation, but, as they’ve discussed with you, they won’t be able to implement your preferred solution. They don’t think that a selection-on-observables approach will work (they’re very sophisticated). They’re also limited by state privacy laws: they will only be able to give you one wave of data (no repeated observations). Given these limitations, describe the type of research design you would try to use to answer their question of interest. Be explicit about the assumptions required for this design to work, describing them in both math and words. ### + +In the ideal RCT experiment, we can recover $\tau^{ATE}$ with a simple difference in means or refression because the key assumption $E[\varepsilon \mid D_i] = 0$ holds. + +However, in the absence of randomization, $E[\varepsilon \mid D_i] \neq 0$, which means that the error term, conditional on treatment is non-zero. Therefore, we have selection bias and our $\tau^{ATE}$ would be biased. Other reasons why $E[\varepsilon \mid D_i] \neq 0$, include omitted variable bias and/or reverse causality. + +As a solution to this problem, and given that CALBEARS provides me with _cross-sectional data_, I would propose implementing an Instrumental Variable approach, part of the Selection on Unobservables (SOU) toolkit. + +This design requires finding an appropriate instrument, which is essentially a source of "good" variation in the treatment status (ie. change groundwater costs) and is _quasi random_. Conceptually, we need to isolate exogenous variation by separating selection into groundwater costs $D$ in two parts: + +$$D_i = B_i\varepsilon_i + C_i$$ + +Where: + +* $D$ is the treatment status, $D = 1$ for farmers subject to a shock in consumption prices and $D = 0$ for farmers with unchanged costs +* $B_i\varepsilon_i$ refers to the self-selection into groundwater costs (treatment) that is correlated with the individual farmer's choice or decision-making +* $C_i$ is an arbitrary component of selection into groundwater costs (treatment) +* $i$ is each individual farmer + +Hypothetically, if we could observer $C_i$, we could regress groundwater consumption $Y_i$ on it to recover $\tau^{ATE}$. In reality, we cannot statistically distinguish $C_i$ from $B_i\varepsilon_i$. Instead, we can carefully select an instrumental variable $Z_i$ that varies the $C_i$ component and is uncorrelated with the error term $\varepsilon_i$. + +**Assumptions required for an IV approach** + +**1. First Stage:** +The instrument $Z_i$ needs to be correlated with treatment status $D_i$, (ie. groundwater costs) which means their covariance is non-zero: + +$$Cov(Z_i,D_i) \neq 0$$ +$\longrightarrow$ This first assumption is **testable**. + +**2. Exclusion Restriction:** +The instrument is _as good as randomly assigned_ because if it only affects groundwater consumption $Y_i$ through treatment, ie. groundwater costs $D_i$. + +\begin{equation} +Z_i \rightarrow D_i \rightarrow Y_i +\end{equation} + +And is not directly related to groundwater consumption $Y_i$: +\begin{equation} +Z_i \not \rightarrow Y_i +\end{equation} + +This occurs when the instrument $Z_i$ is uncorrelated with the error term $\varepsilon_i$. + +$$Cov(Z_i,\varepsilon_i) = 0$$ +$\longrightarrow$ This second assumption is fundamentally **untestable**. + + +### (3) CALBEARS are interested in this research design. It sounds promising. They’d like you to propose a specific approach. Please describe a plausible instrumental variable you could use to evaluate the effect of the cost of groundwater pumping on acre-feet of groundwater consumption. Why is your proposed instrument a good one? Do you have any concerns about your ability to estimate the treatment effect using your instrument? If yes, why? If no, why not?### + +Groundwater extraction costs is a function of 1) electricity prices, 2) water pump efficiency and 3) aquifer depth. Potentially, farmers can be invited to participate in a lottery of efficient groundwater pumps distribution as part of a giveaway state program. + +Since the lottery to distribute efficient pumps would be random, the lottery itself could be a plausible instrument to evaluate the effect of groundwater costs on groundwater consumption. For this instrument to be valid, it needs to satisfy both identifying assumptions. + +**1) Assessing the First Stage Assumption** +The giveaway lottery $Z_i$ would be correlated with treatment $D_i$, where treatment would be _lower groundwater costs as a function of efficient water pump usage_, because only farmers that participate in the lottery would have access to a free efficient pump, which would lower groundwater extraction costs. + +**2) Assessing the Exclusion Restriction Assumption** +If the lottery is in fact random, this assumption is likely to hold. We can expect the lottery to affect groundwater assumption **only** through the distribution of efficient water pump. There is no reason to believe that the lottery itself would be correlated with groundwater consumption. + +I wouldn't be concerned with my ability to estimate $\tau^{ATE}$ because both requires assumptions are likely to hold for this plausible instrument, making it _as good as randomly assigned_ to recover an unbiased estimate of $\hat{\tau}^{IV}$. As long as the lottery is random and transparent, there shouldn't be ethical concerns of why some farmers receive a free efficient water pump over others. The general public may believe that only farmers with lower economic means should be considered for a giveaway, but this depends on how the program is pitched and implemented. + + +### (4) CALBEARS is intrigued by your approach. After an internal discussion, they’ve come back to you with great news! It turns out that two of the California utilities ran a small pilot program where they randomly varied electricity prices to different farms as part of a new policy proposal. With this new information, please describe to CALBEARS how you would estimate the impacts of electricity prices on groundwater consumption, and how you would estimate the impacts of groundwater costs on groundwater consumption. Use both words and math.