claims.ptexw

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{ pdfauthor = {Gyan Sinha},
  pdftitle={Modeling Initial Claims as Quasi-Natural Disasters},
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\title{Modeling Initial Claims as Quasi-Natural Disasters}
\author{Gyan Sinha, Godolphin Capital Management, LLC%
\thanks{\scriptsize \emph{%Godolphin Capital Management, LLC,%
\href{mailto:gsinha@godolphincapital.com}{Email Gyan}. This report
has been prepared by Godolphin Capital Management, LLC
(``Godolphin'') and is provided for informational purposes only and
does not constitute an offer to sell or a solicitation to purchase
any security. The contents of this research report are not intended
to provide investment advice and under no circumstances does this
research report represent a recommendation to buy or sell a security.
The information contained herein reflects the opinions of Godolphin.
Such opinions are based on information received by Godolphin from
independent sources. While Godolphin believes that the information
provided to it by its sources is accurate, Godolphin has not independently
verified such information and does not vouch for its accuracy. Neither
the author, nor Godolphin has undertaken any responsibility to update
any portion of this research report in response to events which may
transpire subsequent to its original publication date. As such, there
can be no guarantee that the information contained herein continues
to be accurate or timely or that Godolphin continues to hold the views
contained herein. Godolphin is an investment adviser. Investments
made by Godolphin are made in the context of various factors including
investment parameters and restrictions. As such, there may not be
a direct correlation between the views expressed in this article and
Godolphin's trading on behalf of its clients. 
<%print(f'Version:{datetime.datetime.now()}') %>}}
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\date{\today}

\begin{document}
\maketitle

\section{Introduction}

Unprecedented is a word often used in the context of the economic data
being generated in the post-COVID world, but are there historical episodes
that may be useful in serving as guide in terms of thinking about 
near-term job losses after the initial shock observed in late March. What
 could the path of initial claims be over the next few months, and what 
 should we expect in terms of the total number of claims during thi period?

In Bram \& Deitz~\cite{bramdeitz}, an interesting idea to answer this 
question is explored --- that the COVID shock to labor markets is more akin 
to a natural disaster like hurricane Katrina or Maria, rather than a classic 
recession-induced hit to employment. As they correctly point out

\begin{quote}
Recessions typically develop gradually over time, reflecting underlying economic
and financial conditions, whereas the current economic situation developed
suddenly as a consequence of a fast-moving global pandemic. A more appropriate
comparison would be to a regional economy suffering the effects of a severe natural
disaster, like Louisiana after Hurricane Katrina or Puerto Rico after Hurricane Maria. 
To illustrate this point, we track the recent path of unemployment claims in the 
United States, finding a much closer match with Louisiana after Katrina than the 
U.S. economy following the Great Recession.
\end{quote}

While the national scope of the COVID shock is clearly different and unprecedented, 
there are similarities with natural disasters which by their nature, tend to be more
localized. As such, they can provide insights into the way labor markets evolve in 
the short-run. We do this by estimating exponential decay models for different episodes, 
both natural-disaster related or recession-driven, and calibrate the parameters of 
this process to the historical data on claims.


\section{Model}
To be precise, we posit that initial claims, normalized to their peak
value during each episode, follow:

\begin{equation}
\label{eqn:ic_process}
y_{t} = A \exp(-\kappa x^{\beta})\epsilon_{t}
\end{equation}
where $x$ is the number of weeks from the peak of initial claims
during the episode and $\epsilon_{t} \sim \mathcal{LN}(0,\, \sigma_{y})$%
\footnote{Estimating the model in logarithms linearizes it and
makes the right-hand side additive.}. 
The parameter $A$ is normalized to 1, so we just have the
2 parameters $\kappa$ and $\beta$ to estimate. We go further and 
specify episode-specific parameters that drive this process, so
these become $\kappa_{j}$ and $\beta_{j}$ where $j$ is an 
episode-specific index. The smaller the $\kappa$ parameter,
the longer it takes for the iniital shock to claims to taper down.

