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Fully Modified Least Squares and Vector Autoregression

Peter C. B. Phillips

Abstract

Fully modified least squares (FM-OLS) regression was originally designed in work by Phillips and Hansen (1990) to provide optimal estimates of cointegrating regressions. The method modifies least squares to account for serial correlation effects and for the endogeneity in the regressors that results from the existence of a cointegrating relationship. This paper provides a general framework which makes it possible to study the asymptotic behavior of FM-OLS in models with full rank $I(1)$ regressors, models with $I(1)$ and $I(0)$ regressors, models with unit roots, and models with only stationary regressors. This framework enables us to consider the use of FM regression in the context of vector autoregressions (VAR's) with some unit roots and some cointegrating relations. The resulting FM-VAR regressions are shown to have some interesting properties. For example, when there is some cointegration in the system, FM-VAR estimation has a limit theory that is normal for all of the stationary coefficients and mixed normal for all of the nonstationary coefficients. Thus, there are no unit root limit distributions even in the case of the unit root coefficient submatrix (i.e., $I_{n - r}$, for an $n$-dimensional VAR with $r$ cointegrating vectors). Moreover, optimal estimation of the cointegration space is attained in FM-VAR regression without prior knowledge of the number of unit roots in the system, without pretesting to determine the dimension of the cointegration space and without the use of restricted regression techniques like reduced rank regression. The paper also develops an asymptotic theory for inference based on FM-OLS and FM-VAR regression. The limit theory for Wald tests that rely on the FM estimator is shown to involve a linear combination of independent chi-squared variates. This limit distribution is bounded above by the conventional chi-squared distribution with degrees of freedom equal to the number of restrictions. Thus, conventional critical values can be used to construct valid (but conservative) asymptotic tests in quite general FM time series regressions. This theory applies to causality testing in VAR's and is therefore potentially useful in empirical applications.