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Sieve Wald and QLR Inferences on Semi/nonparametric Conditional Moment Models1
, 2013
"... This paper considers inference on functionals of semi/nonparametric conditional moment restrictions with possibly nonsmooth generalized residuals, which include all of the (nonlinear) nonparametric instrumental variables (IV) as special cases. For these models it is often difficult to verify wheth ..."
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Cited by 6 (1 self)
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This paper considers inference on functionals of semi/nonparametric conditional moment restrictions with possibly nonsmooth generalized residuals, which include all of the (nonlinear) nonparametric instrumental variables (IV) as special cases. For these models it is often difficult to verify whether a functional is regular (i.e., rootn estimable) or irregular (i.e., slower than rootn estimable). We provide computationally simple, unified inference procedures that are asymptotically valid regardless of whether a functional is regular or not. We establish the following new useful results: (1) the asymptotic normality of a plugin penalized sieve minimum distance (PSMD) estimator of a (possibly irregular) functional; (2) the consistency of simple sieve variance estimators of the plugin PSMD estimator, and hence the asymptotic chisquare distribution of the sieve Wald statistic; (3) the asymptotic chisquare distribution of an optimally weighted sieve quasi likelihood ratio (QLR) test under the null hypothesis; (4) the asymptotic tight distribution of a nonoptimally weighted sieve QLR statistic under the null; (5) the consistency of generalized residual bootstrap sieve Wald and QLR tests; (6) local power properties of sieve Wald and QLR tests and of their bootstrap versions; (7) Wilks phenomenon of the sieve QLR test of hypothesis with increasing di
Adaptive Nonparametric Instrumental Variables Estimation: Empirical Choice of the Regularization Parameter,Northwestern, unpublished working paper
, 2010
"... ABSTRACT In nonparametric instrumental variables estimation, the mapping that identifies the function of interest, g say, is discontinuous and must be regularized (that is, modified) to make consistent estimation possible. The amount of modification is controlled by a regularization parameter. The ..."
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ABSTRACT In nonparametric instrumental variables estimation, the mapping that identifies the function of interest, g say, is discontinuous and must be regularized (that is, modified) to make consistent estimation possible. The amount of modification is controlled by a regularization parameter. The optimal value of this parameter depends on unknown population characteristics and cannot be calculated in applications. Theoretically justified methods for choosing the regularization parameter empirically in applications are not yet available. This paper presents such a method for use in series estimation, where the regularization parameter is the number of terms in a series approximation to g . The method does not require knowledge of the smoothness of g or of other unknown functions. It adapts to their unknown smoothness. The estimator of g based on the empirically selected regularization parameter converges in probability at a rate that is at least as fast as the asymptotically optimal rate multiplied by 1/ 2 (log ) n , where n is the sample size. The asymptotic integrated meansquare error (AIMSE) of the estimator is within a specified factor of the optimal AIMSE.
Nonparametric instrumental variable estimation under monotonicity
, 2014
"... The illposedness of the inverse problem of recovering a regression function in a nonparametric instrumental variable model leads to estimators that may suffer from a very slow, logarithmic rate of convergence. In this paper, we show that restricting the problem to models with monotone regression f ..."
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Cited by 1 (0 self)
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The illposedness of the inverse problem of recovering a regression function in a nonparametric instrumental variable model leads to estimators that may suffer from a very slow, logarithmic rate of convergence. In this paper, we show that restricting the problem to models with monotone regression functions and monotone instruments significantly weakens the illposedness of the problem. Under these two monotonicity assumptions, we establish that the constrained estimator that imposes monotonicity possesses the same asymptotic rate of convergence as the unconstrained estimator, but the finitesample behavior of the constrained estimator (in terms of error bounds) is much better than expected from the asymptotic rate of convergence when the regression function is not too steep. In the absence of the pointidentifying assumption of completeness, we also derive nontrivial identification bounds on the regression function as implied by our two monotonicity assumptions. Finally, we provide a new adaptive test of the monotone instrument assumption and a simulation study that demonstrates significant finitesample performance gains from imposing monotonicity.
Sieve Quasi Likelihood . . . Conditional Moment Models
, 2013
"... This paper considers inference on functionals of semi/nonparametric conditional moment restrictions with possibly nonsmooth generalized residuals. These models belong to the difficult (nonlinear) illposed inverse problems with unknown operators, and include all of the (nonlinear) nonparametric inst ..."
