Moments of Marginal Effects with Multidimensional Unobserved Heterogeneity

Abstract

This paper considers estimation of the moments of heterogeneous marginal effects using panel data. The setting is general in three key respects: (i) time-invariant unobserved heterogeneity is unrestricted in both form and dimension, (ii) time-varying unobserved components are allowed to be nonstationary, and (iii) the structural function accommodates flexible time effects. Within this framework, we identify all integer-order moments of marginal effects, including their variance, and express them explicitly in terms of the data. Only two periods of data are required. We propose simple nonparametric estimators and establish their consistency and asymptotic normality. An application to Engel curves for food at home illustrates the methodology. Our analysis of the variance and skewness of marginal effects reveals substantial heterogeneity: while the average Engel curve is downward-sloping for all expenditure levels, some households exhibit upward-sloping sections at lower expenditure levels.