Stephen G. Donald

Inference with Difference-in-Differences and Other Panel Data

We examine inference in panel data when the number of groups is small, as is typically the case for difference-in-differences estimation and when some variables are fixed within groups. In this case, standard asymptotics based on the number of groups going to infinity provide a poor approximation to the finite sample distribution. We show that in some cases the t-statistic is distributed as t and propose simple two-step estimators for these cases. We apply our analysis to two well-known papers. We confirm our theoretical analysis with Monte Carlo simulations.

Testing Identifiability and Specification in Instrumental Variable Models

The paper develops and explores tests, based on standard moment specifications, for the identifiability of parameters apparently estimable by instrumental variables. An asymptotic expansion under standard restrictive assumptions on the error distribution suggests a correction to the asymptotic distribution. A small sampling experiment indicates that the tests are of use.

CONSISTENT TESTS FOR STOCHASTIC DOMINANCE

Methods are proposed for testing stochastic dominance of any pre--specified order, with primary interest in the distributions of income. We consider consistent tests, that are similar to Kolmogorov--Smirnov tests, of the complete set of restrictions that relate to the various forms of stochastic dominance. For such tests, in the case of tests for stochastic dominance beyond first order, we propose and justify a variety of approaches to inference based on simulation and the bootstrap. We compare these approaches to one another and to alternative approaches based on multiple comparisons in the context of a Monte Carlo experiment and an empirical example. Copyright The Econometric Society 2003.

Choosing the Number of Instruments

Properties of instrumental variable estimators are sensitive to the choice of valid instruments, even in large cross-section applications. In this paper we address this problem by deriving simple mean-square error criteria that can be minimized to choose the instrument set. We develop these criteria for two-stage least squares (2SLS), limited information maximum likelihood (LIML), and a bias adjusted version of 2SLS (B2SLS). We give a theoretical derivation of the mean-square error and show optimality. In Monte Carlo experiments we find that the instrument choice generally yields an improvement in performance. Also, in the Angrist and Krueger (1991) returns to education application, when the instrument set is chosen in the way we consider, it turns out that both 2SLS and LIML give similar (large) returns to education.

Inferring the rank of a matrix

Abstract This paper considers methods of inference concerning the rank of matrix a π - ξ based on an asymptotically normal estimate of π and some identifiable specification for ξ. One such specification is ξ = 0, in which case one is interested in the rank of π. We first propose, and examine the properties of, a test of the hypothesis that the rank is of a given size against the alternative that the rank is larger. We then look at the problem of estimating the rank of this matrix using model selection procedures and sequential hypothesis testing. Conditions for consistency of such procedures are obtained. Some economic applications are discussed and the various methods are compared in a Monte Carlo experiment.

Piecewise Pseudo-maximum Likelihood Estimation in Empirical Models of Auctions

In applications of game theory to auctions, researchers assume that players choose strategies based upon a commo nly known distribution of the latent characteristics. Rational behavior, within an assumed class of distributions for the latent process, imposes testable restrictions upon the data generating process of th e equilibrium strategies. Unfortunately, the support of the distributi on of equilibrium strategies often depends upon all of the parameters o f the distribution of the latent characteristics, making the standard application of maximum likelihood estimation procedures inappropriat e. The authors present a piecewise pseudo-maximum likelihood estimator as well as the conditions for its consistency and its asymptotic distribution. Copyright 1993 by Economics Department of the University of Pennsylvania and the Osaka University Institute of Social and Economic Research Association.

Differences in Wage Distributions Between Canada and the United States: An Application of a Flexible Estimator of Distribution Functions in the Presence of Covariates

We construct a tractable, flexible-functional-form estimator of cumulative distribution functions for non-negative random variables which admits large numbers of covariates. The estimator adopts and extends techniques from the spell-duration literature for estimating hazard functions to distribution functions for wages, earnings, and income. We apply these methods to investigate sources of wage inequality for full-time male workers between Canada and the United States, finding that the Canadian wage density has a thinner left tail because low-educated workers have higher pay and a thinner right tail because of a lower proportion of highly-educated workers. Unions appear to play a large role in these outcomes.

Identification, Estimation, and Testing in Parametric Empirical Models of Auctions within the Independent Private Values Paradigm

Recent advances in the application of game theory to the study of auctions have spawned a growing empirical literature involving both experimental and field data. In this paper, we focus on four different mechanisms (the Dutch, English, first-price sealed-bid, and Vickrey auctions) within one of the most commonly used theoretical models (the independent private values paradigm) to investigate issues of identification, estimation, and testing in parametric structural econometric models of auctions.

Empirical likelihood estimation and consistent tests with conditional moment restrictions

This paper is about efficient estimation and consistent tests of conditional moment restrictions. We use unconditional moment restrictions based on splines or other approximating functions for this purpose. Empirical likelihood estimation is particularly appropriate for this setting, because of its relatively low bias with many moment conditions. We give conditions so that efficiency of estimators and consistency of tests is achieved as the number of restrictions grows with the sample size. We also give results for generalized empirical likelihood, generalized method of moments, and nonlinear instrumental variable estimators.

On the Asymptotic Properties of LDU-Based Tests of the Rank of a Matrix

Abstract Gill and Lewbel recently introduced a test for the rank of a matrix based on the LDU decomposition. Unfortunately, the asymptotic distribution suggested by them is incorrect except in a very limited problem. In general, the asymptotic distribution is that of a highly complicated nonlinear function of a normally distributed random vector that appears to defy useful characterization. The LDU decomposition can be used to produce a valid test asymptotically equivalent to the minimum-X 2 test.