Bayesian approaches to the ‘unit root’ problem: A comment

Peter Phillips has produced a most provocative and useful article that will, I suspect, stimulate further research and debate both on unit roots in macroeconomic time-series and, more generally, on Bayesian techniques in time-series econometrics. Phillips' main results are that the inferences drawn from Bayesian analyses of unit roots are highly sensitive to the choice of the prior and, when the prior is chosen by a Jeffreys ignorance criterion, the evidence against the unit root hypothesis is much weaker than found in flat-prior Bayesian work. I am largely in sympathy with both conclusions and have no important criticism of Phillips' article. I also applaud his willingness to examine previous Bayesian analyses of the unit root problem that have produced considerable confusion among empirical macroeconomists. These remarks therefore raise some additional issues not emphasized by Phillips concerning the papers by, in Sims and Uhligs' (1988) terminology, the Bayesian unit rooters (Sims, 1988; Sims and Uhlig 1988; DeJong and Whiteman, 1989a,b,c; and Schotman and van Dijk, 1989). My argument has three main points. First, as Phillips aptly points out, from the classical (frequentist) perspective the flat-prior analysis of unit roots has some unsettling features, such as severely biased estimators and interval estimates with grossly incorrect asymptotic confidence levels. These puzzling results arise not just because of choice of priors, but more fundamentally because the classical and Bayesian techniques answer different questions. Second, as Sims (1988) and Sims and Uhlig (1988) emphasized, because of this discrepancy between the Bayesian and classical results, researchers must take a stand on whether they are classical or Bayesian statisticians. My personal view is that the classical framework is more satisfying and useful, for two reasons: I find the axiomatic development based on the likelihood principle uncompelling; and the Bayesian paradigm does not provide a useful way to communicate information to scientific audiences. Phillips and the Bayesian unit rooters provide one of the best examples of this second point: their disagreement over priors (of which there appears to be no ready resolution) leads to different posteriors, quite different results, and a general failure in information transfer. These arguments are made in the second section of this discussion. Third, one of Phillips' key arguments for his priors is that they come closest to producing classical confidence sets and produce unbiased (in a frequentist sense) point estimates. This raises the question: why adopt the artifice of a random coefficient and the additional burden of constructing a prior, if the goal is simply to produce results close to the classical ones? Why not simply use classical techniques instead? This is, as it turns out, not particularly difficult to do using recent research by Phillips and others in this area, and I discuss this in Section 3.