Semiparametric Estimation of Consumer Demand Systems in Real Expenditure

Consumer demand microdata typically exhibit a great deal of expenditure variation but not very much price variation. In this paper, we propose a semiparametric approach to the consumer demand problem in which expenditure share equations are nonparametric in the real expenditure direction and parametric (with fixed coefficients) in price directions. Here, Engel curves are unrestricted so that demands may have any rank. We also consider a 'varying coefficients' extension in which price effects depend on real expenditure. Because the demand model is derived from a model of cost, it may be restricted to satisfy integrability and used for consumer surplus calculations. Since real expenditure is not observed, but rather estimated under the model, we face a semiparametric model with a nonparametrically generated regressor. We show efficient convergence rates for parametric and nonparametric components. The estimation procedures are introduced for both cases, under integrability restrictions and without. Further we give specification tests to check these integrability restrictions. An empirical illustration with Canadian price and expenditure data shows that Engel curves display curvature which cannot be encompassed by standard parametric models. In addition, we find that although the rationality restriction of Slutsky symmetry is rejected in our fixed coefficients model, it is not rejected in the varying coefficients extension