Statistics Seminar（2014-03）

Topic:Econometric Estimation with High-Dimensional Moment Equalities

Speaker:Zhentao Shi, Yale University

Time:Friday, 14 February, 10:30-11:30

Location:Room 217, Guanghua Building 2

Abstract：Structural models involving moment conditions are in widespread practical use, and commonly include many moments to capture the stylized facts in large datasets. The number of moments m can explode with the sample size n. The growing complexity of the model challenges familiar estimators like Generalized Method of Moments (GMM) or Empirical Likelihood (EL), whose asymptotic normality demands that n dominate the cube of m. In the extreme case m>n, the two estimators break down even numerically, as the weighting matrix is non-invertible in the criterion function of two-step GMM, and the constraints are infeasible in the primal problem of EL.We consider a structural model in which the number of moments is not limited by the sample size, and where the econometric problem is to estimate and perform inference on a finite-dimensional parameter. We develop a novel two-step estimation procedure. We call the first step the Relaxed Empirical Likelihood (REL), which relaxes the moment constraints of the primal problem of EL. While EL requires that all moment constraints equal zero, REL tolerates a small violation specified by the user as a tuning parameter. The tuning parameter controls, as it shrinks to zero asymptotically, the maximal approximation error of the sample means of the moment functions. Under a high-dimensional asymptotic framework, we derive the consistency of REL and its rate of convergence. As the relaxation introduces first-order bias that slows the rate, the second step selects a small subset of moments in a computationally efficient manner to correct the bias of REL. The algorithm adds one moment in each iteration---the one that maximizes an information criterion evaluated at REL conditional on the moments chosen in the preceding iterations. We establish asymptotic normality and efficiency of bias-corrected REL. To the best of our knowledge, this paper provides the first asymptotically normally distributed estimator in such an environment.The new estimator is shown to have favorable finite sample properties in simulations. Estimating an international trade model with the massive China Customs Database and Annual Survey of Manufacturing Firms, our empirical application investigates the heterogeneity and efficiency of Chinese exporting firms. In comparison with empirical works on their French counterparts, we find the Chinese firms are of lower cost efficiency.