Confidence Distributions for the Autoregressive Parameter

Abstract

The notion of confidence distributions is applied to inference about the parameter in a simple autoregressive model, allowing the parameter to take the value one. This makes it possible to compare to asymptotic approximations in both the stationary and the nonstationary cases at the same time. The main point, however, is to compare to a Bayesian analysis of the same problem. A noninformative prior for a parameter, in the sense of Jeffreys, is given as the ratio of the confidence density and the likelihood. In this way, the similarity between the confidence and noninformative Bayesian frameworks is exploited. It is shown that, in the stationary case, asymptotically the so induced prior is flat. However, if a unit parameter is allowed, the induced prior has to have a spike at one of some size. Simulation studies and two empirical examples illustrate the ideas.

Keywords:

• Bootstrap/resampling
• Inference
• Time series

Supplemental Materials

Three supplemental appendices are given. Appendix 1 gives details on the simulations and numerical calculations that lead to the figures. In appendix 2, a proof of Proposition 1 is found. Finally, Appendix 3 provides some extra figures with simulation results, and comments on these figures.

Acknowledgments

I am grateful to Svante Janson and Shaobo Jin for helpful discussions and comments. I also thank the anonymous referees, the associate editor and the editor for careful reading and comments that greatly helped to improve the article.