<<claims_setup, echo=False>>=
import datetime
import joblib
import pathlib

import pandas as pd
import numpy as np
import pytoml

from fredapi import Fred

import matplotlib.pyplot as plt
import matplotlib.ticker as mtick
import seaborn as sns
sns.set()

plt.rcParams.update({
    "font.family": "Source Sans Pro",
    "font.serif": ["Source Sans Pro"], 
    "font.sans-serif": ["Source Sans Pro"],
    "font.size": 10,
})

models_dir = "/home/gsinha/admin/db/dev/Python/projects/models/"
data_dir = models_dir + "data/"

fname = data_dir + "claims.pkl"
with open(fname, "rb") as f:
    claims_dict = joblib.load(f)

START_DATE = datetime.date(1995, 1, 1)
CRISIS_START_DATE = datetime.date(2020, 3, 14)
HOME_DIR = str(pathlib.Path.home())

with open(HOME_DIR + "/.config/gcm/gcm.toml", "rb") as f:
    config = pytoml.load(f)
    FRED_API_KEY = config["api_keys"]["fred"]

claims_az_data = claims_dict["az_data"]
claims_sum_df = claims_dict["sum_df"]
claims_trace = claims_dict["trace"]
claims_data = claims_dict["data"]
claims_epi_enc = claims_dict["epi_enc"]
claims_sim_dict = claims_dict["sim_dict"]

A = 0
κ = claims_trace["κ"]
β = claims_trace["β"]

def project_claims(state, covid_wt, sum_df, epi_enc, verbose=False):
    ''' get labor market data from STL '''
    
    def states_data(suffix, state, fred):
        ''' gets data from FRED for a list of indices '''

 
        idx = "ICSA" if state == "US" else state + suffix            
        x =  pd.Series(
                fred.get_series(
                    idx, observation_start=START_DATE), name=v
            )

        x.name = state

        return x    
    
    def forecast_claims(initval, initdate, enddate, covid_wt):
        ''' project initial claims '''
    
        μ_β = sum_df.loc["β", "mean"]
        μ_κ = sum_df.loc[["κ: COVID", "κ: Katrina"], "mean"].values
        μ_decay = covid_wt * μ_κ[0] + (1 - covid_wt) * μ_κ[1]
        
        dt_range = (
            pd.date_range(start=initdate, end=enddate, freq="W") - 
            pd.tseries.offsets.Day(1)
        )
        max_x = len(dt_range)
        
        w = np.arange(max_x)
        covid_idx = list(epi_enc.classes_).index("COVID")
        katrina_idx = list(epi_enc.classes_).index("Katrina")
        
        decay = covid_wt * κ[:, covid_idx] + (1 - covid_wt) * κ[:, katrina_idx]
        μ = np.exp(-decay * np.power(w.reshape(-1, 1), β))
        
        μ_df = pd.DataFrame(
            np.percentile(μ, q=[5, 25, 50, 75, 95], axis=1).T, 
            columns=["5th", "25th", "50th", "75th", "95th"]
        ) * initval
        μ_df["period"] = w
           
        ic = np.zeros(max_x)
        ic[0] = 1
        for j in np.arange(1, max_x, 1):
            ic[j] = np.exp(-μ_decay * np.power(j, μ_β))
        
        df = pd.concat(
            [
                pd.Series(np.arange(max_x), name="period"),
                pd.Series(ic, name="ic_ratio"),
                pd.Series(ic * initval, name="ic"),
                pd.Series((ic * initval).cumsum(), name="cum_ic")
            ], axis=1
        )
       
        df.index = dt_range
        μ_df.index = dt_range
    
        return df, μ_df
    
    fred = Fred(api_key=FRED_API_KEY)
    ic_raw = states_data("ICLAIMS", state, fred)

