Cross-validated SNP density estimates

Abstract

Abstract We consider cross-validation strategies for the seminonparametric (SNP) density estimator, which is a truncation (or sieve) estimator based upon a Hermite series expansion with coefficients determined by quasi-maximum likelihood. Our main focus is on the use of SNP density estimators as an adjunct to efficient method of moments (EMM) structural estimation. It is known that for this purpose a desirable truncation point occurs at the last point at which the integrated squared error (ISE) curve of the SNP density estimate declines abruptly. We study the determination of the ISE curve for iid data by means of leave-one-out cross-validation and hold-out-sample cross-validation through an examination of their performance over the Marron–Wand test suite and models related to asset pricing and auction applications. We find that both methods are informative as to the location of abrupt drops, but that neither can reliably determine the minimum of the ISE curve. We validate these findings with a Monte Carlo study. The hold-out-sample method is cheaper to compute because it requires fewer nonlinear optimizations. We consider the asymptotic justification of hold-out-sample cross-validation. For this purpose, we establish rates of convergence of the SNP estimator under the Hellinger norm that are of interest in their own right.