Nonlinear Panel Data Models

Panel or longitudinal data are becoming increasingly popular in applied work as they offer a number of advantages over pure cross-sectional or pure time-series data. A particularly useful feature is that they allow researchers to model unobserved heterogeneity at the level of the observational unit, where the latter may be an individual, a household, a firm or a country. Standard practice in the econometric literature is to model this heterogeneity as an individual-specific effect which enters additively in the model, typically assumed to be linear, that captures the statistical relationship between the dependent and the independent variables. The presence of these individual effects may cause problems in estimation. In particular in short panels, that is, in panels where the time-series dimension is of smaller order than the cross-sectional dimension, their estimation in conjunction with the other parameters of interest usually yields inconsistent estimators for both. (Notable exceptions are the static linear and the Poisson count panel data models, where estimation of the individual effects along with the finite dimensional coefficient vector yields consistent estimators of the latter.) This is the well-known incidental parameters problem (Neyman and Scott, 1948). In linear regression models, this problem may be dealt with by taking transformations of the model, such as first differences or differences from time averages (‘within transformation’), which remove the individual effect from the equation under consideration. However they do not apply to nonlinear econometric models, that is, models which are nonlinear in the parameters of interest and which include models that arise frequently in applied work, such as discrete choice models, limited dependent variable models, and duration models, among others.