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.