Dickey-Fuller Test

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Definition

The Dickey-Fuller (DF) test tests for the presence of an unit root in an AR(1) Process process.

This determines whether a time series is non-stationary or not.

Tip

An AR(p) model is stationary, if all the roots $x$ of its characteristic equation is $|x|>1$.

Unit root means there exists an $|x|=1$, which implies that the AR(p) model is not stationary.

Model

$Y_t=\theta Y_{t-1}+u_t$

If $\theta =1$, the model becomes a random walk (non-stationary).

Reformulation

$\nabla Y_t=\pi Y_{t-1}+u_t\text{where }\pi =\theta -1$

Hypotheses

$$\begin{array}{rl}{H}_0& :\pi =0\text{(unit root present, non-stationary)}\\ H_1& :\pi <0\text{(stationary)}\end{array}$$

Test Statistic

$\tau =\frac{\hat{\pi }}{\text{se}(\hat{\pi })}=\frac{\hat{\theta }-1}{\text{se}(\hat{\theta })}$

Distribution

The test statistic $\tau$ follows the Dickey-Fuller distribution (not standard t-distribution).

Decision Rule

Reject $H_0$ if $|\tau |>|{\tau }_{\text{critical}}|$

WARNING

Use DF critical values, not standard t-distribution tables. The DF distribution has fatter tails.

Model Variants

Model	Equation
No constant/trend	$\nabla Y_t=\pi Y_{t-1}+u_t$
With constant	$\nabla Y_t={\beta }_1+\pi Y_{t-1}+u_t$
With constant & trend	$\nabla Y_t={\beta }_1+{\beta }_{2}t+\pi Y_{t-1}+u_t$

Related

• Augmented Dickey-Fuller Test
• Random Walk
• Weakly Stationary