Efficiency Bounds Implied by Multiperiod Conditional Moment Restrictions

Abstract In this article we study a class of econometric models that imply a set of multiperiod conditional moment restrictions. These restrictions depend on an unknown parameter vector. We construct an extensive class of consistent, asymptotically normal estimators of this parameter vector and calculate the greatest lower bound for the asymptotic covariance matrices of estimators in this class. In so doing, we extend results reported by Hansen (1985) and Stoica, Söderström, and Friedlander (1985), by allowing for more general forms of nonlinearities and temporal dependence. Many dynamic econometric models imply that the expectation of a function of a currently observed data vector and an unknown parameter vector conditioned on information available at some point in the past is 0. We focus on models in which the conditioning information is lagged more than one time period, as in the models considered by Barro (1981), Dunn and Singleton (1986), Eichenbaum and Hansen (1987), Eichenbaum, Hansen, and Singleton (1988), Hansen and Hodrick (1983), Hansen and Singleton (1988), and Hall (1988). Hence we consider econometric models that imply multiperiod conditional moment restrictions that depend on an unknown parameter vector. Within the context of these models, it is possible to estimate the parameter vector without simultaneously estimating the law of motion for the entire set of observable variables. The basic idea is to use the conditional moment restrictions to deduce a set of unconditional moment restrictions. Then estimators of the parameter vector can be obtained by using sample counterparts to the unconditional moment restrictions as described by Sargan (1958) and Hansen (1982). Such estimators are referred to as generalized method of moments (GMM) estimators. For most applications the conditional moment restrictions imply an extensive set of unconditional moment restrictions. As a consequence, there is a vast array of GMM estimators that can be used to estimate consistently the parameter vector of interest. Each member of this set of estimators is constructed using a distinct collection of the unconditional moment restrictions. Hence it is of interest to compare the performances of the alternative GMM estimators. For tractability we investigate only the asymptotic distributions of the estimators in question. More precisely, we use a method suggested by Hansen (1985) for calculating a greatest lower bound for the asymptotic covariance matrices of the alternative GMM estimators, that is, an efficiency bound. We compute the efficiency bound for a rich collection of time series models that imply multiperiod conditional moment restrictions. Hansen (1985) illustrated this method for a time series model with conditionally homoscedastic moving-average disturbance terms for which the moving-average polynomial is invertible. Stoica et al. (1985) calculated efficiency bounds for GMM estimators for autoregressive parameters in autoregressive moving-average models without unit roots. They established that the efficiency bound for GMM estimators of the autoregressive parameters coincides with the asymptotic covariance matrix of the Gaussian maximum likelihood estimators. The models considered by Hansen (1985) in his illustrative example and by Stoica et al. (1885) can be viewed as special cases of the models considered in this article. Although we do not make any direct comparisons to maximum likelihood, we do allow for moving-average disturbances that are conditionally heteroscedastic and moving-average lag polynomials that cannot be inverted.