Panel Cointegration With Cross-section Dependence Using Bayesian Quantile Regression Method

Recently, nonstationary panel data is an important topic in areas of econometrics. An approach in this direction is panel unit root and cointegration, which has attracted considerable interest and is more valuable. Panel unit root and cointegration is known as the extension of unit root and cointegration in time series. In the globalised economy, co-movements of economies are often observed. Global international trends or international business cycles for instance, can cause cross-section dependence in panel time series of macroeconomics, management and finance. Therefore, panel cointegration tests with cross-section dependence may be more valid and realistic. In contrast to classic panel cointegraiton, however, this paper proposes Bayesian quantile panel cointegrating under the assumption of cross-section dependence. In the framework of Bayesian econometrics, cross-section dependence is addressed and Bayesian quantile regression method is used. A distinctive feature of Bayesian quantile cointegrating regression is that it is the combination of Bayesian quantile regression method and cointegration. It considers the uncertainty of parameter, and can capture systematic influences of covariates on the location, scale and shape of the conditional distribution of the response. Thus, Bayesian quantile cointegrating regression could analyze the long run relationship between response variable and covariates. In theory, Bayesian quantile panel cointegration will develop research ideas and methods for panel cointegration. Meanwhile, it will also provide technology and evidence for analysis and decision making in economic and management applications.First, common factor is used to handle the cross-section dependence in panel data. In the context of Bayesian theorem and Bayesian decision theorem, the analytic posterior estimator of conditional quantile function is obtained by a linear mixture representation of asymmetric Laplace distribution. To get the estimate and conduct cointegration test, the Kalman filter and Gibbs sampling algorithms are utilized to simulate the posterior marginal distribution of quantile cointegrating parameters, which resolved the difficulties of the high dimension numerical integral. Monte Carlo simulation also indicates that Bayesian quanitle panel cointegration can conduct more comprehensive analysis cointegration relationship between variables.It is known that the phenomenon of structural breaks is a common place in macroeconomic series. Omitting the effect of structural breaks can thus cause a deceptive inference in time series and panel data testing. This paper proposes panel cointegration model with smooth structural break, which implement the Fourier expansion to explain structural breaks in panel time series. The panel data are demeaned to control potential cross-section dependence, and Bayesian quantile regression method is employed to obtain the analytic posterior estimator of conditional quantile function. MCMC algorithm is then designed to estimate parameters and conduct cointegration test. We also conduct a small Monte Carlo study to illustrate the effect of Bayesian quantile panel cointegration with smooth structural breaks. The results show that, as expected, the Bayesian quantile cointegration methods are more effective and comprehensive to analyze the structural break long run relationship except for some special cases.In contrast to cointegraiton with structural breaks, threshold cointegration aims to explore the asymmetry or asymmetric adjustment behaviors, where the cointgrating regression is linear and the error correction term is asymmetric. In the existing threshold cointegration methods, the jagged and potentially multimodal nature of the likelihood function of threshold model complicates optimization and also makes the identification of unknown nuisance parameters more difficult. This paper proposes Bayesian quantile threshold cointegration model and conduct quantile cointegration tests. Especially, the demeaning approach is used to control the potential cross sectional dependence. The choice of priors is discussed to get robust posterior estimator of conditional quantile function. Based on the posterior conditional distributions of the parameters, MCMC samplers are designed to estimate the parameters and compute the posterior probability for threshold cointegration tests.Finally, the above mentioned Bayesian quantile panel cointegration methods are applied to the analysis of crude oil and stock markets. Comparing with traditional panel cointegration methods, Bayesian quantile cointegraiton methods can describe the relationship between crude oil and stock markets more comprehensively. Therefore, the usefulness of Bayesian quantile cointegration methods is also demonstrated. It shows that Bayesian quantile cointegration performs well and can provide fully information on parameter estimation and cointegration tests.