Unobservable individual effects and efficiency gain for panel data.

• Author: Min, Chung-ki

1. INTRODUCTION

   In panel data, individual effects are unobservable and often correlated with the explanatory variables. A widely-employed approach to control for the unobservable individual effects is the within transformation which is equivalent to including dummy variables. Payne and Mohammadi (2006) control for countryspecific differences in their analysis of 26 transition economies. When eliminating the individual effects from the regression models, these approaches remove the between-units variability and use only the within-unit variability in estimating coefficients, thereby losing efficiency. However, it indicates a possibility that we can improve the efficiency if we keep at least a part of the between-units variability by imposing some restrictions on the coefficients (Hausman and Taylor, 1981).

   This study reports experiment results which show a possibility of gaining efficiency with restrictions on the coefficients. As an exploratory study, we impose a restriction that the unit-specific effects sum to zero with setting the intercept to zero. Although this restriction appears to limit many applications, it has a wide range of applications as most regression models can be transformed to satisfy this restriction. The purpose of this study is to show that some restrictions imposed on the parameters can help us improve the estimation efficiency.

   The following section explains the model and estimation methods. Section 3 presents the estimation results and evaluates the efficiency gain of the proposed method. Section 4 contains the conclusions.

2. THE MODEL AND ESTIMATION METHODS

   Consider the following regression model which includes unit-specific effects (ai ).1 For i = 1,--,M and

   t = 1, ..., T ,

   [y,sub,it] = [x.sub.it] [beta] + [[alpha].sub.i] + [u.sub.it] (1)

   where [x.sub.it] is a kxl vector of explanatory variables which vary over time and across units, and [beta] is a k x 1 vector of regression coefficients. Notice that an intercept is not included in this regression model. The unit-specific effects may be correlated with some or all of the explanatory variables. For unbiased and consistent estimation of p, we have to control for the unit-specific effects [[alpha].sub.i]. The disturbance [u.sub.it], having mean 0 and variance [[rho].sup.2.sub.u], is assumed mutually independent and uncorrelated with the explanatory variables.

   We impose a restriction on the unit-specific effects such that [summation of I][[alpha].sub.i] = 0. Although this restriction appears to limit application, it in fact does not since most regression models can be transformed to satisfy this restriction. (2) In this study we focus on the estimation and efficiency gain of this simple model while we address general cases in future work. This regression model is different from usual regression models as it requires [summation of I] [[alpha[.sub.i] = 0 without including an intercept.

   In Eq.(1), once the values of [[alpha].sub.i]'s are given, the conditional posterior density for [beta] and [[rho].sub.u] can be easily derived. Further, given [[alpha].sub.i], this posterior...