Unit Roots: Bayesian Significance Test

Abstract

The unit root problem plays a central role in empirical applications in the time series econometric literature. However, significance tests developed under the frequentist tradition present various conceptual problems that jeopardize the power of these tests, especially for small samples. Bayesian alternatives, although having interesting interpretations and being precisely defined, experience problems due to the fact that that the hypothesis of interest in this case is sharp or precise. The Bayesian significance test used in this article, for the unit root hypothesis, is based solely on the posterior density function, without the need of imposing positive probabilities to sets of zero Lebesgue measure. Furthermore, it is conducted under strict observance of the likelihood principle. It was designed mainly for testing sharp null hypotheses and it is called FBST for Full Bayesian Significance Test.

Keywords:

• Bayesian inference
• Hypothesis testing
• Time series
• Unit roots

Mathematics Subject Classification:

• Primary 62P20
• Secondary 62F03, 62F15

Notes

See Phillips and Xiao (1998).

See Good (1983).

See (Schervish (1995), p. 436).

See the Appendix.

See Bauwens et al. (1992, p. 162) and comments about Phillips (1991a).

Our reference density is, therefore, the improper density, r(θ, σ) ∝ 1.

In the simulations described below, we combined Monte Carlo sampling with the Laplace approximation techniques described in Sec. 2 to perform the integration step.

See Bauwens et al. (1999).

See Pereira et al. (2008).

See the Appendix.