Arthur Lewbel

Quadratic Engel Curves and Consumer Demand

This paper presents a model of consumer demand that is consistent with the observed expenditure patterns of individual consumers in a long time series of expenditure surveys and is also able to provide a detailed welfare analysis of shifts in relative prices. A nonparametric analysis of consumer expenditure patterns suggests that Engel curves require quadratic terms in the logarithm of expenditure. While popular models of demand such as the Translog or the Almost Ideal Demand Systems do allow flexible price responses within a theoretically coherent structure, they have expenditure share Engel curves that are linear in the logarithm of total expenditure. We derive the complete class of integrable quadratic logarithmic expenditure share systems. A specification from this class is estimated on a large pooled data set of U.K. households. Models that fail to account for Engel curvature are found to generate important distortions in the patterns of welfare losses associated with a tax increase.

Using Heteroscedasticity to Identify and Estimate Mismeasured and Endogenous Regressor Models

This article proposes a new method of obtaining identification in mismeasured regressor models, triangular systems, and simultaneous equation systems. The method may be used in applications where other sources of identification, such as instrumental variables or repeated measurements, are not available. Associated estimators take the form of two-stage least squares or generalized method of moments. Identification comes from a heteroscedastic covariance restriction that is shown to be a feature of many models of endogeneity or mismeasurement. Identification is also obtained for semiparametric partly linear models, and associated estimators are provided. Set identification bounds are derived for cases where point-identifying assumptions fail to hold. An empirical application estimating Engel curves is provided.

Constructing Instruments for Regressions with Measurement Error when no Additional Data are Available, with an Application to Patents and R&D

Constructing instruments for regressions with measurement error when no additional data are available, with an application to patents and R&D

THE INFORMATION-CONTENT OF EQUIVALENCE SCALES

It is well known that equivalence scales, defined as ratios of cost functions between demographically different households, cannot be completely identified from demand data alone. This paper derives the exact components of equivalence scales that are identifiable. It is shown that demand equations alone provide no information about equivalence scales in any one price regime, but that if equivalence scales in any one price regime were known, then demand data can identify the unique true equivalence scales in all other price regimes. Sensible identifying assumptions and implications for the appropriate construction and interpretation of equivalence scales are discussed, and some empirical tests and estimates from UK micro-data are provided.

Semiparametric qualitative response model estimation with unknown heteroscedasticity or instrumental variables

This paper provides estimators of discrete choice models, including binary, ordered, and multinomial response (choice) models. The estimators closely resemble ordinary and two stage least squares. The distribution of the models latent variable error is unknown and may be related to the regressors, e.g., the model could have errors that are heteroscedastic or correlated with regressors. The estimator does not require numerical searches, even for multinomial choice. For ordered and binary choice models the estimator is root N consistent and asymptotically normal. A consistent estimator of the conditional error distribution is also provided.

Household equivalence scales and welfare comparisons

Abstract The ratio of cost functions of different households facing the same prices and attaining the same utility level is a true cost of (demographic) characteristics index. This index is independent of a base level of income or utility (IB) if and only if it equals a ratio of household equivalence scales that are themselves IB. Applied cost index and equivalence scales specifications are usually IB. This paper describes the general restrictions on cost functions and social welfare functions required for IB scales and for their use in welfare analysis.

Tricks With Hicks: The Easi Demand System

We invent Implicit Marshallian demands, which combine desirable features of Hicksian and Marshallian demands. We propose and estimate the Exact Affine Stone Index (EASI) implicit Marshallian demand system. Like the Almost Ideal Demand (AID) system, EASI budget shares are linear in parameters given real expenditures. However, unlike the AID, EASI demands can have any rank and its Engel curves can have any shape over real expenditures. EASI error terms equal random utility parameters to account for unobserved preference heterogeneity. EASI demand functions can be estimated using GMM or three stage least squares, and, like AID, an approximate EASI model can be estimated by linear regression. (JEL D11, D12)

Comparing features of convenient estimators for binary choice models with endogenous regressors

We discuss the relative advantages and disadvantages of four types of convenient estimators of binary choice models when regressors may be endogenous or mismeasured or when errors are likely to be heteroscedastic. For example, such models arise when treatment is not randomly assigned and outcomes are binary. The estimators we compare are the two‐stage least squares linear probability model, maximum likelihood estimation, control function estimators, and special regressor methods. We specifically focus on models and associated estimators that are easy to implement. Also, for calculating choice probabilities and regressor marginal effects, we propose the average index function (AIF), which, unlike the average structural function (ASF), is always easy to estimate. On discute les avantages et desavantages de quatre types d’estimateurs pratiques de modeles de choix binaires quand les regresseurs peuvent etre endogenes ou mal mesures, ou quand les termes d’erreurs peuvent souffrir d’heteroskedasticite. Par exemple, de tels modeles emergent quand le traitement n’est pas assigne de facon aleatoire et que les resultats sont binaires. Les estimateurs qu’on compare sont le modele de probabilite lineaire estime par la methode des moindres carres a deux etapes, l’estimation de maximum de vraisemblance, les estimateurs a fonction de controle, et les methodes de regresseurs speciaux. On met l’accent specifiquement sur les modeles faciles a utiliser et les estimateurs qui s’y aboutent. De plus, pour calculer les probabilites de choix et les effets marginaux des regresseurs, on propose la fonction de l’indice moyen qui, contrairement a la fonction structurelle moyenne, est toujours facile a estimer.

A Simple Estimator for Binary Choice Models With Endogenous Regressors

This paper provides a few variants of a simple estimator for binary choice models with endogenous or mismeasured regressors, or with heteroskedastic errors, or with panel fixed effects. Unlike control function methods, which are generally only valid when endogenous regressors are continuous, the estimators proposed here can be used with limited, censored, continuous, or discrete endogenous regressors, and they allow for latent errors having heteroskedasticity of unknown form, including random coefficients. The variants of special regressor based estimators we provide are numerically trivial to implement. We illustrate these methods with an empirical application estimating migration probabilities within the US.

Identification and Estimation of Equivalence Scales under Weak Separability

This paper shows that most methods of incorporating demographic variation into separable models can be represented in a form that is identical to Barten equivalence scales, except that the scales themselves depend on the exact mix of goods that comprise each group, as well as on demographic variables. This generalization of Barten scales is shown to be more plausible than ordinary scales, can be used to increase the efficiency of demand system estimation, and can overcome Muellbauers under-identification result for cross-sectional estimation of equivalence scales.