Gary Chamberlain

Analysis of Covariance with Qualitative Data

In data with a group structure, incidental parameters are included to control for missing variables. Applications include longitudinal data and sibling data. In general, the joint maximum likelihood estimator of the structural parameters is not consistent as the number of groups increases, with a fixed number of observations per group. Instead a conditional likelihood function is maximized, conditional on sufficient statistics for the incidental parameters. In the logit case, a standard conditional logit program can be used. Another solution is a random effects model, in which the distribution of the incidental parameters may depend upon the exogenous variables. (This abstract was borrowed from another version of this item.)

Multivariate regression models for panel data

Abstract The paper examines the relationship between heterogeneity bias and strict exogeneity in a distributed lag regression of y on x . The relationship is very strong when x is continuous, weaker when x is discrete, and non-existent as the order of the distributed lag becomes infinite. The individual specific random variables introduce nonlinearity and heteroskedasticity; so the paper provides an appropriate framework for the estimation of multivariate linear predictors. Restrictions are imposed using a minimum distance estimator. It is generally more efficient than the conventional estimators such as quasi-maximum likelihood. There are computationally simple generalizations of two- and three-stage least squares that achieve this efficiency gain. Some of these ideas are illustrated using the sample of Young Men in the National Longitudinal Survey. The paper reports regressions on the leads and lags of variables measuring union coverage, SMSA, and region. The results indicate that the leads and lags could have been generated just by a random intercept. This gives some support for analysis of covariance type estimates; these estimates indicate a substantial heterogeneity bias in the union, SMSA, and region coefficients.

Asymptotic efficiency in estimation with conditional moment restrictions

Abstract In this paper, bounds on asymptotic efficiency are derived for a class of non-parametric models. The data are independent and identically distributed according to some unknown distribution F. There is a given function of the data and a parameter. The restrictions are that a conditional expectation of this function is zero at some point in the parameter space; this point is to be estimated. If F is assumed to be a multinomial distribution with known (finite) support, then the problem becomes parametric and the bound can be obtained from the information matrix. This bound turns out to depend only upon certain conditional moments, and not upon the support of the distribution. Since a general F can be approximated by a multinomial distribution, the multinomial bound applies to the general case.

A characterization of the distributions that imply mean—Variance utility functions☆

If there is a riskless asset, then the distribution of every portfolio is determined by its mean and variance if and only if the random returns are a linear transformation of a spherically distributed random vector. If there is no riskless asset, then the spherically distributed random vector is replaced by a random vector in which the last n − 1 components are spherically distributed conditional on the first component, which has an arbitrary distribution. If the number of assets is infinite, then there must exist random variables m, v, y, where the distribution of y conditional on m and v is standard normal, such that every portfolio is distributed as some linear combination of m and vy. If there is a riskless asset, then m has zero variance. These distributions exhibit two-fund separability even if the utility function is not concave.

Efficiency Bounds for Semiparametric Regression

Efficiency bounds for conditional moment restrictions with a nonparametric component are derived. There is a given function of the data (a random sample from a distribution F) and a parameter. The restriction is that a conditional expectation of this function is zero at some point in the parameter space. The parameter has two parts: a finite-dimensional component and a general function evaluated at a subset of the conditioning variables. An example is a regression function that is additive in parametric and nonparametric component, as arises in sample selection models. Copyright 1992 by The Econometric Society.

Asymptotic efficiency in semi-parametric models with censoring

Abstract We consider a regression model subject to bivariate censoring in which the errors are independent of the explanatory variables. Our objective is to drop the assumption of bivaraite normality and obtain an information bound on asymptotic efficiency when the error distribution is unrestricted except for smoothness and regularity conditions. This bound has simple form; if there are no exclusion restrictions on the selection equation, then positive information requires a restriction on the regression slope parameters and a continuous distribution for a component of the explanatory variables. We also consider a binary choice model under the weak assumption that the error distribution has zero median conditional on the explanatory variables. Here the semi-parametric information bound is zero. Hence, although a consistent estimator exists, it is not possible to attain covergence at rate 1 n .

Chapter 22 Panel data

Publisher Summary This chapter discusses the models that are static conditional on a latent variable. The panel aspect of the data has been primarily used to control for the latent variable. Much work needs to be done on models that incorporate uncertainty and interesting dynamics. Exploiting the martingale implications of time additive utility seems fruitful. There is, however, a potentially important distinction between time averages and cross-section averages. A time average of forecast errors over T periods should converge to zero as T→ ∞. But an average of forecast errors across N individuals surely need not converge to zero as N→ ∞; there is a common component in those errors, due to economy-wide innovations. The same point applies when considering covariances of forecast errors with variables that are in the agents information sets. If those conditioning variables are discrete, one can think of averaging over subsets of the forecast errors; as T→ ∞, these averages should converge to zero but not necessarily as N → ∞.

Econometrics and decision theory

Abstract The paper considers the role of econometrics in decision making under uncertainty. This leads to a focus on predictive distributions. The decision makers subjective distribution is only partly specified; it belongs to a set S of distributions. S can also be regarded as a set of plausible data-generating processes. Criteria are needed to evaluate procedures for constructing predictive distributions. We use risk robustness and minimax regret risk relative to S . To obtain procedures for constructing predictive distributions, we use Bayes procedures based on parametric models with approximate prior distributions. The priors are nested, with a first stage that incorporates qualitative information such as exchangeability, and a second stage that is quite diffuse. Special points in the parameter space, such as boundary points, can be accommodated with second-stage priors that have one or more mass points but are otherwise quite diffuse. An application of these ideas is presented, motivated by an individuals consumption decision. The problem is to construct a distribution for that individuals future earnings, based on his earnings history and on a longitudinal data set that provides earnings histories for a sample of individuals.

Nonparametric Applications of Bayesian Inference

This article evaluates the usefulness of a nonparametric approach to Bayesian inference by presenting two applications. Our first application considers an educational choice problem. We focus on obtaining a predictive distribution for earnings corresponding to various levels of schooling. This predictive distribution incorporates the parameter uncertainty, so that it is relevant for decision making under uncertainty in the expected utility framework of microeconomics. The second application is to quantile regression. Our point here is to examine the potential of the nonparametric framework to provide inferences without relying on asymptotic approximations. Unlike in the first application, the standard asymptotic normal approximation turns out not to be a good guide.

Education, income, and ability revisited

Abstract The paper examines two approaches to the omitted variable problem. Both of them try to correct for the omitted variable bias by specifying several equations in which the unobservable appears. The first approach assumes that the common left out variable is the only thing connecting the residuals from these equations, making it possible to extract this common factor and control for it. The second approach relies on building a model of the unobservable, by specifying observable variables which are causally related to it. A combination of these two methods is applied to the 1964 CPS-NORC veterans sample in order to evaluate the bias in income- schooling regressions caused by the omission of an unobservable initial ‘ability’ variable.