On Jeffreys Prior when Using the Exact Likelihood Function
计算了AR(1)过程在有无常数项和时间趋势下使用精确似然函数时的Jeffreys先验，包括爆炸区域，支持了Schotman-van Dijk方法，并调和了文献中关于非平稳时间序列先验的不同观点。
In this paper, we calculate Jeffreys prior for an AR(1) process with and without a constant and a time trend when using the exact likelihood function. We show how this prior can be calculated for the explosive region, even though the unconditional variance of the process is infinite. The calculations lend additional support to the Schotman-van Dijk [6] procedure for restricting the location and the variance of the time trend coefficient. The results show that flat priors are reasonable for the nonexplosive region in an AR(1) without a constant and a time trend where the variance is known and the initial observation is zero, i.e., for the special case studied by Sims and Uhlig [7]. Differences to a flat prior analysis remain in particular for nonzero initial observations, however. For the explosive region, the unconditional prior diverges as the root diverges, supporting findings by Phillips [4]. This paper thus provides a useful perspective as well as some reconciliation for the different stands taken in the literature about priors and Bayesian inference for potentially nonstationary time series.