REGRESSOR DIMENSION REDUCTION WITH ECONOMIC CONSTRAINTS: THE EXAMPLE OF DEMAND SYSTEMS WITH MANY GOODS

Econometric Theory, 28, 2012, 10871120. doi:10.1017/S0266466612000412

STEFAN HODERLEIN AND ARTHUR LEWBEL Boston College

Microeconomic theory often yields models with multiple nonlinear equations, non-separable unobservables, nonlinear cross equation restrictions, and many potentially multicolinear covariates. We show how statistical dimension reduction techniques can be applied in models with these features. In particular, we consider estimation of derivatives of average structural functions in large consumer demand systems, which depend nonlinearly on the prices of many goods. Utility maximization imposes nonlinear cross equation constraints including Slutsky symmetry, and preference heterogeneity yields demand functions that are nonseparable in unobservables. The standard method of achieving dimension reduction in demand systems is to impose strong, empirically questionable economic restrictions such as separability. In contrast, the validity of statistical methods of dimension reduction such as principal components has not hitherto been studied in contexts like these. We derive the restrictions implied by utility maximization on dimension-reduced demand systems and characterize the implications for identication and estimation of structural marginal effects. We illustrate the results by reporting estimates of the effects of gasoline prices on the demands for many goods, without imposing any economic separability assumptions.

1. INTRODUCTION

Consider a policy question such as, What would be the effects of an increase in gasoline taxes or prices on the demands for every good consumers purchase? By standard utility theory, Marshallian demand functions for each good depend (typically nonlinearly) on the prices of every consumption good. The number of different goods that consumers buy is large, and observed price variation across consumers in most data sets is limited (because, absent price discrimination, all consumers in a given time period and region pay similar if not identical prices).

Excellent research assistance by Sonya Mihaleva is gratefully acknowledged. Address correspondence to Arthur Lewbel, Department of Economics, Boston College, 140 Commonwealth Ave., Chestnut Hill, MA 02467, USA; e-mail: [email protected].

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Cambridge University Press 2012 1087

REGRESSOR DIMENSION REDUCTION WITH ECONOMIC CONSTRAINTS: THE EXAMPLE OF DEMAND SYSTEMS WITH

MANY GOODS

1088 STEFAN HODERLEIN AND ARTHUR LEWBEL

As a result, available data sets do not have sufcient price variation to estimate demand functions containing the separate prices of all the dozens or hundreds of different goods that theory says must be considered.

This is a common problem in many microeconometric...