Posterior means and precisions of the coefficients in linear models with highly collinear regressors

When there is exact collinearity between regressors, their individual coefficients are not identified, but given an informative prior their Bayesian posterior means are well defined. The case of high but not exact collinearity is more complicated but similar results follow. Just as exact collinearity causes non-identification of the parameters, high collinearity can be viewed as weak identification of the parameters, which we represent, in line with the weak instrument literature, by the correlation matrix being of full rank for a finite sample size T, but converging to a rank defficient matrix as T goes to infinity. This paper examines the asymptotic behavior of the posterior mean and precision of the parameters of a linear regression model for both the cases of exactly and highly collinear regressors. We show that in both cases the posterior mean remains sensitive to the choice of prior means even if the sample size is sufficiently large, and that the precision rises at a slower rate than the sample size. In the highly collinear case, the posterior means converge to normally distributed random variables whose mean and variance depend on the priors for coefficients and precision. The distribution degenerates to fixed points for either exact collinearity or strong identification. The analysis also suggests a diagnostic statistic for the highly collinear case, which is illustrated with an empirical example.