    init_value, init_date, last_date = (
        ic_raw[ic_raw.idxmax()], ic_raw.idxmax(), ic_raw.index[-1]
    )
    end_date  = (
        last_date + pd.tseries.offsets.QuarterEnd() + pd.tseries.offsets.DateOffset(months=3)
    )
    
    if verbose:
        print(
            f'State: {state}, {init_value}, {init_date}, {end_date}, {last_date}'
        )
    
    ic_fct, ic_pct = forecast_claims(init_value, init_date, end_date, covid_wt)
    ic_fct["state"] = state
    ic_pct["state"] = state
    
    return ic_raw, ic_fct, ic_pct, init_date, end_date
@

\section{Results}
In Table~\ref{tbl:claims_decay_tbl}, we present the results of the
exponential decay model. 
The $\kappa$ parameter ranges from a low
of <%print(f'{claims_sum_df.loc["κ: GFC", "mean"]:.2f}') %> during
the Great Recession (GFC) to 
a high of <%print(f'{claims_sum_df.loc["κ: Katrina", "mean"]:.2f}') %>
in the aftermath of Katrina. The estimate for the current COVID
episode is <%print(f'{claims_sum_df.loc["κ: COVID", "mean"]:.2f}') %>,
in between the 2 but subject to considerable uncertainty given we
are still early in the process. As more data becomes available, 
the 95\% credibility intervals will shrink but we have to be
cognizant of the fact that any projections will of necessity be
subject to fairly wide error bands. Regardless, the initial
results out of the gate suggest a pattern of behavior more in
line with natural disasters rather than the longer tapering
timeframes evidenced during and after recessions%
\footnote{Based on a judgment call given some of the early
fit results, we used a weighted $\kappa$ value 
for the claims projections, using a 90\% weight on the COVID value
and a 10\% weight on the Katrina value.}.

\begin{table}[htb]
\centering
\caption{Initial claims: exponential-decay}
\label{tbl:claims_decay_tbl}
\scalebox{1}{
<<claims_decay_tbl, echo=False, results="tex">>=
print(
  claims_sum_df[["mean", "sd", "hpd_3%", "hpd_97%", "r_hat"]].to_latex(
     column_format="rrrrrr"
  )
)
@
}

\end{table}
In Figure~\ref{fig:claims_model_fits}, we present the model predictions and
the associated 1 and 2 standard deviation ranges versus the observed data
on claims (relative to the peak) for each episode. Overall, the fits match
the historical experience quite well, except for the aftermath of Hurricane
Sandy where the initial data points come down much more sharply than the
model predictions. Note the relatively wide confidence bands for both 
the current ``COVID'' episode and the history of claims after 9/11.

\begin{figure}[htb]
\centering
\caption{Model fits \emph{vs} actuals}
\label{fig:claims_model_fits}
\scalebox{1}{
<<claims_model_fits, echo=False>>=
fig, ax = plt.subplots(2, 3, figsize=(12, 6))

for i, v in enumerate(["κ: " + x for x in claims_epi_enc.classes_]):
    
    xx = claims_data[claims_data.epi_idx==i]["x"].values
    yy = claims_data[claims_data.epi_idx==i]["y"].values
    xx = xx.reshape((xx.max() + 1, 1))
    
    mu = A - κ[:, i].reshape(-1,1) * np.power(xx, β).T
    ic_hat_means = mu.mean(axis=0)
    ic_hat_se = mu.std(axis=0)
    
    j = i % 3
    l = 0 if i < 3 else 1
    ax[l, j].plot(xx, yy, 'C0.')
    ax[l, j].plot(xx, np.exp(ic_hat_means), c='k')

