Quantile cointegrating regression

Quantile regression has important applications in risk management, portfolio optimization, and asset pricing. The current paper studies estimation, inference and financial applications of quantile regression with cointegrated time series. In addition, a new cointegration model with quantile-varying coefficients is proposed. In the proposed model, the value of cointegrating coefficients may be affected 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 effects. Asymptotic properties of the proposed model and limiting distribution of the cointegrating regression quantiles are derived. In the presence of endogenous regressors, fully-modified quantile regression estimators and augmented quantile cointegrating regression are proposed to remove the second order bias and nuisance parameters. Regression Wald tests are constructed based on the fully modified quantile regression estimators. An empirical application to stock index data highlights the potential of the proposed method.

[1]  Clive W. J. Granger,et al.  A cointegration analysis of treasury bill yields , 1992 .

[2]  R. Shiller,et al.  The Dividend-Price Ratio and Expectations of Future Dividends and Discount Factors , 1986 .

[3]  Eve Bofingeb,et al.  ESTIMATION OF A DENSITY FUNCTION USING ORDER STATISTICS1 , 1975 .

[4]  D. Tasche,et al.  Expected Shortfall: a natural coherent alternative to Value at Risk , 2001, cond-mat/0105191.

[5]  C. Granger,et al.  Co-integration and error correction: representation, estimation and testing , 1987 .

[6]  E. J. Hannan,et al.  Multiple time series , 1970 .

[7]  Cheng Hsiao,et al.  Cointegration and Dynamic Simultaneous Equations Model , 1997 .

[8]  P. Robinson,et al.  AUTOMATIC FREQUENCY DOMAIN INFERENCE ON SEMIPARAMETRIC AND NONPARAMETRIC MODELS , 1991 .

[9]  Arthur Havenner,et al.  Cointegration and stock prices: The random walk on wall street revisited , 1988 .

[10]  Victor Chernozhukov,et al.  Conditional value-at-risk: Aspects of modeling and estimation , 2000 .

[11]  D. Hendry,et al.  Co-Integration and Error Correction : Representation , Estimation , and Testing , 2007 .

[12]  Roger Koenker,et al.  Quantile Autoregression , 2006 .

[13]  D. Brillinger.  Time series - data analysis and theory , 1981, Classics in applied mathematics.

[14]  Zhijie Xiao.  Testing the Null Hypothesis of Stationarity Against an Autoregressive Unit Root Alternative , 2001 .

[15]  Pentti Saikkonen,et al.  Asymptotically Efficient Estimation of Cointegration Regressions , 1991, Econometric Theory.

[16]  R. Koenker,et al.  Robust rank tests of the unit root hypothesis , 1997 .

[17]  G. Evans.  Pitfalls in Testing for Explosive Bubbles in Asset Prices , 1991 .

[18]  C. Alexander,et al.  Optimal hedging using cointegration , 1999, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[19]  R. Engle,et al.  CAViaR , 1999 .

[20]  D. Pollard.  Asymptotics for Least Absolute Deviation Regression Estimators , 1991, Econometric Theory.

[21]  Robert B. Barsky,et al.  Why Does the Stock Market Fluctuate? , 1992 .

[22]  D. Andrews.  Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Estimation , 1991 .

[23]  Herschel I. Grossman,et al.  Explosive Rational Bubbles in Stock Prices , 1988 .

[24]  Philippe Artzner,et al.  Coherent Measures of Risk , 1999 .

[25]  R. Rockafellar,et al.  Optimization of conditional value-at risk , 2000 .

[26]  Jinyong Hahn,et al.  Série Scientifique Scientific Series Testing and Comparing Value-at-risk Measures , 2022 .

[27]  Sydney C. Ludvigson,et al.  Consumption, Aggregate Wealth and Expected Stock Returns , 1999 .

[28]  K. Knight.  Limit theory for autoregressive‐parameter estimates in an infinite‐variance random walk , 1989 .

[29]  Robert J. Shiller,et al.  Cointegration and Tests of Present Value Models , 1987, Journal of Political Economy.

[30]  Victor Solo,et al.  Asymptotics for Linear Processes , 1992 .

[31]  M. Herce.  Asymptotic Theory of LAD Estimation in a Unit Root Process with Finite Variance Errors , 1996, Econometric Theory.

[32]  P. Billingsley,et al.  Convergence of Probability Measures , 1970, The Mathematical Gazette.

[33]  Roger Koenker,et al.  Inference on the Quantile Regression Process , 2000 .

[34]  J. Cochrane.  Permanent and Transitory Components of GNP and Stock Prices , 1994 .

[35]  R. Koenker,et al.  Pessimistic Portfolio Allocation and Choquet Expected Utility , 2004 .

[36]  R. Donaldson,et al.  A New Dividend Forecasting Procedure That Rejects Bubbles in Asset Prices: The Case of 1929's Stock Crash , 1996 .

[37]  Roger Koenker,et al.  Conditional Quantile Estimation and Inference for Arch Models , 1996, Econometric Theory.

[38]  M. M. Siddiqui.  Distribution of quantiles in samples from a bivariate population , 1960 .

[39]  Ivana Komunjer,et al.  Evaluation and Combination of Conditional Quantile Forecasts , 2002 .

[40]  Julio J. Rotemberg,et al.  The Comovement of Stock Prices , 1993 .

[41]  A. Lo,et al.  THE ECONOMETRICS OF FINANCIAL MARKETS , 1996, Macroeconomic Dynamics.

[42]  Salih N. Neftçi.  Are Economic Time Series Asymmetric over the Business Cycle? , 1984, Journal of Political Economy.

[43]  Gary Koop,et al.  Do recessions permanently change output , 1993 .

[44]  Walter Krämer,et al.  The CUSUM test for OLS residuals , 1992 .

[45]  A. Chowdhury.  Futures market efficiency: Evidence from cointegration tests , 1991 .

[46]  Peter C. B. Phillips,et al.  Statistical Inference in Instrumental Variables Regression with I(1) Processes , 1990 .

[47]  Clive W. J. Granger,et al.  Unit Root Tests and Asymmetric Adjustment with an Example Using the Term Structure of Interest Rates , 1998 .

[48]  James W. Taylor.  A Quantile Regression Approach to Estimating the Distribution of Multiperiod Returns , 1999 .

[49]  Y. Shin.  A Residual-Based Test of the Null of Cointegration Against the Alternative of No Cointegration , 1994, Econometric Theory.

[50]  Joon Y. Park,et al.  COINTEGRATING REGRESSIONS WITH TIME VARYING COEFFICIENTS , 1999, Econometric Theory.

[51]  C. Granger.  Some properties of time series data and their use in econometric model specification , 1981 .

[52]  P. Phillips,et al.  Asymptotic Properties of Residual Based Tests for Cointegration , 1990 .

[53]  Jianqing Fan,et al.  Quantile autoregression. Commentary , 2006 .

[54]  D. B. Preston.  Spectral Analysis and Time Series , 1983 .

[55]  A. Lucas.  Strategic and tactical asset allocation and the effect of long-run equilibrium relations , 1997 .

[56]  Zhijie Xiao,et al.  A CUSUM Test for Cointegration Using Regression Residuals , 2001 .

[57]  Karolina Koziorowska.  Conditional Value at Risk , 2009 .