HIERARCHICAL BAYESIAN ESTIMATION OF HETEROGENEOUS DYNAMIC PANEL DATA MODEL

Abstract

The estimation of static panel data model assumes homoscedastic error terms that is often violated in most economic models and when this happens, the Dynamic Panel Data Model (DPDM) is specified. The DPDM presumes correlation between lagged dependent variable and individual (unit) specific effects, resulting to heterogeneity among the units. The parameters of DPDM are usually estimated using classical approach which has no control for heterogeneity of the error terms leading to non-consistent estimates of the parameters. This study was aimed at deriving Hierarchical Bayesian Estimator (HBE) capable of handling Heterogeneous Dynamic Panel Data Model (HDPDM).

The HDPDM, , was generalised as for and where indicates that the marginal effect of on varies across the units, is vector of dependent variable, is matrix of unit specific regressors, is vector of parameters, and is vector of error terms. The HBE was derived in two stages. First Stage of Hierarchical (FSH) parameter priors were and , where and are means, is variance-covariance and is degree of freedom with independent Normal-Gamma prior. Second Stage of Hierarchical (SSH) parameter priors were and , where ( and ) are means and ( and ) are variance-covariance with independent Normal-Wishart prior. To account for heterogeneity, the FSH was derived from SSH to produce consistent estimates. Data were simulated using Markov chain Monte Carlo approach with and to obtain Posterior Estimates (PEs) at 10,000 iterations. Three experiments for the individual (N) and time (T) were considered: (20, 50), (50, 50) and (100, 15). The performance of the HBE was assessed using Numerical Standard Error (NSE). Relatively Non-informative Prior (RNP) ( = 0.04, 0.03, 0.02, 0.01) and Informative Prior (IP) ( = 25, 30, 50, 70) were examined to check for the sensitivity of priors on the PEs.

The derived HBE was The PEs of SSH for were 0.1009, 0.1326, 1.0808, 4.0607, NSE were 0.0002, 0.0008, 0.0017, 0.0068 and was 0.0007 for were 0.0154, 0.0061, 3.9674, 1.9943, NSE were 0.0002, 0.0005, 0.0014, 0.0041 and was 0.0001 for and were 0.1535, 0.1635, 2.0456, 2.8847, NSE were 0.0001, 0.0004, 0.0006, 0.0006 and was 0.0000 for . The obtained gave a constant error variance for all the parameters across the units. The option produced the least NSE, hence outperformed the other two. The PEs of FSH for , were = 0.1425, 0.1443, 0.1501, 0.1275, 0.1333. = 1.0172, 0.9123, 0.8553, 1.0172, 0.2225, = 1.5539, 1.5911, 1.5761, 1.5539, 1.4245, = 2.5193, 2.5345, 2.5005, 2.5193, 2.5231. These reflected the marginal effects of on across the units. The RNP with values of for = 1.5400, 1.5404, 1.5413, 1.5431 and = 2.5358, 2.5336, 2.5354, 2.5358, while for IP, = 1.5418, 1.5399, 1.5427, 1.5420 and = 2.6358, 2.6336, 2.6369, 2.6373. The estimated parameters with changes in values were closely identical to the pre-set and values. Thus, indicating the sensitivity of prior information on the PEs.

The Hierarchical Bayesian Estimator facilitated by suitable prior information solved the problem of heterogeneity in the dynamic panel data model. Therefore, will find useful applications in panel data economic models.