Thomas M. Stoker

Semiparametric estimation of index coefficients

This paper gives a solution to the problem of estimating coefficients of index models, through the estimation of the density-weighted average derivative of a general regression function. A normalized version of the density-weighted average derivative can be estimated by certain linear instrumental variables coefficients. The estimators, based on sample analogies of the product moment representation of the average derivative, are constructed using nonparametric kernel estimators of the density of the regressors. Consistent estimators of the asymptotic variance-covariance matrices of the estimators are given, and a limited Monte Carlo simulation is used to study the practical performance of the procedures. Copyright 1989 by The Econometric Society.

Investigating Smooth Multiple Regression by the Method of Average Derivatives

Abstract Let (x 1, …, xk, y) be a random vector where y denotes a response on the vector x of predictor variables. In this article we propose a technique [termed average derivative estimation (ADE)] for studying the mean response m(x) = E(y | x) through the estimation of the k vector of average derivatives δ = E(m′). The ADE procedure involves two stages: first estimate δ using an estimator , and then approximate m(x) by ), where ĝ is an estimator of the univariate regression of y on . We argue that the ADE procedure exhibits several attractive characteristics: data summarization through interpretable coefficients, graphical depiction of the possible nonlinearity between y and , and theoretical properties consistent with dimension reduction. We motivate the ADE procedure using examples of models that take the form . In this framework, δ is shown to be proportional to β and [mcirc](x) infers m(x) exactly. The focus of the procedure is on the estimator , which is based on a simple average of kernel smoother...

World Carbon Dioxide Emissions: 1950–2050

Emissions of carbon dioxide from the combustion of fossil fuels, which may contribute to long-term climate change, are projected through 2050 using reduced-form models estimated with national-level panel data for the period of 19501990. Using the same set of income and population growth assumptions as the Intergovernmental Panel on Climate Change (IPCC), we find that the IPCCs widely used emissions growth projections exhibit significant and substantial departures from the implications of historical experience. Our model employs a flexible form for income effects, along with fixed time and country effects, and we handle forecast uncertainty explicitly. We find clear evidence of an inverse U relation with a within-sample peak between carbon dioxide emissions (and energy use) per capita and per-capita income.

Heterogeneity and Aggregation

This survey covers recent solutions to aggregation problems in three application areas, consumer demand analysis, consumption growth and wealth, and labor participation and wages. Each area involves treatment of heterogeneity and nonlinearity at the individual level. Three types of heterogeneity are highlighted: heterogeneity in individual tastes, heterogeneity in income and wealth risks and heterogeneity in market participation. Work in each area is illustrated using results from empirical data. The overall aim is to show how concerns faced by empirical researchers regarding aggregation can be addressed.

Household gasoline demand in the United States

Supported by the MIT Center for Energy and Environmental Policy Research, the U.S. Dept. of Energy and the National Science Foundation.

Efficiency of Weighted Average Derivative Estimators and Index Models

Weighted average derivatives are useful parameters for semiparametric index models and nonparametric demand analysis. This paper gives efficiency results for average derivative estimators, including formulating estimators that have high efficiency. The authors derive the efficiency bound for weighted average derivatives of conditional location functionals, such as the conditional mean and median. They also derive the efficiency bound for semiparametric index models, where the location measure depends on indices or linear combinations of the regressors. The authors derive the efficient weight function when the distribution of the regressors is elliptically symmetric. They also discuss how to combine estimators with different known weight functions to achieve efficiency. Copyright 1993 by The Econometric Society.

Optimal bandwidth choice for density-weighted averages☆

Abstract This paper characterizes the optimal bandwidth value for estimating density-weighted averages, statistics that arise in semiparametric estimation methods for index models and models of selected samples based on nonparametric kernel estimators. The optimal bandwidth is derived by minimizing the leading terms of mean squared error of the density-weighted average. The optimal bandwidth formulation is developed by comparison to the optimal pointwise bandwidth of a naturally associated nonparametric estimation problem, highlighting the role of sample size and the structure of nonparametric estimation bias. The methods are illustrated by estimators of average density, density-weighted average derivatives, and conditional covariances, and bandwidth values are calculated for normal designs. A simple ‘plug-in’ estimator for the optimal bandwidth is proposed. Finally, the optimal bandwidth for estimating ratios of density-weighted averages is derived, showing that the earlier optimal formulae can be implemented directly using naturally defined ‘residual’ values.

Interpreting Aggregate Wage Growth: The Role of Labor Market Participation

A new and easily implementable framework for the empirical analysis of the relationship between aggregate and individual wages is developed. Aggregate real wages are shown to contain three important bias terms: one associated with the dispersion of individual wages, a second deriving from compositional changes in the (selected) sample of workers, and a third reflecting the distribution of working hours. Their importance for interpreting the path of aggregate wages and of the returns to education for recent experience in Britain is highlighted. A close correspondence between the estimated biases and the patterns of differences shown by aggregate wages is established. (JEL C34, E24, J31)

Bias From Censored Regressors

We study the bias that arises from using censored regressors in estimation of linear models. We present results on bias in ordinary least aquares (OLS) regression estimators with exogenous censoring and in instrumental variable (IV) estimators when the censored regressor is endogenous. Bound censoring such as top-coding results in expansion bias, or effects that are too large. Independent censoring results in bias that varies with the estimation method—attenuation bias in OLS estimators and expansion bias in IV estimators. Severe biases can result when there are several regressors and when a 0–1 variable is used in place of a continuous regressor.

Tailor-made tests for goodness of fit to semiparametric hypotheses

1. Introduction. The practice of statistical testing plays several roles inempirical research. These roles range from the careful assessment of theevidence against specific scientific hypotheses to the judgment of whetheran estimated model displays decent goodness of fit to the empirical data.The paradigmatic situation we consider is one where the investigator viewssome departures from the hypothesized model as being of primary impor-tance, with others of interest if sufficiently gross, but otherwise secondary.For instance, low-frequency departures from a signal hypothesized to beconstant might be considered of interest, even if of low amplitude; whilehigh-frequency departures are less so, unless they are of high amplitude.The optimal testing of a simple hypothesis against a simple alternativeis the cornerstone of modern statistical theory. However, there is no clearnotion of optimality for more complicated situations. The H´ajek–Le Camasymptotic theory proved that there exist strong concepts of asymptotic ef-ficiency in parametric estimation. These ideas have been extended to semi-parametric models—see [3, 14, 22]. However, there is no compelling sense ofan asymptotically optimum test, in either the parametric or the semipara-metric asymptotic theories, save for some simple one-parameter hypotheses.