Dickey-Fuller test

We consider the stochastic process of form

where |φ| ≤ 1 and εi is white noise. If |φ| = 1, we have what we call a unit root. In particular, if φ = 1, we have a random walk (without drift), which is not stationary. In fact, if |φ| = 1, the process is not stationary, whereas if |φ| < 1, the process is stationary. We will not consider the case where |φ| > 1 more since in this case the process is said to be explosive and increases over time.

This process is a first-order autoregressive process, AR(1), which we study in more detail in Autoregressive Processes. We will also see why such processes without a unit root are stationary and why the term "root" is used.

The Dickey-Fuller test is a way to determine whether the above process has a unit root. The approach used is quite simple. First calculate the first difference, i.e.

that's to say

If we use the delta operator, defined by Δyi = yi – yi-1 and define β = φ – 1, then the equation becomes the equation of regression linear

where β ≤ 0 and therefore the test for φ is transformed into a test according to which the slope parameter β = 0. Thus, we have a one-sided test (since β cannot be positive) where

H0: β = 0 (equivalent to φ = 1)

H1: β < 0 (equivalent to φ < 1)

Under the alternative hypothesis, if b is the ordinary least squares (OLS) estimate of β, and therefore φ-bar = 1 + b is the ordinary least squares (OLS) estimate of φ, then for sufficiently large n

or

We can use the usual linear regression approach, except that when the null hypothesis is true, the t coefficient does not follow a normal distribution and therefore we cannot use the usual t test. Instead, this coefficient follows a tau distribution, and so our test is whether the tau statistic τ (which is equivalent to the usual t statistic) is less than τcrit based on a table of critical tau statistic values shown in the Dickey-Fuller table. .

If the calculated tau value is less than the critical value in the critical value table, then we have a significant result; otherwise, we accept the null hypothesis that there is a unit root and that the time series is not stationary.

There are the following three versions of the Dickey-Fuller test:

Type 0 No constant, no trend Δyi = β1 yi-1 + εi
Type 1 Constant, no trend Δyi = β0 + β1 yi-1 + εi
Type 2 Constant and trend Δyi = β0 + β1 yi-1 + β2 i+ εi

Each version of the test uses a different set of critical values, as shown in the Dickey-Fuller table. It is important to select the correct version of the test for the time series being analyzed. Note that the type 2 test assumes that there is a constant term (which can be significantly equal to zero).