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Robust Linear Static Panel Data Models Using -Contamination Badi H. Baltagi, Georges Bresson, Anoop Chaturvedi, and Guy Lacroix

Paper No. 208 September 2017

mailto:ctrpol@syr.eduCENTER FOR POLICY RESEARCH Fall 2017 Leonard M. Lopoo, Director

Professor of Public Administration and International Affairs (PAIA)

Associate Directors

Margaret Austin Associate Director, Budget and Administration

John Yinger Trustee Professor of Economics (ECON) and Public Administration and International Affairs (PAIA)

Associate Director, Metropolitan Studies

SENIOR RESEARCH ASSOCIATES

Badi Baltagi, ECON Robert Bifulco, PAIA Leonard Burman, PAIA Thomas Dennison, PAIA Alfonso Flores-Lagunes, ECON Sarah Hamersma, PAIA Madonna Harrington Meyer, SOC Colleen Heflin, PAIA William Horrace, ECON Yilin Hou, PAIA

Hugo Jales, ECON Jeffrey Kubik, ECON Yoonseok Lee, ECON Amy Lutz, SOC Yingyi Ma, SOC Katherine Michelmore, PAIA Jerry Miner, ECON Shannon Monnat, SOC Jan Ondrich, ECON David Popp, PAIA

Stuart Rosenthal, ECON Michah Rothbart, PAIA Rebecca Schewe, SOC Amy Ellen Schwartz, PAIA/ECON Saba Siddiki, PAIA Perry Singleton, ECON Yulong Wang, ECON Michael Wasylenko, ECON Peter Wilcoxen, PAIA

GRADUATE ASSOCIATES

Rachel Baker, PAIA Ziqiao Chen, PAIA Yoon Jung Choi, PAIA Stephanie Coffey, ECON Henry Dyer Cruzado, PAIA Wancong Fu, ECON Jordana Gilman, PAIA Emily Gutierrez, PAIA Jeehee Han, PAIA Boqian Jiang, ECON Hyunseok Jung, ECON

Yusun Kim, PAIA Hyoung Kwon, PAIA Carolee Lantigua, PAIA Eunice Lee, PAIA Ling Li, ECON Joyce Lin, PAIA Ibrahim Malik, PAIA Christopher Markwood, PAIA Jindong Pang, ECON Christopher Rick, PAIA Laura Rodriguez-Ortiz, PAIA

Fabio Rueda de Vivero, ECON David Schwegman, PAIA Shulin Shen, ECON Iuliia Shybalkina, PAIA Mackenzie Theobald, PAIA Saied Toossi, PAIA Rebecca Wang, SOC Victoria Wright, PAIA Yimin Yi, ECON Bo Zheng, PAIA

STAFF

Joanna Bailey, Research Associate Joseph Boskovski, Maxwell X Lab Emily Henrich, Administrative Assistant Kathleen Nasto, Administrative Assistant

Candi Patterson, Computer Consultant Laura Walsh, Administrative Assistant Katrina Wingle, Administrative Specialist

Abstract

The paper develops a general Bayesian framework for robust linear static panel data models

using -contamination. A two-step approach is employed to derive the conditional type-II maximum

likelihood (ML-II) posterior distribution of the coefficients and individual effects. The ML-II posterior

means are weighted averages of the Bayes estimator under a base prior and the data-dependent empirical

Bayes estimator. Two-stage and three stage hierarchy estimators are developed and their finite sample

performance is investigated through a series of Monte Carlo experiments. These include standard

random effects as well as Mundlak-type, Chamberlain-type and Hausman-Taylor-type models. The

simulation results underscore the relatively good performance of the three-stage hierarchy estimator.

Within a single theoretical framework, our Bayesian approach encompasses a variety of specifications

while conventional methods require separate estimators for each case.

JEL No. C11, C23, C26

Keywords: -Contamination, Hyper g-Priors, Type-II Maximum Likelihood Posterior Density, Panel Data, Robust Bayesian Estimator, Three-Stage Hierarchy.

Authors: Badi H. Baltagi, Center for Policy Research, Department of Economics, Syracuse University, bbaltagi@maxwell.syr.edu; Georges Bresson, Universit Paris II, Sorbonne Universits, bresson-georges@orange.fr; Anoop Chaturvedi, University of Allahabad, anoopchaturv@gmail.com; Guy Lacroix, Dpartment dconomique, Universit Laval, Guy.Lacroix@ecn.ulaval.ca

1. Introduction

The choice of which classic panel data estimator to use in a linear static regression frameworkdepends upon the hypothesized correlation between the individual effects and the regressors. Oneusually assumes either of two polar situations. The random effects model assumes that the regressorsand the individual effects are uncorrelated. On the other hand, the fixed effects model assumes thatall the regressors are correlated with the individual effects (see e.g. Mundlak (1978); Chamberlain(1982)). An intermediate situation arises when both a subset of time-varying and time-invariant re-gressors are assumed correlated with the individual effects, in which case the instrumental variablesestimator of Hausman and Taylor (1981) is an appropriate alternative.

