0.1 Unit Root Tests
Before differencing (as in ARIMA modelling) or testing for Cointegration, we need a formal test for whether a series is non-stationary because \(\phi(z)\) has a root at \(z=1\) (a “unit root”, \(X_t \sim I(1)\)) rather than stationary around a trend.
Consider \(X_t = \rho X_{t-1} + Z_t\). The null \(H_0: \rho = 1\) (unit root, non-stationary random walk) is tested against \(H_1: |\rho|<1\) (stationary) using the regression \[ \nabla X_t = \gamma X_{t-1} + Z_t, \qquad \gamma = \rho - 1, \] and the \(t\)-statistic for \(\hat\gamma\). Under \(H_0\) this statistic does not converge to a Student-\(t\) distribution but to a nonstandard Dickey-Fuller distribution (functional of Brownian motion), requiring specialized critical values.
Adds lagged differences to whiten residual autocorrelation: \[ \nabla X_t = \alpha + \beta t + \gamma X_{t-1} + \sum_{j=1}^k \delta_j \nabla X_{t-j} + Z_t, \] testing \(H_0: \gamma = 0\). Constant/trend terms and lag order \(k\) (chosen via Model Selection Criteria (AIC BIC) or sequential testing) must be specified.
Reverses the null: \(H_0\) is stationarity (trend- or level-stationary), tested against the alternative of a unit root, by testing whether the partial sums of OLS residuals from a trend regression behave like a stationary process. Used alongside ADF since the two tests have opposite null hypotheses — agreement between them strengthens conclusions about the true order of integration.
Other variants: Phillips-Perron (non-parametric correction for serial correlation/heteroskedasticity instead of augmenting with lags), and Zivot-Andrews (allows for a structural break under the alternative).