To Criticize the Critics: An Objective Bayesian Analysis of Stochastic Trends

In two recent articles, Sims (1988) and Sims and Uhlig (1988) question the value of much of the ongoing literature on unit roots and stochastic trends. They characterize the seeds of this literature as “sterile ideas,” the application of nonstationary limit theory as “wrongheaded and unenlightening” and the use of classical methods of inference as “unreasonable” and “logically unsound.” They advocate in place of classical methods an explicit Bayesian approach to inference that utilizes a flat prior on the autoregressive coefficient. DeJong and Whiteman adopt a related Bayesian approach in a group of papers (1989a,b,c) that seek to reevaluate the empirical evidence from historical economic time series. Their results appear to be conclusive in turning around the earlier, influential conclusions of Nelson and Plosser (1982) that most aggregate economic time series have stochastic trends. So far, these criticisms of unit root econometrics have gone unanswered; the assertions about the impropriety of classical methods and the superiority of flat prior Bayesian methods have been unchallenged; and the empirical reevaluation of evidence in support of stochastic trends has been left without comment.

This paper breaks that silence and offers a new perspective. We challenge the methods, the assertions and the conclusions of these articles on the Bayesian analysis of unit roots. Our approach is also Bayesian but we employ objective ignorance priors not flat priors in our analysis. Ignorance priors represent a state of ignorance about the value of a parameter and in many models are very different from flat priors. We demonstrate that in time series models flat priors do not represent ignorance but are actually informative (sic) precisely because they neglect generically available information about how autoregressive coefficients influence observed time series characteristics. Contrary to their apparent intent, flat priors unwittingly bias inferences toward stationary and iid alternatives where they do represent ignorance, as in the linear regression model. This bias helps to explain the outcome of the simulation experiments in Sims and Uhlig and the empirical results of DeJong and Whiteman.

Under flat priors and ignorance priors this paper derives posterior distributions for the parameters in autoregressive models with a deterministic trend and an arbitrary number of lags. Marginal posterior distributions are obtained by using the Laplace approximation for multivariate integrals along the lines suggested by the author (1983) in some earlier work. The bias from the use of flat priors is shown in our simulations to be substantial; and we conclude that it is unacceptably large in models with a fitted deterministic trend, for which the expected posterior probability of a stochastic trend is found to be negligible even though the true data generating mechanism has a unit root. Under ignorance priors, Bayesian inference is shown to accord more closely with the results of classical methods. An interesting outcome of our simulations and our empirical work is the bimodal Bayesian posterior, which demonstrates that Bayesian confidence sets can be disjoint, just like classical confidence intervals that are based on asymptotic theory. The paper concludes with an empirical application of our Bayesian methodology to the Nelson-Plosser series. Seven of the fourteen series show evidence of stochastic trends under ignorance priors, whereas under flat priors on the coefficients all but three of the series appear trend stationary. The latter result corresponds closely with the conclusion reached by DeJong and Whiteman (1989b) (based on truncated flat priors) that all but two of the Nelson-Plosser series are trend stationary. We argue that the DeJong-Whiteman inferences are biased toward trend stationarity through the use of flat priors and that their inferences are fragile (i.e., not robust) not only to the prior but also to the lag length chosen in the time series specification.

Keywords: Bayesian analysis, bimodal posterior density, disjoint confidence set, flat prior, fragile inference, hypergeometric function, ignorance prior, Laplace approximation, asymmetric posterior density

JEL Classification: 211