### + + +Given this information, the random variation in electricity prices could be used as an instrumental variable to measure the effect of groundwater costs on groundwater consumption. + +We can do this by employing two approaches **A) Two Stage Least Squares (2SLS)** or **B) Reduced Form**. + +**A) 2SLS APPROACH:** This method consists of the following two steps. + +1. **First Stage** +We run a regression to estimate the effect of our instrument $Z_i$ (ie. randomized electricity price variation) on treatment $D_i$ (ie. groundwater cost variation). + +$$D_i = \alpha + \gamma Z_i + \beta X_i + \eta_i$$ +Where: + +* $D$ is the treatment status $D = 1$ groundwater cost variation and $D = 0$ if no variation +* $Z_i$ is the instrument, randomized electricity price variation +* $X_i$ refers to baseline covariates +* $i$ is each individual farmer + +After running the regression, we have to store the predicted values of $D_i$ and $\hat D_i$. +Next, we check if the F-stat > 20 to confirm whether the **First Stage Assumption**, ie. $Cov(Z_i,D_i) \neq 0$. + +2. **Second Stage** +We regress the effect of the "good variation" in treatment $\hat D_i$ (ie. the change in groundwater costs that is random, akin to $C_i$), on our outcome, $Y_i$ (ie. groundwater consumption). + +$$Y_i = \alpha+ \tau \hat D_i + \delta X_i + \varepsilon_i$$ +Where: + +* $\tau$ is the instrumental variable estimate +* $Y_i$ is the outcome variable, groundwater consumption +* $\hat D_i$ is treatment, stripped of selection bias +* $Z_i$ is the instrument, ie. randomized electricity price variation +* $X_i$ refers to baseline covariates +* $i$ is each individual farmer + +By the exclusion restriction assumption, the instrument is _as good as randomly assigned_. + +With these two assumption in place, 2SLS lets us recover an unbiased estimate of $\hat{\tau}^{IV}$. + +$\longrightarrow$ When running the 2SLS method in a statistical software, the standard errors need to be adjusted because the software does not know this regression uses predictions from a previous regression. + +**A) REDUCED FORM APPROACH:** + +Unlike the 2SLS approach, which gives us $\tau^{IV}$, the Reduced Form indicates how groundwater consumption $Y_i$ varies with randomized electricity variation, our instrument $Z_i$. + +The Reduced Form can be mathematically expressed as: + +$$Y_i = \alpha + \theta Z_i + \pi X_i + \eta_i$$ +Where: + +* $\theta$ is the effect of the instrument (randomized electricity price variation) on the outcome (groundwater consumption) +* $Y_i$ is the outcome variable, groundwater consumption +* $Z_i$ is the instrument, ie. randomized electricity price variation +* $X_i$ refers to baseline covariates +* $i$ is each individual farmer + +The Reduced Form (RF) should have a causal interpretation and can be very informative. + + +### (5) CALBEARS agree that your approach is a good one. So good, in fact, that they’d like to see it in action! They are willing to share some data with you, in the form of ps3_data.csv. Please report the results of an analysis of the impact of electricity prices on groundwater costs, using electricity_price_pilot as the price variable and groundwater_cost as the cost variable. What parameter does this regression estimate? Interpret your estimate. Will this utility pilot be a helpful way forward to estimating the impacts of groundwater costs on groundwater usage? Why or why not?### + +```{r data, include=FALSE} +#Load data +data <- read_csv("ps3_data.csv") +``` + +```{r} +# Regress electricity price variation on groundwater costs +reg <- lm(groundwater_cost ~ electricity_price_pilot, data = data) +summary(reg) +``` +This regression provides an estimate of ${\hat{\gamma}}$, or the effect of randomized electricity price variation (ie. our instrument) on groundwater costs, our treatment. In other words, this is a **First Stage** regression. + +Every additional USD dollar in electricity price variation, is associated with a 0.74 dollar increase per acre-foot in groundwater cost. This estimate is highly statistically significant at all conventional levels of significance, since the p-value is very small. + +Given that every additional dollar increase in electricity price is estimated to increase groundwater costs by 74 cents, this is a considerable amount and reflects that groundwater costs is in fact a function of electricity prices. + +The F-statistic is 292.7 > 20, so the instrument passes the test first stage assumption that the instrument (ie. randomized electricity priced variation) is correlated with treatment (ie. groundwater costs). As to the exclusion restriction, since we are told the variation in electricity prices was random, we can also assume it holds. + +Therefore, utility pilot is a valid instrument and is in fact helpful in estimating the impacts of groundwater costs on groundwater usage. + + +### (6) CALBEARS wants you to use the pilot in your analysis (they are ignoring any opinion you gave in (5), good or bad). Please report the results of an analysis of the impact of electricity prices on groundwater consumption, using electricity_price_pilot as the price variable and groundwater_use as the usage variable. What parameter does this regression estimate? Interpret your estimate. Is this estimate useful for policy? Why or why not?### + +```{r} +# Regress electricity prices on groundwater costs +reg2 <- lm(groundwater_use ~ electricity_price_pilot, data = data) +summary(reg2) +``` +This regression provides a * ${\hat{\theta}}$ estimate, which captures the effect of the instrument (randomized electricity price variation) on the outcome (groundwater consumption). As described in question (4), this is the **Reduced Form**. + +Every additional USD dollar in electricity price variation, is associated with a - 147.50 acre-feet decrease in +groundwater consumption. This ${\hat{\theta}}$ estimate is highly statistically significant at all conventional levels of significance. The sign of the estimate is logical by economic theory, since we can expect an increase in electricity prices to reduce consumption. + +On the contrary, in a context where the utilities that ran the pilots do not hold an electricity monopoly of in the state, we would expect a substitution effect where farmers switch to cheaper electricity providers. However, assuming the utilities are the main providers, then the negative sign of the Reduced Form estimate is coherent. + +This estimate is useful for public policy. Even though economic theory suggests a negative correlation between electricity prices and groundwater consumption, this estimate provides statistical evidence in support of this claim. This could be helpful in lobbying for electricity-varying policies to reduce groundwater consumption in anticipation of the expected drought. + + + +### (7) CALBEARS would like you to use their pilot to estimate the effect of groundwater costs on groundwater consumption. For full transparency, make sure to show all of your analysis steps. CALBEARS cares about your standard errors, so using a canned routine is a good idea here. Interpret your effect. Do groundwater costs matter for consumption? ### + +First we obtain the **First Stage**: +```{r first_stage} +# First stage regression +first_stage <- lm(groundwater_cost ~ electricity_price_pilot , data = data) +predicted_y <- fitted(first_stage) +gamma <- first_stage$coefficients[[2]] +cat("The effect of instrument on treatment is",gamma) +summary(first_stage) +``` + +This is the same regression as the one estimated in part (5). The first stage ${\hat{\gamma}}$ estimate shows that every additional USD dollar in electricity price variation (our instrument) is associated with a 0.74 dollar increase per acre-foot in groundwater cost (our treatment). The F-statistic 292.7 > 20, shows the instrument satisfies the first stage assumption. The exclusion restriction likely holds because the variation in electricity prices was random. + +Next, we calculate the **second stage**: +```{r second_stage} +# Second Stage regression +second_stage <- lm(groundwater_use ~ predicted_y, data = data) +tau_iv <- second_stage$coefficients[[2]] +cat("The effect of treatment on outcome is", tau_iv) +summary(second_stage) +``` +In the second stage, we obtain our $\hat {\tau}^{IV}$ estimate of the effect of groundwater costs (treatment) on groundwater consumption (outcome variable $Y_i$). + +This means, on average, that each additional dollar per acre-foot in groundwater costs is associated with a decrease of -199.26 dollars per acre-foot on groundwater consumption. This coefficient is highly statistically significant at all conventional levels of significance. + +To note, the standard errors in this second stage estimate are off because R is not aware that this regression is built from the results of another regression (the first stage). + +To correct for this and validate our estimated results, we can use R's AER package: + +```{r} +two_sls <- ivreg(groundwater_use ~ groundwater_cost | electricity_price_pilot, + data = data) +summary(two_sls) +``` +As expected, we obtain the same $\hat {\tau}^{IV}$ estimate as before, ie. -199.26. However, the standard errors decreased from 21.72 to 18.03, showing this estimate is more _precise_. Since the $\hat {\tau}^{IV}$ is highly statistically significant and is of a considerable magnitude, groundwater costs _do_ matter for consumption as an increase in groundwater costs is correlated with a corresponding decrease in groundwater consumption. + +### (8) CALBEARS like your analysis, but they’re a bit worried about the quality of their data on groundwater consumption. The way they normally collect these data is by surveying the farmers. However, they went and did some back-checks in a subsample of data that they gave you, and noticed that the farmer reports seem to be off. They would like you to make a graph showing the relationship between their back-checks (groundwater_use_backchecks) and the farmers estimates (groundwater_use). Describe to them what you find. Is this likely to be a problem for your analysis? Why or why not? Next, estimate the impacts of groundwater costs on groundwater consumption using the backcheck data instead of the farmer estimates. Report what you find. Do your estimates differ? If no, explain why not. If yes, explain why.### + +```{r} +ggplot(data, aes(x=groundwater_use, y=groundwater_use_backchecks)) + +geom_point(shape=18, color="orange") + geom_smooth(method=lm) + +ggtitle("Relationship between back checks and farmer estimates + of groundwater consumption") + theme(plot.title = element_text + (hjust = 0.5)) + xlab("groundwater use") + + ylab("groundwater use backchecks") +``` +From the graph, we can observe that the groundwater back-check variation is very visibly concentrated on lower self-reported groundwater use. In other words, the variation between backchecks and self-reported groundwater use is _not random_, if it were, we wouldn't see the points concentrated in one area as we observe here. It appears that there is systematic discrepancy among farmers that reported lower groundwater use. + +This is a case of measurement error in the outcome variable, $Y$ (groundwater consumption). This is not a problem if and only if two conditions are satisfied: + +**1)** The covariance between the measurement error $\gamma_i$ and, **2)** the error term $\varepsilon_i$ is 0 AND the covariance between the measurement error and treatment is also 0. +$$Cov(\gamma_i, \varepsilon_i)= 0 \text{ and } Cov(\gamma_i, D_i) = 0$$ +When these two conditions hold, even though we don't observe $Y_i$, we do observe : +$$\tilde {Y_i} = Y_i + \gamma_i$$ +Then, the slope doesn't change and the relationship between $X_i$ (groundwater costs) and $Y_i$(groundwater consumption) remains the same because the movement in groundwater consumption is _random_. + +Based on the scatterplot, condition 1) does not hold because the measurement error is not random and it likely does covary with the error term. For example, it may be the case that reporting groundwater use means filing more taxes so there may be an incentive to lie or under-report groundwater consumption. + +As such, this will be a problem for my analysis because the proposed IV design cannot correct for this issue. Alternatively, another instrument variable for the measurement error could be used to account for this problem. + +Now, we estimate the effect of groundwater costs on groundwater consumption using the back-check data: +```{r} +backcheck <- ivreg(groundwater_use_backchecks ~ groundwater_cost | + electricity_price_pilot, data = data) +summary(backcheck) +``` +We obtain a new $\hat {\tau}^{IV}$ of -25.00. On average, each additional dollar per acre-foot in groundwater costs is correlated with a decrease of -25.00 dollars per acre-foot on groundwater consumption. Like before, this estimate is also highly statistically significant. + +In comparison to the IV model in part (7), we find a much smaller estimate (from -199.26 to -25 dollars per acre-foot). This is indicative that the measurement error in reported groundwater consumption is in fact problematic because it was leading us to considerably overestimate the effect of groundwater costs on groundwater consumption. + +```{r} +# Number of groundwater consumption back-checks conducted +sum(!is.na(data$groundwater_use_backchecks)) +``` +Nonetheless, it should be noted that the groundwater checks were done only on a subset of the sample, 1389 out of 4000 farmers, so the magnitude of the estimate may not be entirely accurate. + + +### (9) The challenge with back-checks is that they’re very expensive to do. Fortunately, CALBEARS realized that they have another dataset on groundwater consumption which seems to match the back-checks much better. They’d like you to make a graph showing the relationship between their back-checks and this new measurement (groundwater_use_v2). Describe to them what you find. Is this likely to be a problem for your analysis? Why or why not? Next, estimate the impacts of groundwater costs on groundwater consumption using the backcheck data and using the new estimates. Report what you find. Do your estimates differ? If no, explain why not. If yes, explain why.