Reconsidering heterogeneity in panel data estimators of the stochastic frontier model

Abstract

This paper examines several extensions of the stochastic frontier that account for unmeasured heterogeneity as well as firm inefficiency. The fixed effects model is extended to the stochastic frontier model using results that specifically employ its nonlinear specification. Based on Monte Carlo results, we find that the incidental parameters problem operates on the coefficient estimates in the fixed effects stochastic frontier model in ways that are somewhat at odds with other familiar results. We consider a special case of the random parameters model that produces a random effects model that preserves the central feature of the stochastic frontier model and accommodates heterogeneity. We then examine random parameters and latent class models. In these cases, explicit models for firm heterogeneity are built into the stochastic frontier. Comparisons with received results for these models are presented in an application to the U.S. banking industry.

Introduction

The developments reported in this paper were motivated by a study undertaken by the author with the World Health Organization based on their year 2000 World Health Report (WHR) (see Tandon et al., 2001; Hollingsworth and Wildman, 2002; Greene, 2004). The WHR study is a panel data analysis of health care outcomes in 191 countries for the years 1993–1997. A fixed effects ‘frontier’ model was fit, and countries were ranked on the basis of the Schmidt and Sickles (1984) suggested corrected effects. Readers of the study argued that with a sample as disparate as this one surely is, the ‘fixed effects’ must be picking up a great deal of unmeasured cross country heterogeneity as well as any ‘inefficiency’ in the provision of health care services. One would expect that the confounding of the two effects has the potential seriously to distort the inefficiency measures of interest in the study. Ideally, it is appropriate to model inefficiency and heterogeneity separately in the same model to segregate the two effects. The stochastic frontiers literature that deals with panel data is diffuse (and not particularly verbose) on this issue. Many of the models in common use provide little or no mechanism for disentangling these two effects.¹ Most of the received applications have effectively blended these two characteristics in a single term in the model. This paper will examine several alternative forms of the stochastic frontier model that take different approaches to incorporating heterogeneity. Not surprisingly, they produce markedly different results.

Aigner, Lovell and Schmidt (ALS) proposed the normal-half normal stochastic frontier in their pioneering work in 1977. A stream of research over the succeeding 25 years has produced many innovations in the specification and estimation of their model (see Greene, 1997 and Kumbhakar and Lovell, 2000, for recent surveys). Panel data applications have kept pace with other types of developments in the literature. Many of these estimators have been patterned on familiar fixed and random effects formulations of the linear regression model. This paper will examine several alternative approaches to modeling heterogeneity in panel data in the stochastic frontier model. We propose specifications which can isolate firm heterogeneity while better preserving the mechanism in the stochastic frontier model that produces estimates of technical or cost inefficiency.

This study is organized as follows: Section 2 will lay out the basic platform for all of the specifications of the stochastic frontier model. We will be presenting a large number of empirical applications in the text. These are based on a study of the U.S. banking industry. The data set to be used and the specific cost frontier model that will be used are also presented in Section 2. The succeeding sections will formalize and apply three classes of models, fixed effects, random effects and varying parameter models. In each case, unmeasured heterogeneity makes a different appearance in the model. Section 3 considers fixed effects estimation. This section considers two issues, the practical problem of computing the fixed effects estimator, and the bias and inconsistency of the fixed effects estimator due to the incidental parameters problem. A Monte Carlo study based on the panel from the U.S. banking industry is used to study the incidental parameters problem and its influence on inefficiency estimation. Section 4 presents results for random effects models. We first reconsider the familiar random effects model that has already appeared in the literature, observing once again that familiar approaches have forced one ‘effect’ to carry both heterogeneity and inefficiency. We then propose a modification of the random effects model which disentangles these terms. The fixed and random effects models treat heterogeneity as a firm specific additive constant. Section 5 will present two extensions of the model that allow for more general types of variation. This section will include development of a simulation based random parameters estimator that is a more flexible, general specification than the simple random effects model. We then turn to a latent class specification. Section 5 will develop the model, then apply it to the data on the banking industry considered in the preceding two sections. Finally, Bayesian estimators for fixed and random effects and for random parameters specifications have been proposed for the stochastic frontier model. We will also consider some of these specifications in Section 5. Some conclusions are drawn in Section 6.