Joel L. Horowitz

A Smoothed Maximum Score Estimator for the Binary Response Model

This paper describes a semiparametric estimator for binary response models in which there may be arbitrary heteroskedasticity of unknown form. The estimator is obtained by maximizing a smoothed version of the objective function of C. Manskis maximum score estimator. The smoothing procedure is similar to that used in kernel nonparametric density estimation. The resulting estimators rate of convergence in probability is the fastest possible under the assumptions that are made. The centered, normalized estimator is asymptotically normally distributed. Methods are given for consistently estimating the parameters of the limiting distribution and for selecting the bandwidth required by the smoothing procedure. Copyright 1992 by The Econometric Society.

Bootstrap critical values for tests based on generalized-method-of-moments estimators

Tests based on generalized-method-of-moments estimators often have true levels that differ greatly from their nominal levels when asymptotic critical values are used. This paper gives conditions under which the bootstrap provides asymptotic refinements to the critical values of t tests and the test of overidentifying restrictions. Particular attention is given to the case of dependent data. It is shown that, with such data, the bootstrap must sample blocks of data and that the formulae for the bootstrap versions of the test statistics differ from the formulae that apply with the original data. Copyright 1996 by The Econometric Society.

Asymptotic properties of bridge estimators in sparse high-dimensional regression models

We study the asymptotic properties of bridge estimators in sparse, high-dimensional, linear regression models when the number of covariates may increase to infinity with the sample size. We are particularly interested in the use of bridge estimators to distinguish between covariates whose coefficients are zero and covariates whose coefficients are nonzero. We show that under appropriate conditions, bridge estimators correctly select covariates with nonzero coefficients with probability converging to one and that the estimators of nonzero coefficients have the same asymptotic distribution that they would have if the zero coefficients were known in advance. Thus, bridge estimators have an oracle property in the sense of Fan and Li [J. Amer. Statist. Assoc. 96 (2001) 1348-1360] and Fan and Peng [Ann. Statist. 32 (2004) 928-961]. In general, the oracle property holds only if the number of covariates is smaller than the sample size. However, under a partial orthogonality condition in which the covariates of the zero coefficients are uncorrelated or weakly correlated with the covariates of nonzero coefficients, we show that marginal bridge estimators can correctly distinguish between covariates with nonzero and zero coefficients with probability converging to one even when the number of covariates is greater than the sample size.

The stability of stochastic equilibrium in a two-link transportation network

Most research and applications of network equilibrium models are based on the assumption that traffic volumes on roadways are virtually certain to be at or near their equilibrium values if the equilibrium volumes exist and are unique. However, it has long been known that this assumption can be violated in deterministic models. This paper presents an investigation of the stability of stochastic equilibrium in a two-link network. The stability of deterministic equilibrium also is discussed briefly. Equilibrium is defined to be stable if it is unique and the link volumes converge over time to their equilibrium values regardless of the initial conditions. Three models of route choice decision-making over time are formulated, and the stability of equilibrium is investigated for each. It is shown that even when equilibrium is unique, link volumes may converge to their equilibrium values, oscillate about equilibrium perpetually, or converge to values that may be considerably different from the equilibrium ones, depending on the details of the route choice decision-making process. Moreover, even when convergence of link volumes to equilibrium is assured, the convergence may be too slow to justify the standard assumption that these volumes are usually at or near their equilibrium values. When link volumes converge to non-equilibrium values, the levels at which the volumes stabilize typically depend on the initial link volumes or perceptions of travel costs. Conditions sufficient to assure convergence to equilibrium in two of the three models of route choice decision-making are presented, and these conditions are interpreted in terms of the route choice decision-making process.

An Adaptive, Rate‐Optimal Test of a Parametric Mean‐Regression Model Against a Nonparametric Alternative

We develop a new test of a parametric model of a conditional mean function against a nonparametric alternative. The test adapts to the unknown smoothness of the alternative model and is uniformly consistent against alternatives whose distance from the parametric model converges to zero at the fastest possible rate. This rate is slower than n[superscript -1/2]. Some existing tests have nontrivial power against restricted classes of alternatives whose distance from the parametric model decreases at the rate n[superscript -1/2]. There are, however, sequences of alternatives against which these tests are inconsistent and ours is consistent. As a consequence, there are alternative models for which the finite-sample power of our test greatly exceeds that of existing tests. This conclusion is illustrated by the results of some Monte Carlo experiments.

