9-arima_a.qmd

---
title: "ETC3550/ETC5550 Applied forecasting"
author: "Ch9. ARIMA models"
institute: "OTexts.org/fpp3/"
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---

```{r setup, include=FALSE}
source("setup.R")
library(patchwork)
library(purrr)
```

## ARIMA models

\begin{tabular}{@{}rl}
\textbf{AR}: & autoregressive (lagged observations as inputs)\\
\textbf{I}: & integrated (differencing to make series stationary)\\
\textbf{MA}: & moving average (lagged errors as inputs)
\end{tabular}

###
An ARIMA model is rarely interpretable in terms of visible data structures like trend and seasonality. But it can capture a huge range of time series patterns.

## Stationarity

\vspace*{0.2cm}
\begin{block}{Definition}
If $\{y_t\}$ is a stationary time series, then for all $s$, the distribution of $(y_t,\dots,y_{t+s})$ does not depend on $t$.
\end{block}\pause\vspace*{-0.3cm}

Transformations help to **stabilize the variance**.\newline
For ARIMA modelling, we also need to **stabilize the mean**.

### Differencing

* Differencing helps to **stabilize the mean**.
* First differencing: *change* between consecutive observations: $y'_t = y_t - y_{t-1}$.
* Seasonal differencing: *change* between years: $y'_t = y_t - y_{t-m}$.


## Automatic differencing
\vspace*{0.2cm}

### Using unit root tests for first differencing

  1. Augmented Dickey Fuller test: null hypothesis is that the data are non-stationary and non-seasonal.
  2. Kwiatkowski-Phillips-Schmidt-Shin (KPSS) test: null hypothesis is that the data are stationary and non-seasonal.

### Seasonal strength

STL decomposition: $y_t = T_t+S_t+R_t$

Seasonal strength $F_s = \max\big(0, 1-\frac{\text{Var}(R_t)}{\text{Var}(S_t+R_t)}\big)$

If $F_s > 0.64$, do one seasonal difference.


## Random walk model

If differenced series is white noise with zero mean:

\begin{block}{}
\centerline{$y_t-y_{t-1}=\varepsilon_t$ \hspace{0.4cm} or \hspace{0.4cm} $y_t=y_{t-1}+\varepsilon_t$}
\end{block}\vspace*{-0.3cm}
where $\varepsilon_t \sim NID(0,\sigma^2)$.

* Model behind the \alert{naïve method}.
* Forecast are equal to the last observation (future movements up or down are equally likely).

\vspace*{10cm}

## Random walk with drift model

If differenced series is white noise with non-zero mean:

\begin{block}{}
\centerline{$y_t-y_{t-1}=c+\varepsilon_t$ \hspace{0.4cm} or \hspace{0.4cm} $y_t=c+y_{t-1}+\varepsilon_t$}
\end{block}\vspace*{-0.3cm}
where $\varepsilon_t \sim NID(0,\sigma^2)$.

* $c$ is the \alert{average change} between consecutive observations.
* Model behind the \alert{drift method}.

\vspace*{10cm}