Test for unit root

Summary

Setup

• The AR(1) processes:

$$y_t=β_1+ρy_{t-1}+ε_t, t=1,…,T$$

• where the error terms are white noise.
• Assume that $ρ∈[0,1]$ .
• The process has a unit root if $ρ=1$ .

Dickey-Fuller test

• Rewrite the AR(1) process:

$$Δy_t=β_1+δy_{t-1}+ε_t, t=1,…,T$$

• where $δ=ρ-1$ . $δ∈[-1,0]$ .
• The process has a unit root if $δ=0$ .
• Test for unit root: $H_0:δ=0$ against $H_1:δ<0$ .
• Rejecting $H_0$ implies that the stability condition holds.
• The OLS estimator of $δ$ is consistent. However, under the null, the standard errors are not. The t-statistic will not follow a t-distribution and critical values from the $t$ -distribution or normal distribution will be incorrect.
• Asymptotically correct critical values and $p$ -values can be found using the Dickey-Fuller test in econometrics software.

Versions of the Dickey-Fuller test

• No intercept:

$$Δy_t=δy_{t-1}+ε_t, t=1,…,T$$

• With a trend:

$$Δy_t=β_1+δy_{t-1}+β_2t+ε_t, t=1,…,T$$

• With additional lags:

$$Δy_t=β_1+δy_{t-1}+ρ_1Δy_{t-1}+…+ρ_p{Δy}_{t-p}+ε_t , t=1,…,T$$

• The test is typically called the augmented Dickey-Fuller test if we have additional lags. The augmented test also allows for removal of the intercept and/or adding a time trend.