Bayesian estimation of simultaneous equation model with lagged endogenous variables and first order serially correlated errors

Abstract

Most simultaneous equation models involve the inclusion of lagged endogenous and/or exogenous variables and sometimes it may be misleading to assume that the errors are normally distributed when in reality they exhibit functional formsthat are not normal especially in practical situations. The classical methods of estimating parameters of simultaneous equation models are usually affected by the presence of autocorrelation among the error terms. Unfortunately, in practice the form of correlation between the pairs of the random deviates is unknown.In this paper classical and Bayesian methods for the estimation of simultaneous equation model withlagged endogenous variables and first order serially correlated errors are considered. The smallsample properties of the methods at different levels of correlation for ρ = 0.2, 0.5 and 0.8are compared.Better parameter estimates were produced by the Bayesian estimator with smaller standard errors compared to the classical method. The standard deviations of the Bayesian estimator are consistently better than those of the OLS estimator for the sample sizes considered. For example, the standard deviations of the Bayesian for b₁₄ (the coefficient of the lagged endogenous variable,y _1t-1) when ρ = 0.2 at N = 10, 15, 20 and 25 were 0.07712781, 0.05433923, 0.03230012 and 0.03177252 respectively while those of OLS were 0.0784732, 0.4718914, 0.05701936 and 1.31422868. However, when ρ = 0.8, the standard deviations were 0.0548055, 0.03860254, 0.02572899 and 0.02126175 for Bayesian and 0.0562190, 0.03882345, 0.053676 and 0.0315632 for OLS. Interestingly, notice that even at high correlation level, the estimates produced by the Bayesian method are closer to the parameter values and the standard deviations decrease as the sample size increases. Hence, the Bayesian estimation method might be a better choice when lagged endogenous variables are included in a simultaneous equation model with auto-correlated disturbances since it appeared to give better results compared to the classical approach.

Keywords: Bayesian estimation, Lagged endogenous variables, Simultaneous equations, Monte-Carlo Simulation, First-order autoregressive process.