Strong consistency

Abstract: This paper provides a simulated moments estimator (SME) of the parameters of dynamic models in which the state vector follows a time-homogeneous Markov process. Conditions are provided for both weak and strong consistency as well as asymptotic normality. Various tradeoff's among the regularity conditions underlying the large sample properties of the SME are discussed in the context of an asset pricing model.

Abstract: This paper provides a simulated moments estimator (SME) of the parameters of dynamic models in which the state vector follows a time-homogeneous Markov process. Conditions are provided for both weak and strong consistency as well as asymptotic normality. Various tradeoffs among the regularity conditions underlying the large sample properties of the SME are discussed in the context of an asset-pricing model. THIS PAPER PROVIDES CONDITIONS for the consistency and asymptotic normality of a simulated moments estimator (SME) of the parameters of asset-pricing models with time-homogeneous Markov representations of the stochastic forc- ing process. SME's for economic models have been proposed by McFadden (1989) and Pakes and Pollard (1989) for i.i.d. environments, and by Lee and Ingram (1991) for a time series environment. The SME for time series models examined in this paper is as follows. The state vector Yt that determines asset prices is assumed to follow a time-homogeneous Markov process whose transi- tion function depends on an unknown parameter vector 3)0. Asset prices, and possibly other relevant data, are observed as f(Yt, ,0), for some given function f of the underlying state and parameter vector. In parallel, a simulated state process {Y)} is generated (analytically or numerically) from the economic model and corresponding simulated observations f(YJ3, 13) are taken, for a given parameter choice f3. The parameter , is chosen so as to "match moments," that is, to minimize the distance between sample moments of the data, f(Y,8030), and those of the simulated series f(Yt/, f3), in a sense to be made precise. The proposed SME extends the generalized method-of-moments (GMM) estimator (Hansen (1982)) to a large class of asset-pricing models for which the moment restrictions of interest do not have analytic representations in terms of observable variables and the unknown parameter vector. We provide conditions on the transition function of Yt and the observation function f under which the SME of 030 is consistent, and characterize the normalized asymptotic distribu- tion of the estimator. For two reasons, neither the regularity conditions underly- ing Hansen's (1982) analysis of GMM estimators for time-series models without

Abstract: We prove the strong consistency and asymptotic normality of the quasi-maximum likelihood estimator of the parameters of pure generalized autoregressive conditional heteroscedastic (GARCH) processes, and of autoregressive moving-average models with noise sequence driven by a GARCH model. Results are obtained under mild conditions.