Koop (1992a) 'Objective' Bayesian Unit Root Tests

Summary

This paper develops Bayesian posterior-odds tests for unit roots in macroeconomic time series that require minimal subjective prior input — hence "objective." Motivated by the well-documented low power of classical unit-root tests against trend-stationary alternatives, Koop computes posterior odds comparing a unit-root (difference-stationary) model to stationary and trend-stationary alternatives, using two classes of reference priors that are informative enough to yield well-defined odds but do not depend on the researcher's subjective opinions. Applied to the Nelson-Plosser and Shiller datasets (and a Monte Carlo study), the analysis shows that classical failure to reject a unit root is not evidence that a unit root is present with high probability.

Key Claims

• The right question is comparative. Classical unit-root testing only rejects or fails to reject; it cannot state the probability of the unit-root hypothesis relative to competitors. Bayesian posterior odds between difference-stationary, stationary, and trend-stationary models answer the question researchers actually care about.
• Classical tests are low-powered against trend stationarity. Following DeJong-Nankervis-Savin-Whiteman, traditional tests have very low power to distinguish a unit root from a persistent trend-stationary process, so a non-rejection is weak evidence.
• "Objective" reference priors. To avoid the burden and subjectivity of the informative priors of Koop (1991), the paper uses two classes of reference priors — informative but requiring minimal subjective input — so the resulting tests are computationally easy and reproducible across analysts.
• Posterior odds, not p-values. Model comparison is via marginal likelihoods / posterior odds (a Bayes factor weighted by model priors), which unlike a Dickey-Fuller p-value give a direct probability statement about the hypotheses.
• Empirical verdict. On Nelson-Plosser and Shiller data, the unit-root hypothesis is often far less certain than classical non-rejection suggests; the data frequently do not strongly favor difference stationarity over trend stationarity.

Concepts Introduced or Extended

• Unit Root Inference
• Bayes Factor

Entities Mentioned

• Gary Koop

Quotes

"The failure of classical procedures to reject the unit root hypothesis is not necessarily proof that a unit root is present with high probability."

"There is a clear need to develop Bayesian unit root tests that are computationally easy and do not depend on the researcher's prior opinions."

My Take

The paper is an early, practical entry in the Bayesian unit-root debate that the wiki traces through Sims (1988) and Sims-Uhlig (1991): where those make the conceptual case that flat-prior posteriors behave sensibly at $\rho=1$, Koop supplies a usable, reproducible testing recipe with reference priors so applied researchers need not elicit subjective beliefs. The lasting point is the reframing from "can we reject a unit root?" to "how do the odds of difference- versus trend-stationarity compare?" — which is exactly where the low power of classical tests stops being a nuisance and starts being the answer. The caveat is that "objective" is doing real work: reference priors are a choice, and posterior odds for non-nested trend/difference models can be sensitive to the prior on the trend and on the initial condition, so the objectivity is relative, not absolute.