Cointegrating Polynomial Regressions: Robustness Of Fully Modified Ols

Cointegrating polynomial regressions (CPRs) include deterministic variables, integrated variables, and their powers as explanatory variables. Based on a novel kernel-weighted limit result and a novel functional central limit theorem, this paper shows that the fully modified ordinary least squares (FM-OLS) estimator of Phillips and Hansen (1990, Review of Economic Studies 57, 99–125) is robust to being used in CPRs. Being used in CPRs refers to a widespread empirical practice that treats the integrated variables and their powers, incorrectly, as a vector of integrated variables and uses textbook FM-OLS. Robustness means that this “formal” FM-OLS practice leads to a zero mean Gaussian mixture limiting distribution that coincides with the limiting distribution of the Wagner and Hong (2016, Econometric Theory 32, 1289–1315) application of the FM estimation principle to the CPR case. The only restriction for this result to hold is that all integrated variables to power one are included as regressors. Even though simulation results indicate performance advantages of the Wagner and Hong (2016, Econometric Theory 32, 1289–1315) estimator, partly even in large samples, the results of the paper give an asymptotic foundation to “formal” FM-OLS and thus enlarge the usability of the Phillips and Hansen (1990, Review of Economic Studies 57, 99–125) estimator implemented in many software packages.