    ax[l, j].fill_between(
        xx[:, 0], np.exp(ic_hat_means + 1 * ic_hat_se),
        np.exp(ic_hat_means - 1 * ic_hat_se), alpha=0.6,
        color='C1'
    )
    ax[l, j].fill_between(
        xx[:, 0], np.exp(ic_hat_means + 2 * ic_hat_se),
        np.exp(ic_hat_means - 2 * ic_hat_se), alpha=0.4,
        color='C1'
    )
    ax[l, j].set_xlabel('Weeks since peak')
    ax[l, j].set_ylabel('Pct. of peak')
    ax[l, j].set_title(f'Episode: {v} = {claims_sum_df.loc[v, "mean"]}')
    
fig.tight_layout()
@
}

\end{figure}

\section{Forecasts}
In Figure~\ref{fig:cum_ic_claim}, we present the forecast for the US
over a 13-week horizon. Both weekly and cumulative claim totals
since the start of the episodic peak are presented. 

\begin{figure}[htb]
\centering
\caption{Initial claim projections: US}
\label{fig:cum_ic_claim}
\scalebox{1}{
<<cum_ic_claim, echo=False>>=
covid_wt = 0.9

ic_raw, fct_df, ic_pct, init_date, end_date = project_claims(
  "US", covid_wt, claims_sum_df, claims_epi_enc
)

fig, ax = plt.subplots(1, 2, figsize=(12, 6))

ax[0].plot(ic_pct["period"], ic_pct["50th"].cumsum())
ax[0].scatter(
  claims_data[claims_data.epi_idx==1]["x"], 
  (claims_data[claims_data.epi_idx==1]["y"] * ic_pct.iloc[0, 1]).cumsum()
)
ax[0].fill_between(
    ic_pct["period"], (ic_pct["25th"]).cumsum(), (ic_pct["75th"]).cumsum(), 
    alpha=0.6, color='C1'
)
ax[0].fill_between(
    ic_pct["period"], ic_pct["5th"].cumsum(), ic_pct["95th"].cumsum(), alpha=0.4,
    color='C1'
)
ax[0].yaxis.set_major_formatter(mtick.StrMethodFormatter("{x:,.0f}"))
ax[0].set(xlabel='Weeks from peak', ylabel='Cum. initial claims')

ax[1].plot(ic_pct["period"], ic_pct["50th"])
ax[1].scatter(
  claims_data[claims_data.epi_idx==1]["x"], 
  (claims_data[claims_data.epi_idx==1]["y"] * ic_pct.iloc[0, 1])
)
ax[1].fill_between(
    ic_pct["period"], (ic_pct["25th"]), (ic_pct["75th"]), alpha=0.6, color='C1'
)
ax[1].fill_between(
    ic_pct["period"], ic_pct["5th"], ic_pct["95th"], alpha=0.4,
    color='C1'
)
ax[1].yaxis.set_major_formatter(mtick.StrMethodFormatter("{x:,.0f}"))
ax[1].set(xlabel='Weeks from peak', ylabel='Initial claims')

plt.tight_layout()
@
}
\end{figure}

\textbf{Since the peak on <%print(f'{init_date.strftime("%B %-d, %Y")}') %>, 
the model generates a median cumulative claims estimate of 
<%print(f'{1.e-6*ic_pct["50th"].cumsum()[-1]:,.2f}')%>
million, with a 95\% confidence band of %
<%print(f'{1.e-6*ic_pct["5th"].cumsum()[-1]:.2f}') %> million and %
<%print(f'{1.e-6*ic_pct["95th"].cumsum()[-1]:.2f}') %> million claims. }
The middle half of the projections span the range from
<%print(f'{1.e-6*ic_pct["25th"].cumsum()[-1]:.2f}') %> million to %
<%print(f'{1.e-6*ic_pct["75th"].cumsum()[-1]:.2f}') %> million claims. 
While the 95\% confidence bands for cumulative claims are relatively wide
(over the entire 13 week period), they simply reflect the uncertainty inherent in
the fact that we are still early in the process. Given the current total of roughly
26 million claims already incurred, the median estimate calls for roughly double
the current amount over the weeks remaining in the second quarter.

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\bibliography{corona}

\end{document}