This paper considers tests for seasonal and non-seasonal serial correlation in time series and in the errors of regression models. The problem of testing for white noise against multiplicative seasonal ARMA(l,l)-ARMA(l,l) alternatives is investigated. This testing problem is non-standard due to nuisance parameters that appear under the alternative but not under the null hypothesis. The likelihood ratio (LR), sup Lagrange multiplier (LM), and exponential average LM and LR tests are considered and are shown to be asymptotically admissible for multiplicative seasonal ARMA(l,l)-ARMA(l,l) alternatives. In addition, they are shown to be consistent against all (weakly stationary strong mixing) non-white noise alternatives. Simulation results compare the tests to several tests in the literature. The exponential average test, Exp-LRinfinity, is found to be the best test overall. It performs substantially better than the Box-Pierce, Durbin-Watson, and Wallis tests.
This paper is concerned with tests for serial correlation in time series and in the errors of regression models. In particular, the nonstandard problem of testing for white noise against ARMA(1,1) alternatives is considered. Sup Lagrange multiplier (LM) and exponential average LM tests are introduced and are shown to be asymptotically admissible for ARMA(1,1) alternatives. In addition, they are shown to be consistent against all (weakly stationary strong mixing) non-white noise alternatives. Simulation results compare the tests to several tests in the literature. These results show that the Exp-LMinfinity test has very good all-around power.
This paper establishes the asymptotic admissibility of the likelihood ratio (LR) test for a general class of testing problems in which a nuisance parameter is present only under the alternative hypothesis. The paper also establishes the finite sample admissibility of the LR test for testing problems of this sort that arise in Gaussian linear regression models with known variance.
This paper offers an approach to time series modeling that attempts to reconcile classical and Bayesian methods. The central idea put forward to achieve this reconciliation is that the Bayesian approach relies implicitly on a frame of reference for the data generating mechanism that is quite different from the one that is employed in the classical approach. Differences in inferences from the two approaches are therefore to be expected unless the altered frame of reference is taken into account. We show that the new frame of reference in Bayesian inference is a consequence of a change of measure that arises naturally in the application of Bayes theorem. Our paper explores this change of measure and its consequences using martingale methods. Examples are given to illustrate its practical implications. No assumptions concerning stationarity or rates of convergence are required in the development of our asymptotic theory. Some implications for statistical testing are explored and we suggest new tests, which we call Bayes model tests, for discriminating between models. A posterior odds version of these tests is developed and shown to have good finite sample properties. This is the test that we recommend for practical use. Autoregressive models with multiple lags and deterministic trends are considered and explicit forms are given for the posterior odds tests for the presence of a unit root and for joint tests for the presence of a unit root, drift and trend.
This paper emphasizes the new conceptual framework for thinking about Bayesian methods in time series and provides illustrations of its use in some common models for possibly nonstationary time series. A sequel to the present paper develops a general and more abstract theory that will have a wider range of applications.
The Kalman filter is sued to derive updating equations for the Bayesian data density in discrete time linear regression models with stochastic regressors. The implied “Bayes model” has time varying parameters and conditionally heterogeneous error variances. A sigma-finite “Bayes model” measure is given and used to produce a new model selection criterion (PIC) and objective posterior odds tests for sharp null hypotheses like the presence of a unit root. Simulation results and an empirical application are reported. The simulations show that the new model selection criterion “PIC” works very well and is generally superior to the Schwarz criterion BIC even in stationary systems.
JEL Classification: C11, C51, C52, C53
Keywords: Kalman filter, Bayesian data density, stochastic regressors
This paper derives asymptotically optimal tests for testing problems in which a nuisance parameter exists under the alternative hypothesis but not under the null. The results of the paper are of interest, because the testing problem considered in non-standard and the classical asymptotic optimality results for the Wald, Lagrange multiplier (LM), and likelihood ratio (LR) tests do not apply. In the non-standard cases of main interest, new optimal tests are obtained and the LR test is not found to be an optimal test.
JEL Classification: C12
Keywords: Asymptotics, changepoint, nonstandard testing problem
This paper determines a class of finite sample optimal tests for the existence of a changepoint at an unknown time in a normal linear multiple regression model with known variance. Optimal tests for multiple changepoints are also derived. Power comparisons of several tests are provided based on simulations.
JEL Classification: C12
Keywords: Optimal test, Multiple changepoints, Structural change test
This paper offers a general approach to time series modeling that attempts to reconcile classical and methods. The central idea put forward to achieve reconciliation is that the Bayesian approach relies implicitly a frame of reference for the data generating mechanism that is quite different from the one that is employed in the classical approach. Differences in inferences from the two approaches are therefore to be expected unless the altered frame reference is taken into account. We show that the new frame of reference in Bayesian inference is a consequence of a change of measure that arises naturally in the application of Bayes theorem. Our paper explores this change of measure and its consequences using martingale methods. Examples are give illustrate its practical implications. No assumptions concerning stationarity or rates of convergence are required and techniques of stochastic differential geometry on manifolds are involved. Some implications for statistical testing are explored and suggest new tests, which we call Bayes model tests, for discriminating between models.
Keywords: Time series, modeling, Bayesian analysis, martingale