Abstract
We consider nonlinear moment restriction semiparametric models where both the dimension of the parameter vector and the number of restrictions are divergent with sample size and an unknown smooth function is involved. We propose an estimation method based on the sieve generalized method of moments (sieve-GMM). We establish consistency and asymptotic normality for the estimated quantities when the number of parameters increases modestly with sample size. We also consider the case where the number of potential parameters/covariates is very large, i.e., increases rapidly with sample size, but the true model exhibits sparsity. We use a penalized sieve GMM approach to select the relevant variables, and establish the oracle property of our method in this case. We also provide new results for inference. We propose several new test statistics for the over-identification and establish their large sample properties. We provide a simulation study and an application to data from the NLSY79 used by Carneiro et al. (2011).
Highlights
Introduction and examplesWe consider a class of moment restriction models where there are many Euclidean valued parameters as well as unknown infinite dimensional functional parameters
The setting includes as a special case the partial linear regression model, Robinson [28], except in our case the number of covariates in the linear part may be large i.e., increase to infinity with sample size
Before showing the asymptotic distribution under the null and the consistency under the alternative for the test statistic, we introduce some necessary assumptions
Summary
We consider a class of moment restriction models where there are many Euclidean valued parameters as well as unknown infinite dimensional functional parameters. The moment restriction model (1.1) features high dimensionality in two folds: a high dimensional Euclidean parameter (α) that shows up in a single-index form, and a zero-mean function m(·) with divergent dimension that usually represents an error term It includes an infinite dimensional unknown function g(·). G(z) can be approximated by the partial sum k−1 j=0 βj φj (z) in the norm of the space In this way, the unknown function is completely parameterized, which enables us to estimate the parameter vector α and the function g(·) in model (1.1) simultaneously. The unknown function is completely parameterized, which enables us to estimate the parameter vector α and the function g(·) in model (1.1) simultaneously This procedure avoids high level assumptions in our study. ⊗ denotes Kronecker product for matrices or vectors; := means equal by definition; Ir is the identity matrix of dimension r
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