Why two random walks give a spurious regression

Question

You generate two completely independent random walks, $X_t$ and $Y_t$ , with unrelated shocks, and regress $Y_t$ on $X_t$ in levels. Even though nothing connects them, you routinely get a large $R^2$ and a $t$ -statistic on the slope that looks wildly significant.

Explain why this happens, and describe the diagnostics and remedy.

Accepted Answer

What goes wrong. Ordinary regression assumes the residuals are stationary and roughly independent. But when both Yt and Xt are random walks, each one wanders far from its starting point and drifts in long, slow excursions. Over any single sample, both series happen to trend somewhere, and the regression latches onto that shared wandering as if it were a relationship. The fitted line "explains" the drift of Y using the drift of X, inflating R^2. Why the t-statistic lies. The usual standard errors are computed assuming stationary, uncorrelated residuals. Here the regression residuals are themse…

Why two random walks give a spurious regression

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