### + +```{r} +ggplot(data, aes(x=groundwater_use_v2, y=groundwater_use_backchecks)) + +geom_point(shape=18, color="red") + geom_smooth(method=lm) + +ggtitle("Relationship between back checks and new groundwater +consumption measurement") + theme(plot.title = element_text + (hjust = 0.5)) + xlab("new groundwater use variable") + + ylab("groundwater use backchecks") +``` +From the graph, we can see that the new measurement for groundwater use is extremely close to the back-checks data. +This provides evidence in favor that the back_checks data, even if performed on a subset of the sample, is very close to the new measurement variable which is provided for the 4000 observations. + +This is unlikely a problem for my analysis. On the contrary, this contributes to the analysis by providing a groundwater consumption variable that is more or less clean of non-random measurement error and it can help to validate the $\hat {\tau}^{IV}$ estimated with the back-checks. + +Next, we run the IV model with the new variable: + +```{r} +groundwater_use2 <- ivreg(groundwater_use_v2 ~ groundwater_cost | + electricity_price_pilot, data = data) +summary(groundwater_use2) +``` +As a result, the estimated newly $\hat {\tau}^{IV}$ is -25.01, marginally different from the -25.00 obtained in part (8). Since, from the scatterplot, the back_checks data is extremely close to the new groundwater consumption measurement, it makes sense that the estimates do not change. This validates that the back-checks subset data is representative of the sample as a whole and that these last two $\hat {\tau}^{IV}$ estimates are likely more reliable. + + + +### (10) CALBEARS comes back to you again with yet another data problem. This time, they’re worried that the utilities aren’t reporting electricity prices very well. They’d like you to focus on the effect of electricity price on groundwater consumption (you can ignore groundwater costs for the remainder of the problem set). CALBEARS explain to you that, in one utility (labeled iou == 1 in the data, because #privacy), something was going wrong with the price information they were using. Farms facing low prices had these prices recorded correctly in the data, but the higher the price, the more inflated utility 1’s record is. In the other utility (labeled iou == 2), there are still imperfect measurements, but CALBEARS is convinced that the measurement problems are random. Explain the implications of these data issues in each utility to CALBEARS. Are these measurement issues going to be a problem for your analysis? Use words and math to explain why or why not. Despite any misgivings you might have, run your analysis anyway, separately for each utility this time (using your preferred groundwater consumption variable from the three described above), and report your findings.### + +Given the new focus, our treatment $D_i$ is now electricity price and the outcome $Y_i$ is still groundwater consumption. + +Measurement error in the treatment, as is described in this question, is problematic. There are two types of this measurement error: + +**1. Classical:** This occurs when the measurement error is _random_, in this case, when iou == 2. Here, the estimated relationship between groundwater consumption and electricity prices gets closer to 0 and the slope becomes flatter. + +We are interested in estimating: +$$Y_i = \alpha + \tau D_i + \varepsilon_i$$ +Where: + +* $Y_i$ is the outcome variable, groundwater consumption +* $i$ is the individual farmer + +Because of the classical measurement error, we don't observe $D_i$, but rather $D_i$ plus the measurement error: +$$\tilde D_i= D_i + \gamma_i$$ +Since the measurement error is random, we assume that groundwater consumption and electricity are uncorrelated. Then the covariance term disappears, as follows: +$$ Var(D_i + \gamma_i) = Var(D_i) + Var(\gamma_i) + 2Cov(D_i, \gamma_i)$$ +$$ Var(D_i + \gamma_i) = Var(D_i) + Var(\gamma_i)$$ +As a result, our estimate becomes: +$$ \hat {\tau}^{IV} = \tau \left(\frac{Var(D_i)}{Var(D_i) + Var(\gamma_i)} \right)$$ +The component within the parenthesis is less than 1, so that it biases the $\hat {\tau}^{IV}$ estimate towards 0. This is known as **attenuation bias**, where even though we have a biased estimate, the sign on the estimate is accurate. + +If we can find a new valid instrument, $Z_i$ for the noisy $\tilde {D_i}$, then we can resolve the problematic aspect of this classical measurement error. Otherwise, we will get an attenuated and biased towards 0 $\hat {\tau}^{IV}$ estimate. + +**2. Non-Classical:** This is a very problematic scenario in which the measurement error is _NOT random_ and occurs in a systematic way. This is the case of iou == 1 because the higher the price, the more inflated the record of utility 1. + +In other words, there is correlation between higher electricity prices and measurement error. Then, the covariance term remains and the $\hat {\tau}^{IV}$ is scaled by this term, where we now get: + +$$ \hat {\tau} = \tau \left(\frac{Var(D_i) + Cov(D_i, \gamma_i)}{Var(D_i) + Var(\gamma_i) + 2Cov(D_i, \gamma_i)} \right)$$ +Once again, we obtain a _biased_ estimate except that the sign can flip and is no longer accurate. Here, there is nothing that can be done to recover an unbiased $\hat {\tau}^{IV}$ . + +**Running IV models to estimate the effect of electricity price on groundwater consumption** + +For the following regressions, I am going to use the groundwater_use_v2 variable because it is as accurate as the back_checks measurement with the added bonus that it's provided for every observation in the sample. + +**A) Classical measurement error, utility 2** + +```{r} +utility2 <- filter(data, iou == 2) +classical <- lm(groundwater_use_v2 ~ electricity_price_pilot, data = utility2) +summary(classical) +``` +For the classical measurement error, we obtain that every additional dollar in the electricity price is associated with a -233.42 decrease in acre-feet groundwater consumption. Even though this coefficient is highly statistically significant, this is misleading as the estimated $\hat {\tau}^{IV}$ is _biased_ towards 0, which suggests that the actual $\tau$ has a higher magnitude. + +However, the negative sign of the estimate is reliable, and tells us there is negative correlation between electricity prices and groundwater consumption, as expected by economic