Spurious regression under deterministic and stochastic trends

This paper analyses the asymptotic and finite sample implications of a mixed nonstationary behavior among the dependent and explanatory variables in a linear spurious regression model. We study the cases when the nonstationarity in the dependent variable is deterministic (stochastic), while the nonstationarity in the explanatory variable is stochastic (deterministic). In particular, we derive the asymptotic distribution of statistics in a spurious regression equation when one variable follows a difference stationary process (a random walk with and without drift), while the other is characterized by deterministic nonstationarity (a linear trend model with and without structural breaks in the trend function). We find that the divergence rate is sensitive to the assumed mixture of nonstationarity in the data generating process, and the phenomenon of spurious regression itself, contrary to previous findings, depends on the presence of a linear trend in the regression equation. Simulation experiments and real data confirm our asymptotic results.