Semi-nonparametric Iv Estimation of Shape-invariant Engel

This paper studies a shape-invariant Engel curve system with endogenous total expenditure, in which the shape-invariant speci…cation involves a common shift parameter for each demographic group in a pooled system of nonparametric Engel curves. We focus on the identi…cation and estimation of both the nonparametric shapes of the Engel curves and the parametric speci…cation of the demographic scaling parameters. The identi…cation condition relates to the bounded completeness, and the estimation procedure applies the sieve minimum distance estimation of conditional moment restrictions allowing for endogeneity. We establish a new root mean squared convergence rate for the nonparametric IV regression when the endogenous regressor could have unbounded support. Root-n asymptotic normality and semiparametric e¢ ciency of the parametric components are also given under a set of ‘low-level’su¢ cient conditions. Our empirical application using the UK Family Expenditure Survey shows the importance of adjusting for endogeneity in terms of both the nonparametric curvatures and demographic parameters of systems of Engel curves.