Dickey-Fuller

The Dickey-Fuller (DF) test checks whether a time series behaves like it has a unit root (shocks have persistent effects, like a random walk) versus being stationary/mean-reverting.

The idea in one model

A common starting point is an AR(1) process:

[ y_t = \rho y_{t-1} + \varepsilon_t. ]

If (\rho = 1), the series has a unit root (it is not mean-reverting).
If (|\rho| < 1), the series is stationary around a constant mean (in the weak/covariance sense).

Rewriting the AR(1) in differences gives the regression form used by DF:

[ \Delta y_t = \gamma y_{t-1} + \varepsilon_t, \qquad \gamma = \rho - 1. ]

So testing for a unit root is equivalent to:

[ H_0: \gamma = 0 \quad (\rho=1) \qquad \text{vs} \qquad H_1: \gamma < 0 \quad (|\rho|<1). ]

Constant and trend versions

Empirical work often includes deterministic terms:

[ \Delta y_t = \alpha + \beta t + \gamma y_{t-1} + \varepsilon_t. ]

Adding (\alpha) allows a nonzero mean (drift).
Adding (\beta t) allows a deterministic linear trend.

Which version is appropriate depends on the series and the economic question.

Why the critical values are “non-standard”

Under the unit-root null, the usual t-distribution for the coefficient on (y_{t-1}) does not apply. The DF test uses special critical values (from Dickey-Fuller tabulations) rather than ordinary t-test cutoffs.

The augmented Dickey-Fuller (ADF) extension

Real time series often have serially correlated errors. The augmented DF test adds lagged differences to absorb that correlation:

[ \Delta y_t = \alpha + \beta t + \gamma y_{t-1} + \sum_{i=1}^{k} \delta_i \Delta y_{t-i} + \varepsilon_t. ]

The unit-root null is still (\gamma=0).

How economists use it

If you fail to reject a unit root, you typically treat the series as integrated (often (I(1))) and consider differencing or cointegration methods.
If you reject a unit root, you are more comfortable modeling the series in levels with stationary time-series tools.

A good practice is to pair test results with plots and economic reasoning (structural breaks, regime changes, and measurement revisions can change the conclusion).

Knowledge Check

### In the Dickey-Fuller setup, the unit-root null corresponds to: - [x] \(H_0:\gamma=0\) in \(\Delta y_t=\gamma y_{t-1}+\varepsilon_t\) (equivalently \(\rho=1\)) - [ ] \(H_0:\gamma<0\) (equivalently \(|\rho|<1\)) - [ ] \(H_0:\Delta y_t=0\) for all \(t\) - [ ] \(H_0:\varepsilon_t=0\) for all \(t\) > **Explanation:** A unit root means \(\rho=1\), which implies \(\gamma=\rho-1=0\) in the differenced regression form. ### Why doesn’t the DF test use ordinary t critical values for \(\gamma\)? - [x] Under the unit-root null, the test statistic has a non-standard distribution - [ ] Because the DF test never uses regression - [ ] Because differencing forces the statistic to be chi-squared - [ ] Because the sample mean is always zero > **Explanation:** The unit-root null changes the asymptotic behavior of the regression, so standard t tables do not apply. ### What does the augmented Dickey-Fuller (ADF) test add, and why? - [x] Lagged differences \(\Delta y_{t-i}\) to reduce serial correlation in the errors - [ ] A second dependent variable to improve fit - [ ] A structural break by default - [ ] A moving-average term estimated by maximum likelihood > **Explanation:** Adding lagged differences helps “soak up” autocorrelation so the remaining errors are closer to white noise.