POSTERIOR ODDS TESTING FOR A UNIT-ROOT WITH DATA-BASED MODEL SELECTION

The Kalman filter is used to derive updating equations for the Bayesia n data density in discrete time linear regression models with stochast ic regressors. The implied ''Bayes model'' has time varying parameters and conditionally heterogeneous error variances. A sigma-finite Bayes model measure is given and used to produce a new-model-selection crit erion (PIC) and objective posterior odds tests for sharp null hypothes es like the presence of a unit root. This extends earlier work by Phil lips and Ploberger [18]. Autoregressive-moving average (ARMA) models a re considered, and a general test of trend-stationarity versus differe nce-stationarity is developed in ARMA models that allow for automatic order selection of the stochastic regressors and the degree of the det erministic trend. The tests are completely consistent in that both typ e I and type II errors tend to zero as the sample size tends to infini ty. Simulation results and an empirical application are reported. The simulations show that the PIC works very well and is generally superio r to the Schwarz BIC criterion, even in stationary systems. Empirical application of our methods to the Nelson-Plosser [11] series show that three series (unemployment, industrial production, and the money stoc k) are level- or trend-stationary. The other eleven series are found t o be stochastically nonstationary.