An Empirical Bayesian Approach to Cointegration Rank Selection and Test of the Present Value Model for Stock Prices
This paper provides an empirical Bayesian approach to the problem of jointly estimating the lag order and the cointegrating rank of a partially non-stationary reduced rank regression. The method employed is a variant of the Posterior Information Criterion (PIC) of Phillips and Ploberger (1994, 1995) and is similar to the asymptotic predictive odds version of the PIC criterion given in Phillips (1994). Here, we use a proper (Gaussian) prior whose hyperparameters are estimated from an initial subsample of the data. The form of the prior is suggested by the asymptotic posterior distribution of the parameters of the model, and, hence, the criterion can be interpreted as an approximate predictive odds ratio in the case where the sample size is large. Applying this procedure to the extended Campbell-Shiller data set for stock prices and dividends, we find the present value model for stock prices to be inconsistent with the data.