Some Invariance Principles and Central Limit Theorems for Dependent Heterogeneous Processes

Building on work of McLeish, we present a number of invariance principles for doubly indexed arrays of stochastic processes which may exhibit considerable dependence, heterogeneity, and/or trending moments. In particular, we consider possibly time-varying functions of infinite histories of heterogeneous mixing processes and obtain general invariance results, with central limit theorems following as corollaries. These results are formulated so as to apply to economic time series, which may exhibit some or all of the features allowed in our theorems. Results are given for the case of both scalar and vector stochastic processes. Using an approach recently pioneered by Phillips, and Phillips and Durlauf, we apply our results to least squares estimation of unit root models.