Spurious multivariate regressions under stationary fractionally integrated processes

Abstract

Spurious regression under stationary processes exhibiting long memory was studied by Tsay and Chung (2000) [JoE 96, pp. 155-182] for a univariate model. We extend their findings for the multivariate linear regression and find that inference drawn from the latter is also spurious. Our results hold for any finite number of independent stationary fractionally integrated explanatory variables. It is shown that the t-statistics associated to the estimated parameters diverge if the processes underlying the dependent variable and the particular explanatory variable are sufficiently persistent. It is shown also that inference drawn from test statistics and goodness of fit measures, such as the Wald F statistic and the R2 can be contradictory in the sense that the test of joint significance may reject the null hypothesis if the underlying variables are strongly persistent, indicating incorrectly that at least one of the explanatory variables affects the dependent variable, whereas the latter always converges to zero, supporting the correct assertion that the variables used as regressors do not explain the variable used as regressand. Comprehensive finite sample evidence is consistent with our asymptotic results and shows that they hold even for small sample sizes such as 100 observations.