This paper provides a robust statistical approach to nonstationary time series regression and inference. Fully modiﬁed extensions of traditional robust statistical procedures are developed which allow for endogeneities in the nonstationary regressors and serial dependence in the shocks that drive the regressors and the errors that appear in the equation being estimated. The suggested estimators involve semiparametric corrections to accommodate these possibilities and they belong to the same family as the fully modiﬁed least squares (FM-OLS) estimator of Phillips and Hansen (1990). Speciﬁc attention is given to fully modiﬁed least absolute deviation (FM-LAD) estimation and fully modiﬁed M (FM-M)-estimation. The criterion function for LAD and some M-estimators is not always smooth and the paper develops generalized function methods to cope with this diﬀiculty in the asymptotics. The results given here include a strong law of large numbers and some weak convergence theory for partial sums of generalized functions of random variables. The limit distribution theory for FM-LAD and FM-M estimators that is developed includes the case of ﬁnite variance errors and the case of heavy-tailed (inﬁnite variance) errors. Some simulations and a brief empirical illustration are reported.
Phillips, Peter C.B., "Robust Nonstationary Regression" (1993). Cowles Foundation Discussion Papers. 1307.