## Priors for Unit Root Models

May, 1995

Tech Report

### Author(s)

Joseph B. Kadane, Ngai Hang Chan and Lara J. Wolfson

### Abstract

In the recent debate about unit roots, part of the discussion now centers on what prior is correct to use. From the perspective of subjective Bayesians, this question seems ill posed. To the extent that the likelihood is sharply peaked, the prior will not matter much. To the extent that the likelihood is flat, the posterior depends principally on the prior. A special consideration in unit roots models is that the spaces $$\rho < 1$$, $$\rho = 1$$, and $$\rho > 1$$ have very different economic implications. A better way to asses the role of the prior is to study its connections with these implications through prior elicitations.

In each of the $$\rho$$ intervals, a conjugate prior (i.e. normal-inverse gamma) is fit to the elicited opinion of an expert. Since the likelihood in each interval is normal, the posterior is of the same form as the prior (i.e. piece-wise normal-inverse gamma), so this family of priors is closed under sampling.

A modification of existing methods for the normal linear model permits elicitation of this expanded family of prior distributions. An example elicitation of a macro-economist is discussed.