Excess heterogeneity, endogeneity and index restrictions

An outcome is determined by a structural function in which the e¤ect of some variables of interest is transmitted through a scalar index. Multiple sources of stochastic variation can appear as arguments of the structural function, but not in the index. There may be endogeneity, that is observable and unobservable variables may not be independently distributed. Conditions are provided under which there is local identi…cation of measures of the relative sensitivity of the index to variations in pairs of its possibly endogenous arguments, namely ratios of partial derivatives of the index. 1. Introduction Many questions arising in microeconometric practice lead to the use of models which include more unobservable latent variables than there are observable stochastic outcomes, that is excess heterogeneity. The latent variables often represent unobserved characteristics of individuals and of the environment in which they make decisions. The inclusion of such variables is common in, for example, models of durations (see van den Berg (2001)) such as mixed proportional hazard models, in discrete choice models (see for example Brownstone and Train (1998), Chesher and Santos Silva (2002), McFadden and Train (2000)) and in count data models (see Cameron and Trivedi (1998)). There is a large econometric literature concerned with random coef…cients models which permit this sort of excess heterogeneity. (Chow (1984)). Excess heterogeneity also arises in other cases, for example when there is measurement error and in panel data models. It is common to …nd strong restrictions imposed in models that admit excess heterogeneity. Frequently the speci…cation is fully parametric as in the mixed multinomial logit models of Brownstone and Train (1998). When parametric restrictions are I thank Whitney Newey for remarks on an earlier paper (Chesher (2002)) which stimulated this work and to Lars Nesheim for helpful discussions. I thank the Leverhulme Trust for support through grants to the Centre for Microdata Methods and Practice (CeMMAP) and the research project “Evidence, Inference and Enquiry” and the Economic and Social Research Council for support of CeMMAP since July 1st 2007 through grant RES-589-28-0001.