Dickey–Fuller test

Remarks

Under $H_0$, the series $y_t$ is a random walk, an ARIMA($0$, $1$, $0$) process, and the usual central limit theory fails. If $y_0 = 0$, then as $n \to \infty$ $$\tau \to \frac{W(1)^2 - 1}{2 \left( \int_0^1 W(r)^2 \, dr \right)^{1/2}}$$ in distribution, where $W$ is a standard Wiener process. This limit is skewed to the left, so its critical values lie well below those of the standard normal distribution; they are tabulated. Variants add a constant, or a constant and a linear trend, to the regression, and each variant has its own limiting distribution and critical values. The augmented Dickey–Fuller test adds lagged differences $\Delta y_{t-1}, \dots, \Delta y_{t-k}$ to the regression, to allow for autocorrelated errors.

Example: for a series with $\hat\delta = -0.2$ and $\operatorname{se}(\hat\delta) = 0.05$, the statistic is $\tau = -4$. This lies far in the left tail of the Dickey–Fuller distribution, so the unit root hypothesis is rejected at the usual levels.

Non-example: comparing $\tau$ with the critical values of the Student $t$ distribution is not the Dickey–Fuller test. That rule rejects a true unit root far too often, because the null distribution of $\tau$ is not a $t$ distribution.