Chapter 3 How to deal with a non-stationary process?

The RW is a simple nonstationary process.\footnote{Non-stationary data: time-series data with mean value that can either rise or fall over time.}
 

The accumulation of random shocks, \(\epsilon_t\), creates a stochastic trend.

This causes the non-stationarity.

To solve this problem, a solution is to take the first difference: \(\Delta y_t=y_t - y_{t-1}\).
The first difference of a random walk process would be the white noise: \[\begin{align*} \Delta y_t&=y_t - y_{t-1}=\epsilon_t \end{align*}\]

Assume the data follow an AR(1) process: \[\begin{align*} y_t&=\phi y_{t-1}+\epsilon_t \end{align*}\] Focus the coefficient \(\phi\). Is it equal to 1 (unit root) or smaller than 1?

We can use the Dickey-Fuller (DF) test to check for stationarity and answer this question.

3.1 Dickey-Fuller test

The Dickey-Fuller test tests the null hypothesis of \(\phi\) being equal to 1, against the alternative hypothesis of \(\phi\) being lower than 1. Using a compact notation, DF tests

The DF test is similar to the \(t\)-test \[\begin{align*} DF&=\frac{\hat{\phi}-1}{SE(\hat{\phi})} \end{align*}\] where \(\hat{\phi}\) is the parameter \(\phi\) of AR(1) estimated using OLS.
The \(t\)-test follows a Student t-distribution, while the DF test follows a .