Recent Work Title Estimation and Inference in Panel Structure Models Permalink

This paper proposes and implements a tractable approach to detect group structure in panel data. The mechanism works by means of a panel structure model, which assumes that individuals form a number of homogeneous groups in a heterogeneous population. Within each group, the (linear) regression coe¢ cients are the same, while they may be di¤erent across di¤erent groups. The econometrician is not presumed to know the group structure. Instead, a multinomial logistic regression is used to infer which individuals belong to which groups. The model is estimated via maximum likelihood. We prove the consistency and asymptotic normality of a global MLE under the mild assumption that the time dimension is larger than the number of regressors in the linear regression. We propose a likelihood ratio test to test the null of one group against the alternative of multiple groups. Simulation studies show that the MLE performs quite well and the likelihood ratio test has good size and power properties in nite samples. JEL Classi cation: C13; C23; C51 Keywords: dynamic panel data model; group structure; logistic regression; nonregular test; parameter heterogeneity I am especially grateful to Donald Andrews, Peter Phillips, Chris Udry for their insightful comments. I would like to thank Michael Keane and Zhijie Xiao who have contributed to this paper with constructive suggestions. All errors are mine alone. Email: yisun@ucsd.edu. Address: Yixiao Sun, Department of Economics, #0508, University of California, San Diego, 9500 Gilman Dr., La Jolla, CA 92093-0508, USA. 1 Introduction Panel data techniques have become standard machinery for empirical analysts in many disciplines, including economics, accounting, nance, and marketing. In part, this is because panel data allow us to construct and test more complicated models that can not be identi ed using only cross section or time series data. In part also, panel data give more variability and more degrees of freedom, which leads to more precise estimation. Since panel data usually cover individuals coming from di¤erent backgrounds and living in di¤erent environments, it is of paramount importance to control unobserved heterogeneity in panel data modeling. One way to deal with the unobserved heterogeneity is to assume complete homogeneity in which regression parameters are the same across all individuals. Under this assumption, individual observations can be viewed as random draws from a common population. By pooling observations across all individuals, one can draw more precise inferences on population characteristics. However, a large body of applied work has tested and rejected parameter homogeneity. For example, Baltagi and Gri¢ n (1997) rejected the hypothesis that the gasoline demand elasticities across the OECD countries were equal. Haque, Pesaran and Sharma (2000) rejected the hypothesis that the savings behaviors across di¤erent countries were equal. The other way to deal with the unobserved heterogeneity is to assume complete heterogeneity in which regression parameters may be completely di¤erent for di¤erent individuals. Under this assumption, the number of parameters goes to in nity as the number of individuals approaches in nity. This leads to the incidental parameter problem in that maximum likelihood estimators of population-speci c parameters may be inconsistent for a xed time dimension (c.f. Neyman and Scott (1948), Lancaster (2000)). When the time dimension is small, it is also very di¢ cult to estimate individual-speci c parameters accurately. For example, Baltagi and Gri¢ n (1997) found that completely heterogeneous models led to very imprecise parameter estimates, which in some cases had the wrong sign. In addition, the assumption of complete heterogeneity may render the underlying empirical analysis meaningless. In an empirical investigation of international growth and convergence, Lee, Pesaran and Smith (1997, 1998) allowed complete heterogeneity in not only the speed of convergence but also the steady state growth rate. However, as pointed out by Islam (1998), the extension to allow varying growth rates runs the risk of robbing the concept of convergence of any real economic meaning. Instead of assuming complete heterogeneity or homogeneity, we set up a new model of an intermediate form, in which individuals form a number of homogeneous groups in a heterogeneous population. Within each group, the regression coe¢ cients are the same, while they may be di¤erent across di¤erent groups. The new model does not assume that the group structure is known a priori. Instead, it employs a multinomial logistic regression to model the membership probabilities. The model thus consists of a set of linear response patterns relating the dependent variable to explanatory variables and a logistic regression that classi es these response patterns. From an empirical Bayes perspective, the logistic regression provides prior probabilities that individuals belong to a particular group, whereas the linear response patterns provide information to update the priors in order to obtain posterior probabilities. Based on the posterior probabilities, an individual is assigned to the group of which it is most likely to be a member. The underlying structure can thus be recovered from the posterior probabilities.