Nonparametric Analysis of Randomized Experiments with Missing Covariate and Outcome Data

Abstract Analysis of randomized experiments with missing covariate and outcome data is problematic, because the population parameters of interest are not identified unless one makes untestable assumptions about the distribution of the missing data. This article shows how population parameters can be bounded without making untestable distributional assumptions. Bounds are also derived under the assumption that covariate data are missing completely at random. In each case the bounds are sharp; they exhaust all of the information available given the data and the maintained assumptions. The bounds are illustrated with applications to data obtained from a clinical trial and data relating family structure to the probability that a youth graduates from high school.

Semiparametric methods in econometrics

1. Introduction.- 2. Single-Index Models.- 2.1 Definition of a Single-Index Model.- 2.2 Why Single-Index Models Are Useful.- 2.3 Other Approaches to Dimension Reduction.- 2.4 Identification of Single-Index Models.- 2.5 EstimatingGin a Single-Index Modei.- 2.6 Optimization Estimators ofss.- 2.7 Direct Semiparametric Estimators.- 2.8 Bandwidth Selection.- 2.9 An Empirical Example.- 3. Binary Response Models.- 3.1 Random-Coefficients Models.- 3.2 Identification.- 3.3 Estimation.- 3.4 Extensions of the Maximum Score and Smoothed Maximum Score Estimators.- 3.5 An Empirical Example.- 4. Deconvolution Problems.- 4.1 A Model of Measurement Error.- 4.2 Models for Panel Data.- 4.3 Extensions.- 4.4 An Empirical Example.- 5. Transformation Models.- 5.1 Estimation with ParametricTand NonparametricF.- 5.2 Estimation with NonparametricTand ParametricF.- 5.3 Estimation when BothTandFare Nonparametric.- 5.4 Predicting Y Conditional onX.- 5.5 An Empirical Example.- Appendix: Nonparametric Estimation.- A.1 Nonparametric Density Estimation.- A.2 Nonparametric Mean Regression.- References.

Identification and Robustness with Contaminated and Corrupted Data

Robust estimation aims at developing point estimators that are not highly sensitive to errors in data. However, the population parameters of interest are not identified under the assumptions of robust estimation, so the rationale for point estimation is not apparent. This paper shows that, under error models used in robust estimation, unidentified population parameters can often be bounded. The bounds provide information that is not available in robust estimation. For example, it is possible to bound the population mean under contaminated sampling. It is argued that estimating the bounds is more natural than attempting point estimation of unidentified parameters. Copyright 1995 by The Econometric Society.

Bootstrap Methods in Econometrics: Theory and Numerical Performance

The bootstrap is a method for estimating the distribution of an estimator or test statistic by resampling ones data. It amounts to treating the data as if they were the population for the purpose of evaluating the distribution of interest. Under mild regularity conditions, the bootstrap yields an approximation to the distribution of an estimator or test statistic that is at least as accurate as the approximation obtained from first-order asymptotic theory. Thus, the bootstrap provides a way to substitute computation for mathematical analysis if calculating the asymptotic distribution of an estimator or statistic is difficult. The maximum score estimator Manski (1975, 1985), the statistic developed by Ha..rdle et al. (1991) for testing positive- definiteness of income-effect matrices, and certain functions of time- series data (Blanchard and Quah 1989, Runkle 1987, West 1990) are examples in which evaluating the asymptotic distribution is difficult and bootstrapping has been used as an alternative.1 In fact, the bootstrap is often more accurate in finite samples than first-order asymptotic approximations but does not entail the algebraic complexity of higher-order expansions. Thus, it can provide a practical method for improving upon first-order approximations. First-order asymptotic theory often gives a poor approximation to the distributions of test statistics with the sample sizes available in applications. As a result, the nominal levels of tests based on asymptotic critical values can be very different from the true levels. The information matrix test of White(1982) is a well-known example of a test in which large finite- sample distortions of level can occur when asymptotic critical values are used (Horowitz 1994, Kennan and Neumann 1988, Orme 1990, Taylor 1987). Other illustrations are given later in this chapter. The bootstrap often provides a tractable way to reduce or eliminate finite- sample distortions of the levels of statistical tests.

Censoring of Outcomes and Regressors Due to Survey Nonresponse: Identification and estimation Using Weights and Imputations

Survey nonresponse makes identification of population statistics problematic. Except in special cases, identification is possible only if one makes untestable assumptions about the distribution of the missing data. However, non-response does not preclude identification of bounds on population statistics. This paper shows how identified bounds on unidentified population statistics can be obtained under several forms of nonresponse. Organizations conducting major surveys commonly release public-use data files that provide nonresponse weights or imputations to be used for estimating population statistics. The paper shows how to bound the asymptotic bias of estimates using weights and imputations. The results are illustrated with empirical examples based on the National Longitudinal Survey of Youth.