Stationarity, unit roots and spurious regression

A series is weakly (covariance) stationary if three things do not change over time: its mean is constant, its variance is constant, and the covariance between two values depends only on how far apart they are, not on when. It then looks the same wherever you sample it, which is what lets averages over time estimate anything. (Strict stationarity requires the whole distribution to be time-invariant; weak is the usual working definition, and the two coincide for normal series.)

The building block is white noise: uncorrelated draws with zero mean and constant variance σ². An AR(1) series x_t = φx_t−1 + e_t with |φ| < 1 is stationary: shocks die out geometrically, the series returns to its mean (zero), and its variance is σ² ÷ (1 − φ²). A constant plus white noise (X_t = 2 + Z_t) is stationary around the level 2. A series with a time-varying mean or variance is not.

Stationarity matters because the whole apparatus of regression assumes that sample averages settle down as the sample grows. For a series that does not settle (a trend, a wandering level), they do not, and the t, F and R² statistics of the earlier lessons lose their meaning.

Set φ = 1 in the AR(1) and the series stops returning:

A series with a unit root is one whose characteristic root is 1: the random walk, with or without drift. It is integrated of order one, I(1), because differencing it once gives a stationary series: Δy_t = e_t (or δ + e_t with drift). A stationary series is I(0). Three things to remember:

• Shocks persist. In a random walk the effect of a shock never fades. In a stationary AR(1) it fades by φ each period.
• Drift adds a trend. A random walk with drift δ rises by about δ per period on average, plus the wandering.
• Two kinds of trend. A trend-stationary series is a deterministic trend plus stationary noise: remove the trend and what is left is stationary. A difference-stationary series (a random walk with drift) is stationary only after differencing. Removing a time trend from a difference-stationary series leaves it still wandering. The distinction decides how to detrend, and the tests below tell them apart.

In the lab, put φ at 1 and press “draw a new series” a few times: a random walk has no home, and its Dickey-Fuller statistic falls to the left of the cutoff only about one time in twenty. Pull φ down to 0.7 and the series keeps returning to zero. In between (φ = 0.95) the series is stationary, but the test can rarely tell.

The Dickey-Fuller (DF) test regresses the change in the series on its lagged level:

• The null is a unit root. Rejecting it (a statistic below the critical value) is evidence of stationarity. Failing to reject is not proof of a unit root; the test has low power against persistent stationary series.
• The augmented Dickey-Fuller (ADF) test adds lagged differences, Δy_t−1, Δy_t−2, …, on the right to soak up autocorrelation in the errors, with the same critical values. Choose enough lags that the residuals are white noise.
• Which deterministic terms? Match the form of the series: a trending series needs the constant-and-trend version; a series that fluctuates around a level needs only the constant. The critical value changes with the choice.
• The KPSS test turns the hypotheses round: its null is stationarity (and the alternative a unit root). The Phillips-Perron and ERS (DF-GLS) tests have a unit root as the null, like ADF. Used together, ADF and KPSS agree when both say stationary or both say unit root, and conflict otherwise.

Real data. India’s log real GDP, 1960 to 2025 (World Bank, read 2026-10-24), is trending steadily upward. The ADF statistic with a constant, a trend and one lagged difference is −1.73, against a simulated 5% critical value of −3.51 for this sample length: cannot reject a unit root. The series in first differences (annual growth) gives −5.65 against −2.92: a clear rejection, so growth is stationary. Log GDP is I(1); its growth rate is I(0). Like all unit-root tests, this one has limited power in 65 annual observations, so the conclusion is “consistent with a unit root”, not “proof”.

Regress one random walk on another that is completely unrelated to it. You would expect a slope near zero and a t statistic that exceeds 2 about one time in twenty. Instead, in the lab’s first setting at 100 observations the t statistic exceeds 1.96 in roughly three samples out of four, the R² is often large, and the Durbin-Watson statistic is tiny: the residuals are themselves a random walk. This is spurious regression (a phenomenon associated with Granger and Newbold). The cause is that with a unit root the sums that make up the t statistic do not settle down, so the t distribution does not apply, and a regression of a trending series on a trending series fits the trends.

The Granger-Newbold rule of thumb: suspect that a regression is spurious if R² is greater than the Durbin-Watson statistic. A high R² with a low d is the warning sign. The lab shows that it is a signal, not a test: it catches only about half of the spurious regressions. Remedies:

• Difference the series and regress growth rates (or changes) on each other: the third setting of the lab behaves perfectly. The price is that you lose information about long-run levels.
• Test for cointegration (lesson 16): two I(1) series can have a genuine long-run relationship, in which case a regression in levels is meaningful, and the residual of the regression is stationary.
• Add lagged variables or a trend, which can help but is not a general cure.

The Durbin-Watson statistic used in the Granger-Newbold rule is in lesson 10, and the next lesson shows when two unit-root series do have a long-run relationship.