The distribution of statistics testing restrictions on the coefficients in time series regressions can depend on the order of integration of the regressors. In practice, the order of integration is rarely known. We examine two conventional approaches to this problem — simply to ignore unit root problems or to use unit root pretests to determine the critical values for second-stage inference—and show that both exhibit substantial size distortions in empirically plausible situations. We then propose an alternative approach in which the second-stage critical values depend continuously on a first-stage statistic that is informative about the order of integration of the regressor. This procedure has the correct size asymptotically and good local asymptotic power.

Bayesian procedures for evaluating linear restrictions imposed by economic theory on dynamic econometric models are applied to a simple class of presentvalue models of stock prices. The procedures generate inferences that are not conditional on ancillary assumptions regarding the nature of the nonstationarity that characterizes the data. Inferences are influenced by prior views concerning nonstationarity, but these views are formally incorporated into the analysis, and alternative views are easily adopted. Viewed in light of relatively tight prior distributions that have proved useful in forecasting, the present-value model seems at odds with the data. Researchers less certain of the interaction between dividends and prices would find little reason to look beyond the present-value model.

This paper proposes residual-based tests for the null of level- and trend-stationarity, which are analogs of the LM test for an MA unit root. Asymptotic distributions of the tests are nonstandard, but they are expressed in a unified manner by expressing stochastic integrals. In addition, the tests are shown to be consistent. By expressing the distributions expressed as a function of a chi-square variable with one degree of freedom, the exact limiting probability density and cumulative distribution functions are obtained, and the exact limiting cumulative distribution functions are tabulated. Finite sample performance of the proposed tests is studied by simulation. The tests display stable size when the lag truncation number for the long-run variance estimation is chosen appropriately. But the power of the tests is generally not high at selected sample sizes. The test for the null of trend-stationarity is applied to the U.S. macroeconomic time series along with the Phillips-Perron Z(⋯) test. For some monthly and annual series, the two tests provide consistent inferential results. But for most series, the two contradictory nulls of trend-stationarity and a unit root cannot be rejected at the conventional significance levels.

The object of this paper is to report, for a simple testing problem of a unit root hypothesis, some experience regarding the numerical problems involved by using a Bayesian encompassing test, i.e., a Bayesian procedure that treats the null and the alternative hypotheses as different models, the null one and the alternative one, that share a same sample space but with different parameter spaces. Numerical procedures and efficient simulations are discussed briefly, and the numerical results so obtained are used to evaluate the meaning of the prior specification and of the empirical evidence about a unit root inference.

Asymptotic normality of the Bayesian posterior is a well-known result for stationary dynamic models or nondynamic models. This paper extends the analysis to a time series model with a possible nonstationary process. We spell out conditions under which asymptotic normality of the posterior is obtained even if the true data-generation process is a nonstationary process.