摘要:

AbstractIn two recent articles, Sims (1988) and Sims and Uhlig (1988/1991) question the value of much of the ongoing literature on unit roots and stochastic trends. They characterize the seeds of this literature as ‘sterile ideas’, the application of nonstationary limit theory as ‘wrongheaded and unenlightening’, and the use of classical methods of inference as ‘unreasonable’ and ‘logically unsound’. They advocate in place of classical methods an explicit Bayesian approach to inference that utilizes a flat prior on the autoregressive coefficient. DeJong and Whiteman adopt a related Bayesian approach in a group of papers (1989a,b,c) that seek to re‐evaluate the empirical evidence from historical economic time series. Their results appear to be conclusive in turning around the earlier, influential conclusions of Nelson and Plosser (1982) that most aggregate economic time series have stochastic trends. So far these criticisms of unit root econometrics have gone unanswered; the assertions about the impropriety of classical methods and the superiority of flat prior Bayesian methods have been unchallenged; and the empirical re‐evaluation of evidence in support of stochastic trends has been left without comment.This paper breaks that silence and offers a new perspective. We challenge the methods, the assertions, and the conclusions of these articles on the Bayesian analysis of unit roots. Our approach is also Bayesian but we employ what are known in the statistical literature as objective ignorance priors in our analysis. These are developed in the paper to accommodate explicitly time series models in which no stationarity assumption is made. Ignorance priors are intended to represent a state of ignorance about the value of a parameter and in many models are very different from flat priors. We demonstrate that in time series models flat priors do not represent ignorance but are actually informative (sic) precisely because they neglect generically available information about how autoregressive coefficients influence observed time series characteristics. Contrary to their apparent intent, flat priors unwittingly bias inferences towards stationary and i.i.d. alternatives where they do represent ignorance, as in the linear regression model. This bias helps to explain the outcome of the simulation experiments in Sims and Uhlig and some of the empirical results of DeJong and Whiteman.Under both flat priors and ignorance priors this paper derives posterior distributions for the parameters in autoregressive models with a deterministic trend and an arbitrary number of lags. Marginal posterior distributions are obtained by using the Laplace approximation for multivariate integrals along the lines suggested by the author (Phillips, 1983) in some earlier work. The bias towards stationary models that arises from the use of flat priors is shown in our simulations to be substantial; and we conclude that it is unacceptably large in models with a fitted deterministic trend, for which the expected posterior probability of a stochastic trend is found to be negligible even though the true data generating mechanism has a unit root. Under ignorance priors, Bayesian inference is shown to accord more closely with the results of classical methods. An interesting outcome of our simulations and our empirical work is the bimodal Bayesian posterior, which demonstrates that Bayesian confidence sets can be disjoint, just like classical confidence intervals that are based on asymptotic theory. The paper concludes with an empirical application of our Bayesian methodology to the Nelson‐Plosser series. Seven of the 14 series show evidence of stochastic trends under ignorance priors, whereas under flat priors on the coefficients all but three of the series appear trend stationary. The latter result corresponds closely with the conclusion reached by DeJong and Whiteman (1989b) (based on truncated flat priors). We argue that the DeJong‐Whiteman inferences are biased towards trend stationarity through the use of flat priors on the autoregressive coefficients, and that their inferences for some of the series (especially stock prices) are fragile (i.e. not robust) not only to the prior but also to the lag length chosen in the ti

ISSN:0883-7252    

DOI:10.1002/jae.3950060402

出版商:Wiley Subscription Services, Inc., A Wiley Company    

年代:1991

数据来源: WILEY

ISSN:0883-7252    

DOI:10.1002/jae.3950060403

出版商:Wiley Subscription Services, Inc., A Wiley Company    

年代:1991

数据来源: WILEY

ISSN:0883-7252    

DOI:10.1002/jae.3950060404

出版商:Wiley Subscription Services, Inc., A Wiley Company    

年代:1991

数据来源: WILEY

摘要:

