Cointegration and unit root tests: A fully Bayesian approach

To perform statistical inference for time series, one should be able to assess if they present deterministic or stochastic trends. For univariate analysis one way to detect stochastic trends is to test if the series has unit roots, and for multivariate studies it is often relevant to search for stationary linear relationships between the series, or if they cointegrate. The main goal of this article is to briefly review the shortcomings of unit root and cointegration tests proposed by the Bayesian approach of statistical inference and to show how they can be overcome by the fully Bayesian significance test (FBST), a procedure designed to test sharp or precise hypothesis. We will compare its performance with the most used frequentist alternatives, namely, the Augmented Dickey-Fuller for unit roots and the maximum eigenvalue test for cointegration. Keywords: Time series; Bayesian inference; Hypothesis testing; Unit root; Cointegration

Bayesian Model Averaging and Identification of Structural Breaks in Time Series

Bayesian model averaging is used for testing for multiple break points in uni- variate series using conjugate normal-gamma priors. This approach can test for the number of structural breaks and produce posterior probabilities for a break at each point in time. Results are averaged over speciÖcations including: station- ary; stationary around trend; and, unit root models, each containing di§ erent types and numbers of breaks and di§ erent lag lengths. The procedures are used to test for structural breaks on 14 annual macroeconomic series and 11 natural resource price series. The results indicate that there are structural breaks in al l of the natural resource series and most of the macroeconomic series. Many of the series had multiple breaks. Our Öndings regarding the existence of unit roots, having al lowed for structural breaks in the data, are largely consistent with previous work.Bayesian Model Averaging, Structural Breaks, Unit Root, Macro- economic Data, Natural Resource data

Bayesian Conditional Cointegration

Cointegration is an important topic for time-series, and describes a relationship between two series in which a linear combination is stationary. Classically, the test for cointegration is based on a two stage process in which first the linear relation between the series is estimated by Ordinary Least Squares. Subsequently a unit root test is performed on the residuals. A well-known deficiency of this classical approach is that it can lead to erroneous conclusions about the presence of cointegration. As an alternative, we present a framework for estimating whether cointegration exists using Bayesian inference which is empirically superior to the classical approach. Finally, we apply our technique to model segmented cointegration in which cointegration may exist only for limited time. In contrast to previous approaches our model makes no restriction on the number of possible cointegration segments.Comment: Appears in Proceedings of the 29th International Conference on Machine Learning (ICML 2012

Time Series Modeling with a Bayesian Frame of Reference: Concepts, Illustrations and Asymptotics

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.Autoregression, Bayes model, Bayes measure, Bayes test, Bayesian inference, data density process, Deleans exponential, exponential Bayes measure, likelihood, martingale, posterior process, prior density, quadratic variation process, stochastic differential equation, unit root

A New Bayesian Unit Root Test in Stochastic Volatility Models

A new posterior odds analysis is proposed to test for a unit root in volatility dynamics in the context of stochastic volatility models. This analysis extends the Bayesian unit root test of So and Li (1999, Journal of Business Economic Statistics) in two important ways. First, a numerically more stable algorithm is introduced to compute the Bayes factor, taking into account the special structure of the competing models. Owing to its numerical stability, the algorithm overcomes the problem of diverged “size ” in the marginal likelihood approach. Second, to improve the “power ” of the unit root test, a mixed prior specification with random weights is employed. It is shown that the posterior odds ratio is the by-product of Bayesian estimation and can be easily computed by MCMC methods. A simulation study examines the “size” and “power” performances of the new method. An empirical study, based on time series data covering the subprime crisis, reveals some interesting results

Improving the accuracy of asset price bubble start and end date estimators

Recent research has proposed using recursive right-tailed unit root tests to date the start and end of asset price bubbles. In this paper an alternative approach is proposed that utilises model-based minimum sum of squared residuals estimators combined with Bayesian Information Criterion model selection. Conditional on the presence of a bubble, the dating procedures suggested are shown to offer consistent estimation of the start and end dates of a fixed magnitude bubble, and can also be used to distinguish between different types of bubble process, i.e. a bubble that does or does not end in collapse, or a bubble that is ongoing at the end of the sample. Monte Carlo simulations show that the proposed dating approach out-performs the recursive unit root test methods for dating periods of explosive autoregressive behaviour in finite samples, particularly in terms of accurate identification of a bubble's end point. An empirical application involving Nasdaq stock prices is discussed

Time Series Modeling with a Bayesian Frame of Reference: Concepts, Illustrations and Asymptotics

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

Bayesian Analysis of Stochastic Trends and Seasonality

A fully Bayesian analysis of seasonal and nonseasonal forms of nonstationarity is presented. The thesis consists of three parts, which are structured as separate research articles. In the first paper a Bayesian approach to model selection in testing regressions for a zero frequency unit root with multiple structural breaks is proposed. For this purpose the number of breaks, the corresponding break dates as well as the number of autoregressive lags are treated as model indicators, whose posterior distributions are computed using a hybrid Markov chain Monte Carlo (MCMC) approach that allows to generate random draws from parameter spaces of varying dimension. The second part of the thesis is devoted to seasonal forms of nonstationarity. Here a Bayesian testing approach for a periodic unit root in the presence of a break at unknown time for quarterly and monthly data is presented and the required posterior distribution is derived. In addition, a Bayesian F-test is suggested to test for seasonal and nonseasonal unit roots again controlling for a possible break. Instead of resorting to a model selection approach by choosing one particular model specification for testing, a Bayesian model averaging (BMA) approach is proposed to capture the model uncertainty associated with a specic parametrization of the test regression. In the third part of the thesis a Bayesian periodic autoregressive (PAR) model is then utilized for the prediction of quarterly and monthly time series data. A model averaging prediction approach for PAR models of unknown lag orders, number of breaks and break dates is proposed in order to improve the forecasting accuracy compared to conditional approaches. Further the joint posterior distribution of the multistep ahead forecasts is derived and an MCMC approach, based on data augmentation, is presented to generate random draws from the marginal posterior predictive distributions. In each of the three articles a Monte Carlo study is conducted to analyze the presented methods under different data generating processes. In the first two parts the presented testing approaches are utilized to examine if there is empirical evidence for persistence or hysteresis in the annual unemployment rates of OECD countries. In the last part of the thesis it is demonstrated how the suggested BMA prediction approach can improve forecasting accuracy compared to conditional, i.e. model selected, Bayesian PAR models using unadjusted monthly unemployment rates of East- and West-Germany and of the 16 German federal states

A decision theoretic analysis of the unit root hypothesis using mixtures of elliptical models

This paper develops a formal decision theoretic approach to testing for a unit root in economic time series. The approach is empirically implemented by specifying a loss function based on predictive variances; models are chosen so as to minimize expected loss. In addition, the paper broadens the class of likelihood functions traditionally considered in the Bayesian unit root literature by: i) Allowing for departures from normality via the specification of a likelihood based on general elliptical densities; ii) allowing for structural breaks to occur; iii) allowing for moving average errors; and iv) using mixtures of various submodels to create a very flexible overall likelihood. Empirical results indicate that, while the posterior probability of trend-stationarity is quite high for most of the series considered, the unit root model is often selected in the decision theoretic analysis