COWLES FOUNDATION FOR RESEARCH IN
ECONOMICS Box 208281
COWLES FOUNDATION DISCUSSION PAPER NO. 1047 "Fully Modified Least Squares and Vector Autoregression" Peter C. B. Phillips May 1993 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., In - 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. |
