You regress one random walk on an independent random walk over 500 observations and get t = 8.4 with R-squared 0.65. Explain what has gone wrong, say what the t-statistic actually converges to, and give the two tests to run before believing any regression of levels on levels.

You regress one random walk on an independent random walk over 500 observations and get t = 8.4 with R-squared 0.65. Explain what has gone wrong, say what the t-statistic actually converges to, and give the two tests to run before believing any regression of levels on levels.

Approach: Check whether the residual of the levels regression satisfies the conditions the usual asymptotics need, then trace what happens to the standard error when it does not.

The regression is spurious: with two independent unit-root series the t-statistic grows like sqrt(T) instead of converging, so it rejects almost always and R-squared has a non-degenerate limiting distribution rather than tending to zero. The standard asymptotics need stationary residuals with a variance a sample average can estimate. Here the residual is itself a random walk, its sample variance grows with T, the reported standard error is inconsistent, and the t-statistic is sqrt(T) times a functional of Brownian motions with no finite limit. At T = 500 a nominal 5% test rejects close to nine times in ten and a value of 8.4 is unremarkable. First test each series for a unit root with an augmented Dickey-Fuller regression of the difference on the lagged level and enough lagged differences, comparing the coefficient on the lagged level against Dickey-Fuller critical values and never against normal ones. Then, if both are integrated of order one, test whether the residual is stationary, which is the Engle-Granger cointegration test; if it is, the levels relation is real, and if it is not, regress the differences.

Follow-up: Two price series are genuinely cointegrated. Why is a regression of the differences on the differences then misspecified, and what term is missing?

Key concepts: spurious regression, unit root, Dickey-Fuller test, cointegration.