Unit Roots and Stationarity Tests

Whether a series reverts to a level or wanders from it decides which models apply. A stationary series holds a fixed mean and a finite variance, so a regression on it behaves and its past informs its future. An integrated series carries a unit root, its variance growing without bound and its level drifting, so a regression on it is spurious and its sample moments never settle. This is the boundary the autoregression left off, where the dominant root reaches the unit circle. Separating the two regimes is the first task, and it is done by a pair of complementary tests built on the limit theory of integrated processes, one taking the unit root as its null and one taking stationarity [1], [2].

#The scaling limit of a random walk

Both tests rest on what a normalised random walk converges to. Write $\cvW$ for weak convergence of the sample path on $[0,1]$, $\cvP$ for convergence in probability, and $W$ for standard Brownian motion.

::: lemma [lem

] Let $\{u_t\}$ be iid with mean zero and variance $\sigma^2 < \infty$, and let $S_{\lfloor Tr\rfloor} = \sum_{t=1}^{\lfloor Tr\rfloor} u_t$ be the partial-sum process. Then $T^{-1/2}S_{\lfloor Tr\rfloor}\cvW\sigma\,W(r)$ on $[0,1]$. :::

The normalised partial sum of the shocks converges to a Brownian path, so a random walk $y_t = \sum_{s\le t}u_s$ scaled by $T^{-1/2}$ is asymptotically $\sigma W$. Two functionals of that path generate every unit-root law below.

::: lemma [lem

] Let $y_t=\sum_{s=1}^{t}u_s$ with $u_t$ iid mean zero variance $\sigma^2$. Then

:::

::: proof For the first, $y_{\lfloor Tr\rfloor}=S_{\lfloor Tr\rfloor}$, so $T^{-1/2}y_{\lfloor Tr\rfloor}\cvW\sigma W(r)$ by ??. The sum is a Riemann average of its square, $T^{-2}\sum_{t=1}^T y_{t-1}^2 = T^{-1}\sum_{t=1}^T\bigl(T^{-1/2}y_{t-1}\bigr)^2$, which converges under the continuous map $r\mapsto W(r)^2$ to $\sigma^2\int_0^1 W^2$.

For the second, telescoping $y_t^2-y_{t-1}^2 = 2y_{t-1}u_t + u_t^2$ gives $\sum_{t=1}^T y_{t-1}u_t = \tfrac12\bigl(y_T^2 - \sum_{t=1}^T u_t^2\bigr)$. Dividing by $T$, the first term is $\tfrac12(T^{-1/2}y_T)^2\cvW\tfrac12\sigma^2 W(1)^2$ and the second is $\tfrac12 T^{-1}\sum u_t^2\cvP\tfrac12\sigma^2$ by the law of large numbers, so the difference tends to $\tfrac{\sigma^2}{2}(W(1)^2-1)$, which is the Ito integral $\sigma^2\int_0^1 W\,\dd W$. :::

#The Dickey-Fuller test

The first test reads the unit root off the estimated autoregressive coefficient.

::: definition [def

] For $y_t=\rho y_{t-1}+u_t$ with ordinary least squares estimator $\hat\rho$, the Dickey-Fuller coefficient and $t$-statistics are $T(\hat\rho-1)$ and $\tau=(\hat\rho-1)/\operatorname{se}(\hat\rho)$. :::

::: theorem [thm

] Under $H_0\!:\rho=1$ with $u_t$ iid mean zero variance $\sigma^2$,

:::

::: proof Least squares gives $\hat\rho-1=\sum y_{t-1}u_t/\sum y_{t-1}^2$, so

and Equation (1) supplies numerator and denominator with the $\sigma^2$ cancelling, giving the first limit. For $\tau$, the standard error obeys $\operatorname{se}(\hat\rho)^2=\hat\sigma^2/\sum y_{t-1}^2$ with $\hat\sigma^2\cvP\sigma^2$, so

one factor of $\sigma$ cancelling against $\hat\sigma$ and one against the square root of the denominator. :::

::: remark Augmenting the regression with $\sum_{k=1}^{\ell}\gamma_k\Delta y_{t-k}$ whitens serial correlation, so the limits hold with $\sigma^2$ the innovation variance, which is the augmented Dickey-Fuller test [3]. A fitted constant or trend replaces $W$ by the demeaned $W-\int_0^1 W$ or a detrended analogue, shifting the quantiles without changing the form. The Phillips-Perron variant corrects the statistic nonparametrically for serial correlation rather than by augmentation [4]. :::

In practice the test fits the augmented regression, selects the lag length by an information criterion, and compares the $t$ ratio to the tabulated Dickey-Fuller critical values rather than the normal ones.

Augmented Dickey-Fuller test
  input   series y, maximum augmenting lags, deterministic case
  1  choose lags p by minimising an information criterion
  2  regress dy_t on y_{t-1}, the lags dy_{t-1}..dy_{t-p},
     and the deterministic terms
  3  tau <- t-statistic of the coefficient on y_{t-1}
  4  reject the unit root when tau is below the Dickey-Fuller
     critical value for the chosen deterministic case

The tabulated values sit well to the left of the normal ones, the practical mark of the skewed limit in Equation (2). The five percent quantile near $-1.95$ sits left of the normal $-1.64$, so reading a unit-root statistic off normal tables would reject far too often [5].

#The confirmatory test

Failing to reject a unit root is not evidence for one, so the coefficient test is not conclusive alone. The confirmatory test runs the hypotheses the other way, taking stationarity as its null.

::: definition [def

] With residuals $\hat e_t$ from regressing $y_t$ on a constant, or on a constant and trend, partial sums $S_t=\sum_{i=1}^t\hat e_i$, and a heteroskedasticity- and autocorrelation-consistent estimator $\hat\lambda^2$ of the long-run variance, the statistic is $\eta=T^{-2}\sum_{t=1}^T S_t^2/\hat\lambda^2$. :::

::: theorem [thm

] Under stationarity $\eta\cvW\int_0^1 V(r)^2\,\dd r$, where $V$ is a standard Brownian bridge in the level case and a second-level bridge in the trend case. :::

::: proof Demeaning maps the functional limit $W$ to the bridge $V(r)=W(r)-rW(1)$, since the residuals sum to zero, so $T^{-1/2}S_{\lfloor Tr\rfloor}\cvW\lambda V(r)$ with $\lambda^2$ the long-run variance. The continuous map $\int V^2$ together with $\hat\lambda^2\cvP\lambda^2$ gives $\eta\cvW\int_0^1 V^2$ [6]. :::

Run on a genuinely integrated series the two verdicts agree, the level failing to reject a unit root under Dickey-Fuller yet rejecting stationarity under KPSS, so the series is integrated of order one and is differenced before any stationary model is fitted. That order of integration is exactly the input the next step needs. When several series each carry a unit root, a linear combination of them can still be stationary, which is the cointegration that underlies statistical arbitrage and the error-correction structure of the cointegration post.