theory. + + +**B) Non-Classical measurement error, utility 1** + +```{r} +utility1 <- filter(data, iou == 1) +non_classical <- lm(groundwater_use_v2 ~ electricity_price_pilot, + data = utility1) +summary(non_classical) +``` + +In the non-classical measurement error, we find that every additional dollar in the electricity price is associated with a -32.03 decrease in acre-feet groundwater consumption. The coefficient is highly statistically significant but that is irrelevant because the estimated $\hat {\tau}^{IV}$ is biased. In fact, the size of the coefficient is much, much smaller than with the classical measurement error (-32.03 vs. -233.42). Since we know that the estimate with classical measurement is biased towards 0, we can infer that the real $\tau$ has a bigger size than 233.42, which further suggests how biased this non-classical estimate is. + + +### (11) CALBEARS conducted a survey of farmers to understand their experience with the pricing pilot, and asked the farms to report their electricity prices (survey_price). Describe how you could use these data to correct any issues you reported in (10). What conditions need to be satisfied in order or this to work? Are these conditions satisfied in utility 1, utility 2, both, or neither? Carry out your proposed analysis in the sample where it will work (utility 1, utility 2, both, or neither). Report your results, and describe how they compare to your estimates in (10), or explain why you didn’t produce any. Which estimates would you send to CALBEARS as your final results?### + +In light of the new variable, we can use an Instrumental Variable approach to solve for the classical attenuation bias we obtained for utility 2. Essentially, we would need a new instrument $Z_i$ for the noisy measure of electricity prices we had before. + +Mathematically, we would now estimate: +$$\mathring{D_i} = D_i + \zeta_i$$ +Where: + +* $\mathring{D_i}$ is the measurement error in the price survey +* $D_i$ is electricity prices +* $\zeta_i$ is the error +* $i$ is the individual farmer + +This approach works as long as the new noisy instrument of electricity prices (ie. price survey) is uncorrelated with the previous noisy measure of treatment (ie. random classical measurement in utility #2's records): +$$\tilde{D_i} \neq \mathring{D_i}$$ +Specifically, three assumptions are required: + +1. The measurement error $\zeta_i$ needs to be uncorrelated with electricity prices +$$Cov(\zeta_i, D_i) = 0 $$ +2. The measurement error $\zeta_i$ in the price survey instrument $Z_i$ must be uncorrelated with the error in $\tilde{D_i}$ (ie. random classical measurement in utility #2's records of electricity prices) +$$Cov(\zeta_i, \gamma_i) = 0 $$ +3. The measurement error $\zeta_i$ has to be uncorrelated with the original error +$$Cov(\zeta_i, \varepsilon_i) = 0 $$ +If these assumptions are met, then $\hat {\tau}^{IV} = \tau$ and our estimate will be _unbiased_. + +Since the measurement error is random for **utility #2** only, these conditions are satisfied because the price survey instrument $Z_i$ and the noisy measure of electricity prices $\tilde{D_i}$ only have the true electricity prices $D_i$ in common. Then, the first stage is: + +$$\tilde{D_i} = \alpha + \pi \mathring{D_i} + \epsilon_i$$ + +This also works in this case because the the treatment, (ie. electricity prices) is a continuous variable. If it were binary, then the measurement error could only be -1 or 0 for $D_i = 1$ and 0 or 1 for $D_i = 0$. + +Now, we run the IV analysis to correct for classical measurement error in utility #2: +```{r} +classical_unbiased <- ivreg(groundwater_use_v2 ~ electricity_price_pilot | + survey_price, data = utility2) +summary(classical_unbiased) +``` +Here, we find that every additional dollar in the electricity price is associated with a -460.1 decrease in acre-feet groundwater consumption. The coefficient is highly statistically significant at all conventional levels and the negative sign is consistent with economic theory. + +As expected, the magnitude of the estimate is bigger than the obtained for the classical measurement in part (10), which was biased towards zero. This unbiased estimate, -460.1, is practically double the size of the previous biased estimate of -233.42. 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zGt1hWTX>S_v5>H_@fyH0t66)0(Q9nH@XT5ydaNYuB&=V-Zw?NwUqP;~BfkA-$9IRX z)qi-&|EwfP^aR)?**PU7C3(3x#l<+d#JPDmIYfC_c)58vcv!hOczA_K{_m8pdlW3} zEnTfhIJvq0n{WGn0B)2E6Ap2z7{facmv_yQ8db*!P2N3~nrTIgmGUQmnj&mBnb9Qn+w#LQ>NSVTk$iex2)SQ#Ah`&Na#YQM+N?#5|dhxn}3E%S5*@27!Z zP+@HJaTdYrc-=6s+TT@r8?fC640*sH9Z{i8V3BVIrD&2dd!o62hdk7QN1NcMenH1h z=IE(nxVGWWD-e35D3}ePJkKIvWdYu4q14h7i@S{pKlWdk50*(ww;6b(hQb`DHni*l zT-^f5zvg>pP9?e6u__m#N~`yGD=Xjz4W;GvCCn9iEh?Du-2xX8M60(?&$iezAPIi8 z`>c3syqs4HBSFyY3J1wsR>DC+>PF-bf^hR5Ei*3yrCuCFdBAToaw#+qAw=nI8iEj2 zYYQPzZaCb@ScbMzzbp^H(pb)W3BPlmBiLyFX6}xKyz6J|Mx!G#Q-AgSp}*~9p8!KI zKs%J)QP%s75o8cv^+(9Q8ES%Ik7;?EPyX_#?0nCF$ff99e_gg-I+{8V%m*s7){7Ja zyvsTG1Zw^yKShZ;S4lY+!v`)F0?m!=^nQA*!UNpIt?ZOyNj4N+kGe%^rVF300A{on zor1boDKOX52Px4Y;e1Ipe|!&SiGE5$YO$cnmA_J|6sB2~lUC3hO{39y@u}#C>Luvt zmu{NRzkpUnoz% + filter(air_quality_regulation_year == 2004) + +data_2004$post <- ifelse(data_2004$year >= 2004, 1, 0) +reg2 <- lm(particulate_matter ~ post, data = data_2004) +summary(reg2) +``` +Next, for the time-series estimator, we find that _overtime_, the estimated difference between a municipality that has air-quality regulations in place and a municipality that does not, is approximately -21.47 micrograms per cubic meter of PM 2.5. This ATE estimate is also highly statistically significant at all conventional levels of significance and like before, retains the negative sign on the coefficient. + +Compared to the previous naive estimate, the size of the estimated ATE is somewhat smaller (-21.47 vs. the -24.45 previously obtained). In other words, there is not much of a difference and this may be due to the problems previously described in part (2) and part (3) of selection bias and inexistence of time-varying characteristics. + +**3.