Semiparametric Two-Step GMM with Dependent Data

Abstract

Paper 1 -- (pdf file) "Sieve Inference on Possibly Misspecified Semi-nonparametric Time Series Models"(authors: Xiaohong Chen, Zhipeng Liao and Yixiao Sun) . This paper provides a general theory on the asymptotic normality of plug-in sieve M estimators of possibly irregular functionals of semi-nonparametric time series models. We show that, even when the sieve score process is not a martingale difference, the asymptotic variances of plug-in sieve M estimators of irregular (i.e., slower than root-T estimable) functionals are the same as those for independent data. Nevertheless, ignoring the temporal dependence in finite samples may not lead to accurate inference. We then propose an easy-to-compute and more accurate inference procedure based on a ``pre-asymptotic" sieve variance estimator that captures temporal dependence of unknown forms. We construct a ``pre-asymptotic" Wald statistic using an orthonormal series long run variance (OS-LRV) estimator. For sieve M estimators of both regular (i.e., root-T estimable) and irregular functionals, a scaled ``pre-asymptotic" Wald statistic is asymptotically F distributed when the series number of terms in the OS-LRV estimator is held fixed. Simulations indicate that our scaled ``pre-asymptotic" Wald test with F critical values has more accurate size in finite samples than the conventional Wald test with chi-square critical values. --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- Paper 2 (no pdf file) -- "Semiparametric Two-step GMM with Weakly Dependent Data" (authors: Xiaohong Chen, Jinyong Hahn and Zhipeng Liao) . This paper considers semiparametric two-step GMM estimation and inference with weakly dependent data, where unknown nuisance functions are estimated via sieve extremum estimation in the first step. We provide a verifiable necessary condition for the root-n consistency of the second step GMM estimator, and characterize its semiparametric asymptotic variance. We show that although the asymptotic variance of the second step GMM estimator may not have a closed form expression, it can be well approximated by sieve variances of the second step GMM estimator. We present confidence sets construction, Wald tests and overidentification tests for the second step GMM that properly reflect the first step estimated functions and the weak dependence of the data. We also derive numerical equivalence results, which suggest that empirical researchers can ignore the nonparametric first step of the model and conduct inference using existing softwares as if the first step were parametric. The paper provides a formal justification for flexible parametric estimation and inferernce in empirical work with weakly dependent data.