AbstractThe paper compares, by a Monte‐Carlo study based on an AR(1) model, the performance of the flat prior and the ignorance prior suggested by Phillips. It argues that the ignorance prior gives heavy weight to values of the autoregressive parameter p higher than 1, and hence distorts the sample evidence as summarized in the likelihood function. It yields bimodal posterior distributions, with the second mode at p higher than 1, even when the true value of p is substantially less than

ISSN:0883-7252    

DOI:10.1002/jae.3950060405

出版商:Wiley Subscription Services, Inc., A Wiley Company    

年代:1991

数据来源: WILEY

ISSN:0883-7252    

DOI:10.1002/jae.3950060406

出版商:Wiley Subscription Services, Inc., A Wiley Company    

年代:1991

数据来源: WILEY

摘要:

AbstractThis paper is a comment on P. C. B. Phillips, ‘To criticise the critics: an objective Bayesian analysis of stochastic trends’ [Phillips, (1991)]. Departing from the likelihood of an univariate autoregressive model different routes that lead to a posterior odds analysis of the unit root hypothesis are explored, where the differences in routes are due to the different choices of the prior. Improper priors like the uniform and the Jeffreys prior are less suited for Bayesian inference on a sharp null hypothesis as the unit root. A proper normal prior on the mean of the process is analysed and empirical results using extended Nelson‐Plosser data are pres

ISSN:0883-7252    

DOI:10.1002/jae.3950060407

出版商:Wiley Subscription Services, Inc., A Wiley Company    

年代:1991

数据来源: WILEY

ISSN:0883-7252    

DOI:10.1002/jae.3950060408

出版商:Wiley Subscription Services, Inc., A Wiley Company    

年代:1991

数据来源: WILEY

摘要:

AbstractIn DeJong and Whiteman (1991a), we concluded that 11 of the 14 macroeconomic time‐series originally studied by Nelson and Plosser (1982) supported trend‐stationarity. Phillips (1991) criticizes this inference, claiming that our procedure is biased against integration, and that our results are sensitive to model and prior specification. However, Phillips' alternative models and priors bias his results in favour of integration; despite these biases, Phillips' own findings indicate that the data provide the greatest relative support to trend‐stationarity. This result is similar to our own (1989, 1990, 1991b) findings concerning the sensitivity of our results; the trend‐stationarity inference is remarkably

ISSN:0883-7252    

DOI:10.1002/jae.3950060409

出版商:Wiley Subscription Services, Inc., A Wiley Company    

年代:1991

数据来源: WILEY

ISSN:0883-7252    

DOI:10.1002/jae.3950060410

出版商:Wiley Subscription Services, Inc., A Wiley Company    

年代:1991

数据来源: WILEY

摘要:

AbstractThis paper provides detailed responses to the following eight discussants of my paper ‘To criticize the critics: an objective Bayesian analysis of stochastic trends’: Gary Koop and Mark Steel; Edward Learner; In‐Moo Kim and G. S. Maddala; Dale J. Poirier; Peter C. Schotman and Herman K. van Dijk; James H. Stock; David DeJong and Charles H. Whiteman; and Christopher Sims. This reply puts new emphasis on the call made in the earlier paper for objective Bayesian analysis in time‐series; it underlines the need for a new approach, especially with regard to posterior odds testing; and it draws attention to a new methodology of Bayesian analysis developed in a recent paper by Phillips and Ploberger (1991a). Some new simulations that shed light on certain comments of the discussants are provided; new empirical evidence is reported with the extended Nelson‐Plosser data supplied by Schotman and van Dijk; and the new Phillips‐Ploberger posterior odds test is given a brief empirical i

ISSN:0883-7252    

DOI:10.1002/jae.3950060411

出版商:Wiley Subscription Services, Inc., A Wiley Company    

年代:1991

数据来源: WILEY