Dickey–Fuller test

Explanation

A simple AR model is

${\displaystyle y_t=\rho y_{t-1}+u_t}$

where ${\displaystyle y_t}$ is the variable of interest, ${\displaystyle t}$ is the time index, ${\displaystyle \rho }$ is a coefficient, and ${\displaystyle u_t}$ is the error term (assumed to be white noise). A unit root is present if ${\displaystyle \rho =1}$. The model would be non-stationary in this case.

The regression model can be written as

${\displaystyle \mathrm{\Delta }{y}_t=(\rho -1)y_{t-1}+u_t=\delta y_{t-1}+u_t}$

where ${\displaystyle \mathrm{\Delta }}$ is the first difference operator and ${\displaystyle \delta \equiv \rho -1}$. This model can be estimated, and testing for a unit root is equivalent to testing ${\displaystyle \delta =0}$. Since the test is done over the residual term rather than raw data, it is not possible to use standard t-distribution to provide critical values. Therefore, this statistic ${\displaystyle t}$ has a specific distribution simply known as the Dickey–Fuller table.

There are three main versions of the test:

1. Test for a unit root:

${\displaystyle \mathrm{\Delta }{y}_t=\delta y_{t-1}+u_t}$

2. Test for a unit root with constant:

${\displaystyle \mathrm{\Delta }{y}_t=a_0+\delta y_{t-1}+u_t}$

3. Test for a unit root with constant and deterministic time trend:

${\displaystyle \mathrm{\Delta }{y}_t=a_0+a_{1}t+\delta y_{t-1}+u_t}$

Each version of the test has its own critical value which depends on the size of the sample. In each case, the null hypothesis is that there is a unit root, ${\displaystyle \delta =0}$. The tests have low statistical power in that they often cannot distinguish between true unit-root processes (${\displaystyle \delta =0}$) and near unit-root processes (${\displaystyle \delta }$ is close to zero). This is called the "near observation equivalence" problem.

The intuition behind the test is as follows. If the series ${\displaystyle y}$ is stationary (or trend-stationary), then it has a tendency to return to a constant (or deterministically trending) mean. Therefore, large values will tend to be followed by smaller values (negative changes), and small values by larger values (positive changes). Accordingly, the level of the series will be a significant predictor of next period's change, and will have a negative coefficient. If, on the other hand, the series is integrated, then positive changes and negative changes will occur with probabilities that do not depend on the current level of the series; in a random walk, where you are now does not affect which way you will go next.

It is notable that

${\displaystyle \mathrm{\Delta }{y}_t=a_0+u_t}$

may be rewritten as

${\displaystyle y_t=y_0+\sum _{i=1}^t{u}_i+a_{0}t}$

with a deterministic trend coming from ${\displaystyle a_{0}t}$ and a stochastic intercept term coming from ${\displaystyle y_0+\sum _{i=1}^t{u}_i}$, resulting in what is referred to as a stochastic trend.

There is also an extension of the Dickey–Fuller (DF) test called the augmented Dickey–Fuller test (ADF), which removes all the structural effects (autocorrelation) in the time series and then tests using the same procedure.