** We now plot particulate matter against time for municipalities that imposed air quality restrictions in 2004. +```{r} +ggplot(data_2004, aes(x=year, y=particulate_matter)) + +geom_point(shape=18, color="red") + geom_smooth(method=lm) + + scale_x_continuous(breaks = seq(2003, 2012, 1)) + +ggtitle("Particulate matter against time for municipalities that imposed + air quality restrictions in 2004") + theme(plot.title = element_text + (hjust = 0.5)) + xlab("year") + + ylab("particulate matter") +``` +What we can see is that in the given years, for municipalities that imposed air quality regulations in 2004, there is an upwards trend particulate matter over time. In other words, there is a positive trend in pollution over time. It looks like right after 2004, the levels of particulate matter are much lower than the levels in particulate matter before 2004. The levels of particulate matter decrease even more in 2005. However, this effect is dissipated, as moving forward from 2005, there is an overall increase in particulate matter, some years lower some years higher, but overall much higher than in 2004 and 2005. + +```{r} +grouped <- data_2004 %>% + group_by(year) %>% + summarise(particulate_matter = mean(particulate_matter)) + +ggplot(grouped, aes(x=year, y=particulate_matter)) + +geom_point(shape=18, color="red") + geom_smooth(method=lm) + + scale_x_continuous(breaks = seq(2003, 2012, 1)) + +ggtitle("Average Particulate matter against time for municipalities that imposed + air quality restrictions in 2004") + theme(plot.title = element_text + (hjust = 0.5)) + xlab("time") + + ylab("particulate matter") + + geom_smooth(formula = y ~ x, method = "lm") + +``` +If instead, we plot the mean particulaate matter consumption across municipalities, we can make similar conclusions as the previous graph. In 2004 there is a sharp decrease in mean PM 2.5 consumption that decreases further in 2005. However, after 2005, there is an upwards trend. In some years there is much higher consumption like 2006, 2007 and 2009, whereas other years like 2008 and 2010 have a comparatively lower PM 2.5 but not as low as in 2004-2005 right when and right after the air-quality regulations were imposed. Only looking at these graphs, we might believe the regulations were effective right after being imposed with effects dissipating overtime. + + +### (7) Plot (average) particulate matter against time for municipalities who never imposed air quality regulations. Assess the viability of using these municipalities as a control group for the 2004 regulators. Plot (average) particulate matter against time for municipalities who passed air quality regulation in 2006. Assess the viability of using the non-regulating municipalities as a control group for the 2006 regulators.### + +**1** Particulate matter against time for municipalities who never imposed air quality regulations +```{r} +# Filter of municipalities that never imposed regulations +no_regulation <- data %>% + filter(treat_municipalities == 0) + +# Plot +ggplot(no_regulation, aes(x=year, y=particulate_matter)) + +geom_point(shape=18, color="green") + geom_smooth(method=lm) + + scale_x_continuous(breaks = seq(2003, 2012, 1)) + +ggtitle("Particulate matter against time for municipalities that NEVER imposed + air quality restrictions") + theme(plot.title = element_text + (hjust = 0.5)) + xlab("time") + + ylab("particulate matter") +``` +```{r} +grouped <- no_regulation %>% + group_by(year) %>% + summarise(particulate_matter = mean(particulate_matter)) + +ggplot(grouped, aes(x=year, y=particulate_matter)) + +geom_point(shape=18, color="green") + geom_smooth(method=lm) + + scale_x_continuous(breaks = seq(2003, 2012, 1)) + +ggtitle("Average Particulate matter against time for municipalities that NEVER imposed + air quality restrictions") + theme(plot.title = element_text + (hjust = 0.5)) + xlab("time") + + ylab("particulate matter") + + geom_smooth(formula = y ~ x, method = "lm") +``` + + +Now, when we look at particulate matter consumption for municipalities that never imposed air quality restrictions, we find that the trend is actually pretty similar as that for for municipalities that impose air quality restrictions. The decline in 2004 and 2005 is also present for unregulated municipalities and the overall positive trend overtime, particularly after 2005 remains the same. This provides evidence in favor that there are common parallel trends between unregulated and regulated municipalities and motivates the use of a DD estimator as its identifying assumption is satisfied. + +We might notice that the levels of the trends are different. For example, the range of the scale for unregulated municipalities is lower than the range in scale for regulated municipalities. Nonetheless, they are trending the same, which is the key DD identifying assumption to retrieve an unbiased ATE estimate. Therefore, using municipalities that never imposed air-quality regulations is a good control group for the 2004 regulators. + +**2** Plot average particulate matter against time for municipalities who passed air quality regulation in 2006 + +```{r} +# Create filter of 2006 treated municipalities +data_2006 <- data %>% + filter(air_quality_regulation_year == 2006) %>% + mutate(post = ifelse(year >= 2006, 1, 0)) + +grouped <- data_2006 %>% + group_by(year) %>% + summarise(particulate_matter = mean(particulate_matter)) + + +ggplot(grouped, aes(x=year, y=particulate_matter)) + +geom_point(shape=18, color="orange") + geom_smooth(method=lm) + + scale_x_continuous(breaks = seq(2003, 2012, 1)) + +ggtitle("Average Particulate matter against time for municipalities that imposed + air quality restrictions in 2006") + theme(plot.title = element_text + (hjust = 0.5)) + xlab("time") + + ylab("particulate matter") + + geom_smooth(formula = y ~ x, method = "lm", color="purple") +``` +Looking at this plot, we can observe a _very different_ trend than the previous two graphs. The slope of the trend is much steeper compared to the previous graphs where it was relatively more flat, albeit positive trending. This means that using non-regulators as a control group for municipalities that imposed air-quality regulations in 2006 violates the common parallel trends assumption and would not be valid. + +### (8) Using just the non-regulators and the 2006 regulators, estimate the causal impact of imposing air quality regulation on particulate matter. To do this, begin with a simple difference in means (rather than regression). Next, use a simple regression (no fixed effects). Finally, use fixed effects to control for common time shocks and time-invariant municipality characteristics (you can do this either via dummy variables or de-meaning). Report what you find. Be sure to adjust your standard errors appropriately in the regression-based estimates. Describe how this compares to what you estimated in (6) and (7).### + +**1.