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This paper considers inference on functionals of semi/nonparametric conditional moment restrictions with possibly nonsmooth generalized residuals. These models belong to the difficult (nonlinear) illposed inverse problems with unknown operators, and include all of the (nonlinear) nonparametric instrumental variables (IV) as special cases. For these models it is generally difficult to verify whether a functional is regular (i.e., rootn estimable) or irregular (i.e., slower than rootn estimable). In this paper we provide computationally simple, unified inference procedures that are asymptotically valid regardless of whether a functional is regular or irregular. We establish the following new results: (1) the asymptotic normality of the plugin penalized sieve minimum distance (PSMD) estimators of the (possibly iregular) functionals; (2) the consistency of sieve variance estimators of the plugin PSMD estimators; (3) the asymptotic chisquare distribution of an optimally weighted sieve quasi likelihood ratio (SQLR) statistic; (4) the asymptotic tight distribution of a possibly nonoptimally weighted SQLR statistic; (5) the consistency of the nonparametric bootstrap and the weighted bootstrap (possibly nonoptimally weighted) SQLR and sieve Wald statistics, which are proved under virtually the same conditions as those for the originalsample statistics. Small simulation studies and an empirical illustration of a nonparametric quantile IV regression are presented.
On completeness and consistency in nonparametric instrumental variable models∗
, 2015
"... This paper provides a first test for the identification condition in a nonparametric instrumental variable model, known as completeness, by linking the outcome of the test to consistency of an estimator. In particular, I show that uniformly over all distributions for which the test rejects with pro ..."
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This paper provides a first test for the identification condition in a nonparametric instrumental variable model, known as completeness, by linking the outcome of the test to consistency of an estimator. In particular, I show that uniformly over all distributions for which the test rejects with probability bounded away from 0, an estimator of the structural function is consistent. This is the case for a large class of complete distributions as well as certain sequences of incomplete distributions. As a byproduct of this result, the paper makes two additional contributions. First, I present a definition of weak instruments in the nonparametric instrumental variable model, which is equivalent to the failure of a restricted version of completeness. Second, I show that the null hypothesis of weak instruments, and thus failure of a restricted version of completeness, is testable and I provide a test statistic and a bootstrap procedure to obtain the critical values. Finally, I demonstrate the finite sample properties of the tests and the estimator in Monte Carlo simulations.
OPTIMAL SUPNORM RATES, ADAPTIVITY AND INFERENCE IN NONPARAMETRIC INSTRUMENTAL VARIABLES ESTIMATION By
, 2015
"... This paper makes several contributions to the literature on the important yet difficult problem of estimating functions nonparametrically using instrumental variables. First, we derive the minimax optimal supnorm convergence rates for nonparametric instrumental variables (NPIV) estimation of the s ..."
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This paper makes several contributions to the literature on the important yet difficult problem of estimating functions nonparametrically using instrumental variables. First, we derive the minimax optimal supnorm convergence rates for nonparametric instrumental variables (NPIV) estimation of the structural function h0 and its derivatives. Second, we show that a computationally simple sieve NPIV estimator can attain the optimal supnorm rates for h0 and its derivatives when h0 is approximated via a spline or wavelet sieve. Our optimal supnorm rates surprisingly coincide with the optimal L2norm rates for severely illposed problems, and are only up to a [log(n)] (with < 1/2) factor slower than the optimal L2norm rates for mildly illposed problems. Third, we introduce a novel datadriven procedure for choosing the sieve dimension optimally. Our datadriven procedure is supnorm rateadaptive: the resulting estimator of h0 and its derivatives converge at their optimal supnorm rates even though the smoothness of h0 and the degree of illposedness of the NPIV model are unknown. Finally, we present two nontrivial applications of the supnorm rates to inference on nonlinear functionals of h0 under lowlevel conditions. The first is to derive the asymptotic normality of sieve tstatistics for exact
Orthogonal Polynomials for Seminonparametric Instrumental Variables Model∗
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Sieve Quasi Likelihood Ratio Inference on Semi/nonparametric Conditional Moment Models1
, 2013
"... This paper considers inference on functionals of semi/nonparametric conditional moment restrictions with possibly nonsmooth generalized residuals. These models belong to the difficult (nonlinear) illposed inverse problems with unknown operators, and include all of the (nonlinear) nonparametric ins ..."
Abstract
 Add to MetaCart
This paper considers inference on functionals of semi/nonparametric conditional moment restrictions with possibly nonsmooth generalized residuals. These models belong to the difficult (nonlinear) illposed inverse problems with unknown operators, and include all of the (nonlinear) nonparametric instrumental variables (IV) as special cases. For these models it is generally difficult to verify whether a functional is regular (i.e., rootn estimable) or irregular (i.e., slower than rootn estimable). In this paper we provide computationally simple, unified inference procedures that are asymptotically valid regardless of whether a functional is regular or irregular. We establish the following new results: (1) the asymptotic normality of the plugin penalized sieve minimum distance (PSMD) estimators of the (possibly iregular) functionals; (2) the consistency of sieve variance estimators of the plugin PSMD estimators; (3) the asymptotic chisquare distribution of an optimally weighted sieve quasi likelihood ratio (SQLR) statistic; (4) the asymptotic tight distribution of a possibly nonoptimally weighted SQLR statistic; (5) the consistency of the nonparametric bootstrap and the weighted bootstrap (possibly nonoptimally weighted) SQLR and sieve Wald statistics, which are proved under virtually the same conditions as those for the originalsample