Asymptotic Inference for Nonstationary Fractionally Integrated Processes

This paper studies the asymptotics of nonstationary fractionally integrated (NFI) multivariate processes with memory parameter d > 0.5 . We provide conditions to establish a functional central limit theorem and weak convergence of stochastic integrals for NFI processes under the assumption that the innovations are linear processes. Several applications of these results are given. More specifically, we prove the rates of convergence of the OLS estimators of cointegrating vectors in triangular representations. When the regressors are strongly exogenous with respect to the corresponding parameters of the cointegrating vector, then the limiting OLS distribution is a mixture of normals from which standard inference can be implemented. On the other hand, we also extend Sims, Stock and Watson's (1990) study on estimation and hypothesis testing in vector autoregressions with integrated processes and deterministic components to the more general fractional framework. We show how their main conclusions remain valid when dealing with NFI processes, namely, that whenever a block of coefficients can be written as coefficients on zero mean I(0) regressors in a model that includes a constant term, they will have a joint asymptotic normal distribution, so that the corresponding restrictions can be tested using standard asymptotic chi-squared distribution theory. Otherwise, in general, the associated statistics will have nonstandard limiting distributions.