3,736 research outputs found
Optimal Uniform Convergence Rates for Sieve Nonparametric Instrumental Variables Regression
We study the problem of nonparametric regression when the regressor is
endogenous, which is an important nonparametric instrumental variables (NPIV)
regression in econometrics and a difficult ill-posed inverse problem with
unknown operator in statistics. We first establish a general upper bound on the
sup-norm (uniform) convergence rate of a sieve estimator, allowing for
endogenous regressors and weakly dependent data. This result leads to the
optimal sup-norm convergence rates for spline and wavelet least squares
regression estimators under weakly dependent data and heavy-tailed error terms.
This upper bound also yields the sup-norm convergence rates for sieve NPIV
estimators under i.i.d. data: the rates coincide with the known optimal
-norm rates for severely ill-posed problems, and are power of
slower than the optimal -norm rates for mildly ill-posed problems. We then
establish the minimax risk lower bound in sup-norm loss, which coincides with
our upper bounds on sup-norm rates for the spline and wavelet sieve NPIV
estimators. This sup-norm rate optimality provides another justification for
the wide application of sieve NPIV estimators. Useful results on
weakly-dependent random matrices are also provided
Optimal Uniform Convergence Rates and Asymptotic Normality for Series Estimators Under Weak Dependence and Weak Conditions
We show that spline and wavelet series regression estimators for weakly
dependent regressors attain the optimal uniform (i.e. sup-norm) convergence
rate of Stone (1982), where is the number of
regressors and is the smoothness of the regression function. The optimal
rate is achieved even for heavy-tailed martingale difference errors with finite
th absolute moment for . We also establish the asymptotic
normality of t statistics for possibly nonlinear, irregular functionals of the
conditional mean function under weak conditions. The results are proved by
deriving a new exponential inequality for sums of weakly dependent random
matrices, which is of independent interest.Comment: forthcoming in Journal of Econometric
On rate optimality for ill-posed inverse problems in econometrics
In this paper, we clarify the relations between the existing sets of
regularity conditions for convergence rates of nonparametric indirect
regression (NPIR) and nonparametric instrumental variables (NPIV) regression
models. We establish minimax risk lower bounds in mean integrated squared error
loss for the NPIR and the NPIV models under two basic regularity conditions
that allow for both mildly ill-posed and severely ill-posed cases. We show that
both a simple projection estimator for the NPIR model, and a sieve minimum
distance estimator for the NPIV model, can achieve the minimax risk lower
bounds, and are rate-optimal uniformly over a large class of structure
functions, allowing for mildly ill-posed and severely ill-posed cases.Comment: 27 page
Optimal Sup-norm Rates and Uniform Inference on Nonlinear Functionals of Nonparametric IV Regression
This paper makes several important contributions to the literature about
nonparametric instrumental variables (NPIV) estimation and inference on a
structural function and its functionals. First, we derive sup-norm
convergence rates for computationally simple sieve NPIV (series 2SLS)
estimators of and its derivatives. Second, we derive a lower bound that
describes the best possible (minimax) sup-norm rates of estimating and
its derivatives, and show that the sieve NPIV estimator can attain the minimax
rates when is approximated via a spline or wavelet sieve. Our optimal
sup-norm rates surprisingly coincide with the optimal root-mean-squared rates
for severely ill-posed problems, and are only a logarithmic factor slower than
the optimal root-mean-squared rates for mildly ill-posed problems. Third, we
use our sup-norm rates to establish the uniform Gaussian process strong
approximations and the score bootstrap uniform confidence bands (UCBs) for
collections of nonlinear functionals of under primitive conditions,
allowing for mildly and severely ill-posed problems. Fourth, as applications,
we obtain the first asymptotic pointwise and uniform inference results for
plug-in sieve t-statistics of exact consumer surplus (CS) and deadweight loss
(DL) welfare functionals under low-level conditions when demand is estimated
via sieve NPIV. Empiricists could read our real data application of UCBs for
exact CS and DL functionals of gasoline demand that reveals interesting
patterns and is applicable to other markets.Comment: This paper is a major extension of Sections 2 and 3 of our Cowles
Foundation Discussion Paper CFDP1923, Cemmap Working Paper CWP56/13 and arXiv
preprint arXiv:1311.0412 [math.ST]. Section 3 of the previous version of this
paper (dealing with data-driven choice of sieve dimension) is currently being
revised as a separate pape
High dimensional generalized empirical likelihood for moment restrictions with dependent data
This paper considers the maximum generalized empirical likelihood (GEL)
estimation and inference on parameters identified by high dimensional moment
restrictions with weakly dependent data when the dimensions of the moment
restrictions and the parameters diverge along with the sample size. The
consistency with rates and the asymptotic normality of the GEL estimator are
obtained by properly restricting the growth rates of the dimensions of the
parameters and the moment restrictions, as well as the degree of data
dependence. It is shown that even in the high dimensional time series setting,
the GEL ratio can still behave like a chi-square random variable
asymptotically. A consistent test for the over-identification is proposed. A
penalized GEL method is also provided for estimation under sparsity setting
Efficient Estimation of Semiparametric Conditional Moment Models with Possibly Nonsmooth Residuals
