A note on spurious regression and random walks with zero, local, or constant drifts

This note investigates the spurious regression where each of the regressand and the regressor follows a random walk with zero, nonzero local, or nonzero constant drift. In the existing literature of spurious regression, both the regressand and regressor have zero or constant drifts. We consider more general cases, and derive the order of convergence or divergence of the estimated slope coefficient and the squared t-statistic, as well as their asymptotic distributions. We find that the estimated slope coefficient may converge, diverge, or neither depending on the case. Further, the asymptotic distribution of the scaled slope estimator takes on various interesting shapes such as a bimodal and asymmetric distribution. We also reveal that the squared t-statistic diverges at different rates across cases.