THE FORM OF THE OPTIMAL NONLINEAR INSTRUMENT FOR MULTIPERIOD CONDITIONAL MOMENT RESTRICTIONS

Abstract

We present the form of the optimal instrument in a system of multiperiod conditional moment restrictions in the presence of conditional heteroskedasticity. Using Hansen's ( 1985 , Journal of Econometrics 30, 203–228) and Hansen, Heaton, and Ogaki's ( 1988 , Journal of the American Statistical Association 83, 863–871) work on efficiency bounds for generalized method of moments estimators, we show that this form is an autoregressive recurrence parametrized by a few auxiliary processes that are defined through a system of nonlinear stochastic restrictions, with a stability condition among them. In general, the system does not allow inversion and obtaining an explicit solution for the auxiliary parameters. This is a part of my Ph.D. dissertation ( Anatolyev, 2000 ). I am grateful to Kenneth West for introducing me to the subject and providing advice and encouragement. I also thank Bruce Hansen, Yuichi Kitamura, seminar participants at Indiana University, Pennsylvania State University, Rutgers University, University of British Columbia, University of Virginia, and University of Wisconsin, the co-editor Richard Smith, and two referees for helpful comments and suggestions. Research for this paper was supported in part by a fellowship from the Economics Education and Research Consortium, with funds provided by the government of Sweden through the Eurasia Foundation. The opinions expressed in this paper are those of the author and do not necessarily reflect the views of the government of Sweden, the Eurasia Foundation, or any other member of the Economics Education and Research Consortium.