Econometrica: Nov, 1991, Volume 59, Issue 6
Understanding Unit Rooters: A Helicopter Tour
Christopher A. Sims, Harald Uhlig
While technically $p$-values should not be interpreted as probablities, they often are, and their usual asymptotic equivalence to Bayesian posterior tail probabilities provides an approximate justification for doing so. In inference about possibly nonstationary dynamic models the usual asymptotic equivalence fails, however. We show with three-dimensional graphs how it is possible that in autoregressive models the distribution of the estimator is skewed asymptotically, while the likelihood and hence the posterior pdf remains symmetric. We show that no single prior can rationalize treating $p$-values as probabilities in these models, and we display examples of the sample-dependent "priors" that would do so. We argue that these results imply at a minimum that the usual test statistics and covariance matrices for autoregressions, which characterize the likelihood shape in dynamic models just as in static regression models, should be reported without any corrections for the special unit root distribution theory, even if the corrected classical $p$-values are reported as well.