To a Bayesian analyst, the distinction between fixed, random and mixed models boils downto the specification of the number of stages in a given hierarchical model. While intuitively moreattractive, the Bayesian approach nevertheless rests upon hypothesized prior distributions (andpossibly on their hyperparameters). The choice of specific distributions is often made out of con-venience rather than being based upon relevant subjective information.1 Yet, it is well-known thatthe estimators can be sensitive to misspecification of the latter. Fortunately, this difficulty canbe partly circumvented by use of the robust Bayesian approach which relies upon a class of priordistributions and selects an appropriate one in a data dependent fashion. This paper studies therobustness of Bayesian panel data models to possible misspecification of the prior distribution inthe spirit of the works of Good (1965), Dempster (1977), Rubin (1977), Hill (1980), Berger (1985),Berger and Berliner (1984) and Berger and Berliner (1986) to mention a few. In particular, it isconcerned with what they call the posterior robustness which is different from the robustness a laWhite (1980). The idea is to acknowledge the prior uncertainty by specifying a class of possibleprior distributions and then investigating the robustness of the posterior distribution as the priorvaries over . Several classes of priors have been proposed in the literature but the most commonlyused one is the -contamination class. As mentioned by Berger (1985), the -contamination classof priors is particularly attractive to work with when investigating posterior robustness. The -contamination class combines the elicited prior, termed the base prior, with a contamination class ofpriors. This approach implements the Type-II maximum likelihood (ML-II) procedure for selectingthe appropriate prior distribution. The primary benefit of using such a contamination class of priordistributions is that the resulting estimator performs well even if the elicited base prior distributiondiffers from the prior distribution that is consistent with the data.

The objective of our paper is to propose a robust Bayesian approach for linear static paneldata models which departs from the standard Bayesian one in two ways. First, we consider the -contamination class of prior distributions for the model parameters (and for the individual effects).Second, both the base elicited priors and the -contamination priors use Zellner (1986)s g-priorsrather than the standard Wishart distributions for the variance-covariance matrices. The papercontributes to the panel data literature by proposing a general robust Bayesian framework whichencompasses all the above mentioned conventional frequentist specifications and their associatedestimation methods as special cases.

Section 2 gives the general framework, while Section 3 derives the Type-II maximum likeli-

1For instance, conventional proper priors in the normal linear model have been based on the conjugate Normal-Gamma family essentially because all the marginal likelihoods have closed-form solutions. Likewise, statisticianscustomarily assume that the variance-covariance matrix of the slope parameters follow a Wishart distribution becauseit is convenient from an analytical point of view.

1

hood posterior mean and the variance-covariance matrix of the coefficients in a two-stage hierarchymodel. Section 4 introduces a three-stage hierarchy with generalized hyper-g priors on the variance-covariance matrix of the individual effects. The predictive densities corresponding to the base priorsand the -contaminated priors turn out to be Gaussian and Appell hypergeometric functions, respec-tively. The main difference between the two-stage and the three-stage hierarchy models pertains tothe definition of the Bayes estimators, the empirical Bayes estimators and the weights of the ML-IIposterior means. Section 5 investigates the finite sample performance of our robust Bayesian esti-mator through extensive Monte Carlo experiments. The simulation results underscore the relativelygood performance of the three-stage hierarchy estimator as compared to the standard frequentistestimation methods. Two applications on an earnings equation and a crime model, available inthe supplementary appendix, illustrate and confirm the simulation results. Section 6 gives ourconclusion.

2. The general setup

Let the Gaussian linear mixed model be written as:

yit = Xit +W

itbi + uit , i = 1, ..., N , t = 1, ..., T, (1)

where X it is a (1K1) vector of explanatory variables including the intercept, and is a (K1 1)vector of parameters. Furthermore, let W it denote a (1K2) vector of covariates and bi a (K2 1)vector of parameters. The subscript i of bi indicates that the model allows for heterogeneityon the W( variables.) Finally, uit is a remainder term assumed to be normally distributed, i.e.uit N 0, 1 . The distribution of uit is parametrized in terms of its precision rather thanits variance 2u (= 1/) . In the statistics literature, the elements of do not differ across i and arereferred to as fixed effects whereas the bis are referred to as random effects.

2 This terminologydiffers from the one used in econometrics. In t

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