Is Stock Price Correlated with Oil Price? Spurious Regressions with Moderately Explosive Processes

This study explores the spurious effects in linear regressions with moderately explosive processes. Asymptotic results are developed for the least square estimator, the typical t‐statistic, the Durbin–Watson statistic, and the coefficient of determination. The typical t‐statistic is unable to detect the presence of a spurious relationship, due to the presence of nuisance parameters that characterize deviations from unity. Moreover, the t‐statistic for common explosive processes has different asymptotics compared to that for distinct explosive processes. Such differences further complicate the use of the t‐statistic. We demonstrate that two popular methods available in the literature are incapable for this purpose due to similar difficulties. To overcome these limitations, we propose a t‐test based upon balanced regressions that induces asymptotic inference based on the standard normal distribution, which is therefore robust to deviations from unity. These results are further generalized to spurious regressions with multivariate mildly explosive processes. Simulation results confirm that our test is effective in finite samples, while other alternatives are not. An empirical example that demonstrates the phenomenon of spurious correlation between the NASDAQ stock index and crude oil price in the US is provided to show the practical merit of our proposed method.