This paper considers semiparametric efficient estimation of conditional moment models with possibly nonsmooth residuals in unknown parametric components (theta) and unknown functions (h) of endogenous variables. We show that: (1) the penalized sieve minimum distance (PSMD) estimator (theta\hat,h\hat) can simultaneously achieve root-n asymptotic normality of theta\hat and nonparametric optimal convergence rate of h\hat, allowing for noncompact function parameter spaces; (2) a simple weighted bootstrap procedure consistently estimates the limiting distribution of the PSMD theta\hat; (3) the semiparametric efficiency bound formula of Ai and Chen (2003) remains valid for conditional models with nonsmooth residuals, and the optimally weighted PSMD estimator achieves the bound; (4) the centered, profiled optimally weighted PSMD criterion is asymptotically chi-square distributed. We illustrate our theories using a partially linear quantile instrumental variables (IV) regression, a Monte Carlo study, and an empirical estimation of the shape-invariant quantile IV Engel curves.Penalized sieve minimum distance, Nonsmooth generalized residuals, Nonlinear nonparametric endogeneity, Weighted bootstrap, Semiparametric efficiency, Confidence region, Partially linear quantile IV regression, Shape-invariant quantile IV Engel curves
Efficient estimation of semiparametric conditional moment models with possibly nonsmooth residuals
This paper considers semiparametric efficient estimation of conditional moment models with possibly nonsmooth residuals in unknown parametric components (Ī) and unknown functions (h)of endogenous variables. We show that: (1) the penalized sieve minimum distance(PSMD) estimator (ĖĪ, Ėh) can simultaneously achieve root-n asymptotic normality of ĖĪ and nonparametric optimal convergence rate of Ėh, allowing for noncompact function parameter spaces; (2) a simple weighted bootstrap procedure consistently estimates the limiting distribution of the PSMD ĖĪ; (3) the semiparametric efficiency bound formula of Ai and Chen (2003) remains valid for conditional models with nonsmooth residuals, and the optimally weighted PSMD estimator achieves the bound; (4) the centered, profiled optimally weighted PSMD criterion is asymptotically chi-square distributed. We illustrate our theories using a partially linear quantile instrumental variables (IV) regression, a Monte Carlo study, and an empirical estimation of the shape-invariant quantile IV Engel curves. This is an updated version of CWP09/08.
Identification and Inference of Nonlinear Models Using Two Samples with Arbitrary Measurement Errors
This paper considers identification and inference of a general latent nonlinear model using two samples, where a covariate contains arbitrary measurement errors in both samples, and neither sample contains an accurate measurement of the corresponding true variable. The primary sample consists of some dependent variables, some error-free covariates and an error-ridden covariate, where the measurement error has unknown distribution and could be arbitrarily correlated with the latent true values. The auxiliary sample consists of another noisy measurement of the mismeasured covariate and some error-free covariates. We first show that a general latent nonlinear model is nonparametrically identified using the two samples when both could have nonclassical errors, with no requirement of instrumental variables nor independence between the two samples. When the two samples are independent and the latent nonlinear model is parameterized, we propose sieve quasi maximum likelihood estimation (MLE) for the parameter of interest, and establish its root-n consistency and asymptotic normality under possible misspecification, and its semiparametric efficiency under correct specification. We also provide a sieve likelihood ratio model selection test to compare two possibly misspecified parametric latent models. A small Monte Carlo simulation and an empirical example are presented.Data combination, Nonlinear errors-in-variables model, Nonclassical measurement error, Nonparametric identification, Misspecified parametric latent model, Sieve likelihood estimation and inference
Semiparametric efficiency bound for models of sequential moment restrictions containing unknown functions
This paper computes the semiparametric efficiency bound for finite dimensional parameters identified by models of sequential moment restrictions containing unknown functions. Our results extend those of Chamberlain (1992b) and Ai and Chen (2003) for semiparametric conditional moment restriction models with identical information sets to the case of nested information sets, and those of Chamberlain (1992a) and Brown and Newey (1998) for models of sequential moment restrictions without unknown functions to cases with unknown functions of possibly endogenous variables. Our bound results are applicable to semiparametric panel data models and semiparametric two stage plug-in problems. As an example, we compute the efficiency bound for a weighted average derivative of a nonparametric instrumental variables (IV) regression, and find that the simple plug-in estimator is not efficient. Finally, we present an optimally weighted, orthogonalized, sieve minimum distance estimator that achieves the semiparametric efficiency bound.
On rate optimality for ill-posed inverse problems in econometrics
In this paper, we clarify the relations between the existing sets of regularity conditions for convergence rates of nonparametric indirect regression (NPIR) and nonparametric instrumental variables (NPIV) regression models. We establish minimax risk lower bounds in mean integrated squared error loss for the NPIR and the NPIV models under two basic regularity conditions that allow for both mildly ill-posed and severely ill-posed cases.We show that both a simple projection estimator for the NPIR model, and a sieve minimum distance estimator for the NPIV model,can achieve the minimax risk lower bounds, and are rate-optimal uniformly over a large class of structure functions, allowing for mildly ill-posed and severely ill-posed cases.
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