Quantile Cointegrating Regression

Quantile regression has important applications in risk management, portfolio optimization, and asset pricing. The current paper studies estimation, inference and …nancial applications of quantile regression with cointegrated time series. In addition, a new cointegration model with varying coe¢ cients is proposed. In the proposed model, the value of cointegrating coe¢ cients may be a¤ected by the shocks and thus may vary over the innovation quantile. The proposed model may be viewed as a stochastic cointegration model which includes the conventional cointegration model as a special case. It also provides a useful complement to cointegration models with (G)ARCH e¤ects. Asymptotic properties of the proposed model and limiting distribution of the cointegrating regression quantiles are derived. In the presence of endogenous regressors, fully-modi…ed quantile regression estimators and augmented quantile cointegrating regression are proposed to remove the second order bias and nuisance parameters. Regression Wald test are constructed based on the fully modi…ed quantile regression estimators. An empirical application to stock index data highlights the potential of the proposed method. JEL: C22, G1. KeyWords: ARCH/GARCH, Cointegration, Portfolio Optimization, Quantile Regression, Time Varying. 1 Introduction Since Granger (1981) and Engle and Granger (1987), cointegration has become a common econometric tool for empirical analysis in numerous areas (see, inter alia, Phillips and Ouliaris 1988; Johansen 1995; and Hsiao 1997, among others), especially in macroeconomic and …nancial applications. Well-known …nancial applications of Version 4.0. Address correspondence: Department of Economics, Boston College, Chestnut Hill, MA 02467. Tel: 617-552-1709. Fax: 617-5522308. Email: [email protected]. The author wish to thank the guest editors, two referees, Konstantin Tyurin, Roger Koenker, Peter Phillips and seminar participants at the …rst symposium on econometric theory and applications for their helpful comments.