** Simple difference in means +```{r} + +dd<-data %>% + filter(air_quality_regulation_year==2006|is.na(air_quality_regulation_year)) +dd[is.na(dd)]<-0 +dd<-dd %>% + mutate(treated=ifelse(air_quality_regulation_year==2006,1,0)) +dd<-dd %>% + mutate(post=ifelse(year>=2006,1,0)) +dd<-dd %>% + mutate(treat_post=treated*post) + +naive_estimator<-mean(dd$particulate_matter[dd$air_quality_regulation_year== +2006])-mean(dd$particulate_matter[dd$air_quality_regulation_year==0]) +naive_estimator +``` +If we only do a difference in means berween regulated and non-regulated municipalities in 2006, we obtain a difference of 124.27 micrograms per cubic meter of PM 2.5. This is can be argued to be large in size but also, it has a positive sign, which is concerning. It suggests air quality control regulations is correlated with an increase in particulate matter. This is the naive estimator and the problems with selection bias previously identified hold, which is why it is unreliable + +**2.** Simple regression with no FE +```{r} +reg3<-felm(particulate_matter ~ treated + post + treat_post|0|0| + municipality_id, + data = dd) +summary(reg3) +``` +Here we find, that the estimated ATE for municipalities after they were regulated, compared to unregulated municipalities, is is 67.47 micrograms per cubic meter of PM 2.5. This coefficient is statistically significant at all conventional levels. It has decreased in magnitude from the naive estimator but it also has a positive sign, which is counter intuitive. + +**3.** +```{r} +reg4<-plm(particulate_matter~ treated + post + treated*post,data =dd, + model = "within", + effect = "twoway", + index = c("municipality_id", "year")) +summary(reg4) +``` +Here, with the DD estimator, we find that the estimated ATE is is 67.47 micrograms per cubic meter of PM 2.5. This coefficient is statistically significant at all conventional levels and is exactly the same as the one obtained previously. The positive sign remains, suggesting there is a positive relation between air-quality regulations and particulate matter, which is contrary to what we would expect. While it is likely to imagine that air-quality regulations may have no effect on pollution (for example if they are not enforced), it seems strange to image a scenario where they might actually lead to an increase in pollution, rather than a decrease. This may be due to some noise in the data that is inducing bias in our estimates. + +### (9) Plot average particulate matter over time, separately (but on the same graph) for municipalities that imposed air quality regulations in each of the years from 2003 to 2007. Drop the municipalities that regulated air quality in the year that looks different from the rest of the years, and explain why you can’t estimate a credible causal effect for these municipalities. Using the remaining municipalities to estimate a panel fixed effects regression to identify the causal effect of air quality regulation on particulate matter. Describe what you find. How does this compare to what you estimated in (8)? Use an event study regression to estimate how this treatment effect varies over time. Note that you will have to omit one of the event study treatment dummies (otherwise everything will be collinear). Standard practice is to leave out the T-1 dummy. Plot the resulting event study point estimates and 95 percent confidence intervals. Describe how the treatment effect varies over time, if at all. ### + +**1.** Plot average particulate matter over time for municipalities that imposed air-quality regulations +```{r} + +regulators <- data %>% + group_by(air_quality_regulation_year, year) %>% + summarize(mean_particulate = mean(particulate_matter), + .groups = 'keep') %>% ungroup() + +regulators %>% +ggplot(aes(y = mean_particulate, x = year, color = + as.factor(air_quality_regulation_year))) + +geom_line() + scale_x_continuous(breaks = seq(2003, 2012, 1)) + +scale_y_continuous(labels = scales::comma) + + theme(axis.text=element_text(colour="black")) + +ggtitle("Average particulate matter overtime for municipalities that +imposed regulations in 2004") + theme(plot.title = element_text + (hjust = 0.5)) + xlab("time") + + ylab("mean particulate matter") +``` +From this graph, we can observe that the blue line that represents the year 2006 is trending _very differently_ from all other years. All other years follow the same trend and, even at the same level from 2007 onwards. Overall, all other years show a sharp decline up to 2005, followed by an general increase with ups and downs. However, 2006 has a very steep increase overtime that is on a different trend entirely. + +We need to omit the year 2006 from the dataset because it violated the **parallel trends** assumption we need for a DD estimator when using panel data. As explained in part (4), we cannot recover a credible unbiased ATE estimate for 2006 municipalities because, given that the identifying assumption is violated, our counterfactual is no longer valid. Essentially, after imposing regulations, we would be incorrectly _under-estimating_ an effect that cannot be attributed solely to air-quality regulations but rather that effect PLUS noise from other factors that are correlated with particulate matter levels. It would result in a **biased estimate**. + +**2.** Estimate panel fixed effects regression of air quality regulation on particulate matter +```{r} +# Remove municipalities from the year 2006, include treat and post dummy +# variables: + +data_clean <- data %>% +filter(air_quality_regulation_year != 2006 | is.na(air_quality_regulation_year)) %>% +mutate(treat = ifelse(is.na(air_quality_regulation_year), 0, 1), + post = ifelse(year >= air_quality_regulation_year & + air_quality_regulation_year!=0, 1, 0), + dd = treat*post) + +# Run fixed effects regression +reg5 <- felm (particulate_matter ~ dd|municipality_id + + year|0|municipality_id, data = data_clean) +summary(reg5) +``` +From the panel fixed effects regression, we find our DD coefficient is -14.84, which means that the predicted local particulate matter for municipalities that impose air-quality regulations, in comparison with unregulated municipalities, is predicted to be a decrease in -14.84 micrograms per cubic meter of PM 2.5. This estimate is statistically significant at all conventional levels. Contrary to the previous estimates in part 8, this time our estimate has a negative sign, suggesting a negative correlation between air quality regulations and pollution, which is more in line with what we would expect. + +In comparison to part (8), this is a more reliable estimate because we dropped municipalities that imposed regulations in 2006 from the data, which was inducing noise in our estimate by violating the parallel trends assumption. Those observations were pushing the coefficient to have a positive sign, which was an unusual result. Having gotten rid of that noise, has improved the reliability of our estimate. + + +**3.