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Penalized Sieve GEL for Weighted Average Derivatives of Nonparametric Quantile IV Regressions

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Document pages: 63 pages

Abstract: This paper considers estimation and inference for a weighted averagederivative (WAD) of a nonparametric quantile instrumental variables regression(NPQIV). NPQIV is a non-separable and nonlinear ill-posed inverse problem,which might be why there is no published work on the asymptotic properties ofany estimator of its WAD. We first characterize the semiparametric efficiencybound for a WAD of a NPQIV, which, unfortunately, depends on an unknownconditional derivative operator and hence an unknown degree of ill-posedness,making it difficult to know if the information bound is singular or not. Ineither case, we propose a penalized sieve generalized empirical likelihood(GEL) estimation and inference procedure, which is based on the unconditionalWAD moment restriction and an increasing number of unconditional moments thatare implied by the conditional NPQIV restriction, where the unknown quantilefunction is approximated by a penalized sieve. Under some regularityconditions, we show that the self-normalized penalized sieve GEL estimator ofthe WAD of a NPQIV is asymptotically standard normal. We also show that thequasi likelihood ratio statistic based on the penalized sieve GEL criterion isasymptotically chi-square distributed regardless of whether or not theinformation bound is singular.

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