Data-driven rate-optimal specification testing in regression models

We propose new data-driven smooth tests for a parametric regression function. The smoothing parameter is selected through a new criterion that favors a large smoothing parameter under the null hypothesis. The resulting test is adaptive rate-optimal and consistent against Pitman local alternatives approaching the parametric model at a rate arbitrarily close to 1/\sqrtn. Asymptotic critical values come from the standard normal distribution and the bootstrap can be used in small samples. A general formalization allows one to consider a large class of linear smoothing methods, which can be tailored for detection of additive alternatives.Comment: Published at http://dx.doi.org/10.1214/009053604000001200 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Uniform Bias Study and Bahadur Representation for Local Polynomial Estimators of the Conditional Quantile Function

This paper investigates the bias and the Bahadur representation of a local polynomial estimator of the conditional quantile function and its derivatives. The bias and Bahadur remainder term are studied uniformly with respect to the quantile level, the covariates and the smoothing parameter. The order of the local polynomial estimator can be higher that the differentiability order of the conditional quantile function. Applications of the results deal with global optimal consistency rates of the local polynomial quantile estimator, performance of random bandwidths and estimation of the conditional quantile density function. The latter allows to obtain a simple estimator of the conditional quantile function of the private values in a first price sealed bids auctions under the independent private values paradigm and risk neutrality.Bahadur representation, Conditional quantile function, Local polynomial estimation, Econometrics of auctions

DATA-DRIVEN RATE-OPTIMAL SPECIFICATION TESTING IN REGRESSION MODELS

We propose new data-driven smooth tests for a parametric regression function. The smoothing parameter is selected through a new criterion that favors a large smoothing parameter under the null hypothesis. The resulting test is adaptive rate-optimal and consistent against Pitman local alternatives approaching the parametric model at a rate arbitrarily close to 1/\sqrt{n}. Asymptotic critical values come from the standard normal distribution and bootstrap can be used in small samples. A general formalization allows to consider a large class of linear smoothing methods, which can be tailored for detection of additive alternatives.Hypothesis testing, nonparametric adaptive tests, selection methods

Adaptive Rate-optimal Detection of Small Autocorrelation Coefficients

A new test is proposed for the null of absence of serial correlation. The test uses a data-driven smoothing parameter. The resulting test statistic has a standard limit distribution under the null. The smoothing parameter is calibrated to achieve rate-optimality against several classes of alternatives. The test can detect alternatives with many small correlation coefficients that can go to zero with an optimal adaptive rate which is faster than the parametric rate. The adaptive rate-optimality against smooth alternatives of the new test is established as well. The test can also detect ARMA and local Pitman alternatives converging to the null with a rate close or equal to the parametric one. A simulation experiment and an application to monthly financial square returns illustrate the usefulness of the proposed approach.Absence of serial correlation; Data-driven nonparametric tests; Adaptive rate-optimality; Small alternatives; Time series

Design Adaptive Nearest Neighbor Regression Estimation

AbstractThis paper deals with nonparametric regression estimation under arbitrary sampling with an unknown distribution. The effect of the distribution of the design, which is a nuisance parameter, can be eliminated by conditioning. An upper bound for the conditional mean squared error of k−NN estimates leads us to consider an optimal number of neighbors, which is a random function of the sampling. The corresponding estimate can be used for nonasymptotic inference and is also consistent under a minimal recurrence condition. Some deterministic equivalents are found for the random rate of convergence of this optimal estimate, for deterministic and random designs with vanishing or diverging densities. The proposed estimate is rate optimal for standard designs

Optimal Minimax Rates for Nonparametric Specification Testing in Regression Models

In the context of testing the specification of a nonlinear parametric regression function, we adopt a nonparametric minimax approach to determine the maximum rate at which a set of smooth alternatives can approach the null hypothesis while ensuring that a test can uniformly detect any alternative in this set with some predetermined power. We show that a smooth nonparametric test has optimal asymptotic minimax properties for regular alternatives. As a by-product, we obtain the rate of the smoothing parameter that ensures rate-optimality of the test. We show that, in contrast, a class of nonsmooth tests, which includes the integrated conditional moment test of Bierens (1982, Journal of Econometrics 20, 105–134), has suboptimal asymptotic minimax properties

UNIFORM BIAS STUDY AND BAHADUR REPRESENTATION FOR LOCAL POLYNOMIAL ESTIMATORS OF THE CONDITIONAL QUANTILE FUNCTION

forthcoming in Econometric Theoryforthcoming in Econometric Theor

Multivariate local polynomial estimators: uniform boundary properties and asymptotic linear representation

The asymptotic bias and variance of a general class of local polynomial estimators of M-regression functions are studied over the whole compact support of the multivariate covariate under a minimal assumption on the support. The support assumption ensures that the vicinity of the boundary of the support will be visited by the multivariate covariate. The results show that like in the univariate case, multivariate local polynomial estimators have good bias and variance properties near the boundary. For the local polynomial regression estimator, we establish its asymptotic normality near the boundary and the usual optimal uniform convergence rate over the whole support. For local polynomial quantile regression, we establish a uniform linearization result which allows us to obtain similar results to the local polynomial regression. We demonstrate both theoretically and numerically that with our uniform results, the common practice of trimming local polynomial regression or quantile estimators to avoid “the boundary effect” is not needed

Nonparametric identification of an interdependent value model with buyer covariates from first-price auction bids

This paper introduces a version of the interdependent value model of Milgrom and Weber (1982), where the signals are given by an index gathering signal shifters observed by the econometrician and private ones specific to each bidders. The model primitives are shown to be nonparametrically identified from first-price auction bids under a testable mild rank condition. Identification holds for all possible signal values. This allows to consider a wide range of counterfactuals where this is important, as expected revenue in second-price auction. An estimation procedure is briefly discussed