** Event study regression + +```{r} +event_study <- data_clean %>% + mutate(air_quality_regulation_year = ifelse(air_quality_regulation_year == 0, NA, air_quality_regulation_year)) %>% mutate(years_from_treatment = year - air_quality_regulation_year, +t_min_1 = year - air_quality_regulation_year) + +# Exclude periods t minus to prevent collinearity +tmin1 <- event_study%>% + filter(t_min_1 != -1) + +# Run Event Study regression with time and municipality fixed effects +reg6<- plm(particulate_matter~ as.factor(t_min_1)*post, + data = tmin1, + model = "within", + effect = "twoway", + index = c("municipality_id", "year")) +summary(reg6) + +results <- coeftest(reg6, vcov = vcovHC(reg6, type = "HC0", cluster = "group")) +results + +coef <- results[1:12] +se <- results[13:24] +es <- data.frame(t_min_1 = c(c(-3,-2), c(0:9)), coef,se, + lb = coef - 1.96 * se, ub = coef + 1.96 * se) +``` +As described in part (5), an event study allows us to _line-up_ air-quality regulations for all municipalities, so that we can compare and estimate the ATE of regulations on particulate matter, regardless of municipalities imposing regulations at different points in time. + +This design lets us compare muncipalities PRE and POST treatment (ie. imposing air-quality regulations). What we can observe from the estimated coefficients is that the effects are statistically significant for all years for which we have data, that is in range from -3 years from regulations being imposed up to 9 years after they were imposed. In particular, the effects are highly statistically significant for years 0-8. + +We can also observe that before treatment, at years -2 and -3, the coefficients were positive but in the year the regulations are imposed, on average, there is an estimated -10.97 decrease in per cubic meter of PM 2.5. This effect can be argued to be sizable, considering the change from 2.54 to -10.97 per cubic meter of PM 2.5, on average. However, the effects dissipate over time by approximately 1 full unit per year, where the magnitude of the ATE decreases all the way to -1.57 per cubic meter of PM 2.5. This result suggests air-quality regulations are effective right after they are imposed but overtime, their effect on pollution decreases substantially. + +**4.** Plot event study estimates with 95% confidence intervals +```{r} +ggplot(es, aes(x = t_min_1, y = coef)) + + geom_point(size = 3, color = "blue") + + geom_errorbar(aes(ymin = lb, ymax = ub)) + + xlab('years since air-quality regulation imposed') + + ylab('ATE on particulate matter') + + ggtitle(label = 'ATE and years since Air-Quality regulations') + + theme(plot.title = element_text + (hjust = 0.5)) +``` + +The plot shows the results previously discussed in a much more visual way. The ATE before air-quality regulations are imposed has a positive sign 2 and 3 years before these regulations are implemented. The year the regulations are implemented is where we see the most marked decline in local particulate matter. Each year after the regulation is imposed, on average, there is an increase in PM 2.5, showing the waning effects of this policy. + + +### (10) PROGRAMEVAL would like a summary of your findings. Explain which of the results you’ve come up with is your preferred estimate, and why. Make sure to describe at least one remaining potential shortcoming with these results. Finally, interpret the magnitude of your estimated effects: do your results suggest that the PROGRAMEVAL should be strongly promoting air quality regulations? ### + +From the results so far, my preferred estimate is the Event Study Design in part (9) because, after dropping municipalities that imposed regulations in 2006, which was violating the parallel trends assumption, we were able to recover a more reliable ATE that has a negative sign. + +We can reasonably expect the sign to be negative or for the coefficient to be close to 0, in which case it would mean air-control regulations have no effect on particulate matter (for example if the policy is not enforced or has such high pollution caps that it is ineffective). However, it seems difficult to come up with reasoning that would sustain a positive correlation. It would be a situation in which, there are defying companies that would counter regulations with purposefully more polluting just to spite the government, which seems highly unlikely, particularly in the political context of China. + +Unlike the panel fixed effects, the Event Design estimates are much more powerful in analyzing the effect of air-quality control regulations on PM 2.5 not only on average, but rather, overtime. The accompanying plot provides a more substantial and complete analysis that, even though the effect is negative as would be desirable, this effect dissipates over time, to the point it has barely any effect after 9 years of these regulations being implemented. This would allow PROGRAMEVAL to make more informed decisions about implementing air-quality control policies. + +Based on these results, I would recommend to PROGRAMEVAL to promote air-quality regulations and show that they should expect the biggest effects to happen near the time they are imposed. However, I would also advise they put measures in place that allow them to maintain these effects over time, like tightening air-quality regulations overtime, or increasing/maintaining enforcement so that they effects are sustained. + +One potential short-coming of these results is we have very limited data before air-quality regulations were imposed. We are comparing up to 9 years after regulations vs only 3 years before these regulations. It's possible that going back there are other considerations in the way these municipalities were trending that could be related with the effectiveness of regulations on pollution. For instance, if it just so happens that those two years happened to be years with relatively low pollution and in the last decade the levels of pollution have in reality been much higher, then we might be over stating the economic significance of these results. + diff --git a/ps-4/PS4.pdf b/ps-4/PS4.pdf new file mode 100644 index 0000000000000000000000000000000000000000..dd0ac9622c0f4cb6e02fabf4cc3697f5e5468f41 GIT binary patch literal 512312 zcma&MQ+F;*pk^D}w(aDN?G@X$R&3k0ZQHhO+qSdPU!QUAdSC3isy|Sp=6uE^R}dAa zXJ%l7A)j9!S%+aGVj{9LvV!5|g<+I2wKaFPAY$QSW+VFl4h*BXrM0uEBN3yxwV|`A zsHw4?i75;}Ka7*Jqp6_{jQi$}mW|!fDDuBs{Q|oJpr|n);tpOSsM-l~i3E|lh^74Y zV^I1942Q&X(h8?-{XKJ6Gis&NjkSFGd0*y6hF04IQwBBgQ1&Rdg{{2n$F?egAa-3UbWF?22-gy4>l-6enQRur zPhoUM=R&q5^NGUnC>e9noZ*n6iV0zQ$DVN%be7t@O z@3*5Wj=JAg6>+gT-~Einn)ecZBuV`i)iw1j!{`Wvou9?@a;vK$diaj=y~)ljv$nq$ zk9wz%=R?R|%QFWjj;2wppQhTcjrwDAh7LV_v}-II+wtM%!X>rgx@H>XU8{u84gA$t zj$x`9QInk|nB2|VD||Bo_cGc$1a9OH3(R6p`_T1Wf#p~ABf1-}FPRp!6x(Th&;3co 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