Fully modified IV, GIVE and GMM estimation with possibly non-stationary regressors and instruments

This paper develops a general theory of instrumental variables (IV) estimation that allows for both I(1) and I(0) regressors and instruments. The main goal of this paper is to develop a theory in which one does not need to know the integration properties of the regressors in order to obtain efficient estimators. The estimation techniques involve an extension of the fully modified (FM) regression procedure that was introduced in earlier work by Phillips and Hansen (1990). FM versions of the generalized instrumental variable estimation (GIVE) method and the generalized method of moments (GMM) estimator are developed. In models with both stationary and nonstationary components, the FM-GIVE and FM-GMM techniques provide efficiency gains over FM-IV in the estimation of the stationary components of a model that has both stationary and non-stationary regressors. The paper exploits a result of Phillips (1991a) that we can apply FM techniques in models with cointegrated regressors and even in stationary regression models without losing the methodʹs good asymptotic properties. The present paper shows how to take advantage jointly of the good asymptotic properties of FM estimators with respect to the non-stationary elements of a model and the good asymptotic properties of the GIVE and GMM estimators with respect to the stationary components. The theory applies even when there is no prior knowledge of the number of unit roots in the system or the dimension or the location of the cointegration space. An FM extension of the Sargan (1958) test for the validity of the instruments is proposed.