The Bayesian Approach to Estimation of Multi-Equation Econometric Models in the Presence of Multicollinearity

Abstract

The Bayesian approach conveys information not available in the data but on prior knowledge of the subject matter, which enables one to make probability statements about the parameters of interest, while the classical approaches deals solely with the data. Several researches on the classical approaches have shown them to be sensitive to multicollinearity, a violation of one of the assumptions of multi-equation models which often plagues economic variables. Studies on the performance of the Bayesian method in this context are however limited. This study was aimed at investigating the performance of the Bayesian approach in estimating multi-equation models in the presence of multicollinearity. The purely just- and over-identified multi-equation models were considered. In both cases, the normal distribution with zero mean and large variance served as locally uniform prior for the regression coefficients. Three Bayesian Method Prior Variances (BMPV) were specified as 10, 100 and 1000 in a Monte Carlo prior variance sensitivity analysis. The Wishart distribution with zero degree of freedom served as prior distribution for inverse of error variance-covariance matrix, being its conjugate. The posterior distributions for the two models were then derived from the prior distributions and the likelihood functions as a bivariate Student-t and generalized Student-t distributions respectively. The estimates were then compared with those from the classical estimators; Ordinary Least Squares (OLS), Two stage Least Squares (2SLS), Three stage Least Squares (3SLS) and Limited Information Maximum Likelihood (LIML). Samples of sizes T=20, 40, 60, and 100 in 5000 replicates were generated based on eight specified research scenario. The Mean Squared Error (MSE) of the estimates were computed and used as evaluation criteria. The BMPV 10 produced the least MSE in the prior variance sensitivity analysis for the over-identified model, whereas for the just-identified model without multicollinearity, BMPV 100 was the smallest. The Bayesian method was better in the small sample cases T  40 than the classical estimators for  (the coefficient of the exogenous variable in the just-identified model); when T=20, MSE for BMPV 10, 100 and 1000 were 0.169, 0.168 and 0.171 respectively, whereas OLS, 2SLS, 3SLS and LIML yielded same results; 0.244, when T=40, BMPV 10, 100 and 1000 were 0.1220, 0.1272, 0.1361 respectively and 0.1262 for the classical methods. The 2SLS and 3SLS estimates of (coefficient of the endogenous explanatory variable) which were the same in the over identified model had smaller MSE than the Bayesian method; when T=20, MSE for 2SLS/3SLS =0.0280, whereas BMPV 10=0.0286, BMPV 100 = 0.0300, and BMPV 1000 = 0.033. The Bayesian method was less sensitive to multicollinearity in estimating coefficients of the correlated exogenous variables; MSE (T=20) for BMPV 10, 100, and 1000 were 0.4529, 0.5220, 0.5290 respectively, while it was 0.7492 for the classical estimators. The MSE of LIML (0.0036) was similar to that of BMPV 100 (0.0036) and BMPV 1000 (0.0036) in large sample caseT 100 for. Bayesian approach was suitable for estimating the parameters of exogenous variables in the small sample cases when the model is purely just-identified, and in over identified model in the presence of multicollinearity