On the Futility of Testing the Error Term Assumptions in a Spurious Regression

A spurious regression model is one in which the dependent and independent variables are non-stationary, but not cointegrated, and the data are not filtered (e.g., by differencing) before the model is estimated. It is well known that in this case the asymptotic behaviour of the least squares parameter estimates, their "t-ratios", the Durbin-Watson statistic and the R-squared, are all non-standard. In particular, the parameter estimates and R-squared converge weakly to functionals of standard Brownian motions; the "t-ratios" diverge in distribution; and the Durbin-Watson statistic converges in probability to zero. In this paper we show that similar results apply to other common tests of a spurious regression model's specification. In particular, standard tests of the Normality and homoskedasticity of the error term are doomed to always reject the null hypotheses, asymptotically. These results further reinforce the need to avoid the estimation of spurious regressions.