You backtest a momentum signal on equity prices over five years. The strategy looks profitable. You run the regression on returns instead of prices and the edge vanishes. The difference is stationarity. And the exam tests whether you can spot the trap before your trade does.
Time series modeling assumes the data-generating process is stable across the sample. If the mean or variance shifts, your regression on past data does not predict future data. Worse, two unrelated non-stationary series can look highly correlated by accident, the spurious-regression problem.
A series is covariance stationary when three conditions hold. First, the unconditional mean is constant. Second, the unconditional variance is constant and finite. Third, the autocovariance between Y at time t and Y at time t-k depends only on k, not on t. Trends violate the first; random walks violate the second; structural breaks violate any of the three.
KEY: Stationarity is a property of the data-generating process, not of the data alone. A short sample from a non-stationary process can look stationary; a long sample from a stationary...
Common mistakes
- Treating high autocorrelation as evidence of forecastability. A random walk has autocorrelation near 1 at every lag, but is not forecastable beyond "today's value." Persistence is not the same as predictability of returns. Trap: a question shows ACF flat at 0.99 across 20 lags and asks if Y is highly forecastable.
- Confusing trend stationary with difference stationary. A trend-stationary series has a deterministic time trend; detrend by subtracting the fitted line. A difference-stationary series has a unit root; difference once. The two require different fixes. Trap: ADF rejects unit root but Y-t still trends, indicates trend-stationary, not difference-stationary.
- Using OLS critical values for ADF. Under the null hypothesis of a unit root, the lagged level Y-t-1 is non-stationary, so the t-statistic does not follow the standard t distribution. Use the Dickey-Fuller tables (more negative critical values).
Bottom line
- Covariance stationarity requires constant mean, constant variance, and autocovariances that depend only on the lag; deterministic trends and unit roots break it.
- AR(1) is stationary when |φ| < 1, with mean-reverting level α / (1 − φ); at φ = 1 you have a random walk with no mean to revert to.
- MA(q) is always stationary and its ACF cuts off at lag q; AR(p) is stationary only if all roots of the characteristic polynomial lie outside the unit circle.
- Random walks have unit roots: variance grows with time and OLS on non-stationary levels gives spurious regressions, so work with returns or first differences.
Exam shortcut
When a question gives you φ in an AR(1) and asks for the long-run mean, divide alpha by one minus phi. When it gives you ACF cuts off at q and PACF decays, the answer is MA(q). When it gives you a series with R-squared of 0.7 between two unrelated random walks, the trap answer is "strongly correlated". The right answer is spurious regression. ADF first, then model.
The full lesson (about 2,834 words, 19 min read) adds 2 worked examples, all 6 common mistakes, a self-check, free in the app.
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