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A CONSISTENT NONPARAMETRIC TEST FOR CAUSALITY IN QUANTILE

Published online by Cambridge University Press:  19 January 2012

Kiho Jeong ,
Wolfgang K. Härdle  and
Song Song
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Abstract

This paper proposes a nonparametric test of Granger causality in quantile. Zheng (1998, Econometric Theory 14, 123–138) studied the idea to reduce the problem of testing a quantile restriction to a problem of testing a particular type of mean restriction in independent data. We extend Zheng’s approach to the case of dependent data, particularly to the test of Granger causality in quantile. Combining the results of Zheng (1998) and Fan and Li (1999, Journal of Nonparametric Statistics 10, 245–271), we establish the asymptotic normal distribution of the test statistic under a β-mixing process. The test is consistent against all fixed alternatives and detects local alternatives approaching the null at proper rates. Simulations are carried out to illustrate the behavior of the test under the null and also the power of the test under plausible alternatives. An economic application considers the causal relations between the crude oil price, the USD/GBP exchange rate, and the gold price in the gold market.

Type
Research Article
Information
Econometric Theory , Volume 28 , Issue 4 , August 2012 , pp. 861 - 887
Copyright
Copyright © Cambridge University Press 2012

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Footnotes

The research was conducted while Jeong was visiting C.A.S.E.—Center for Applied Statistics and Economics—Humboldt-Universität zu Berlin in the summers of 2005 and 2007. Jeong is grateful for their hospitality during the visits. Jeong’s work was supported by a Korean Research Foundation grant funded by the Korean government (MOEHRD) (KRF-2006-B00002), and Härdle and Song’s work was supported by the Deutsche Forschungsgemeinschaft through the SFB 649 “Economic Risk.” We thank the editor, two anonymous referees, and Holger Dette for concrete suggestions on improving the manuscript and restructuring the paper. Their valuable comments and suggestions are gratefully acknowledged.

References

REFERENCES

Auestad, B. & Tjøstheim, D. (1990) Identification of nonlinear time series: First order characterisation and order determination. Biometrica 77, 669–687.10.1093/biomet/77.4.669CrossRefGoogle Scholar
Bollerslev, T. (1986) Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics 31, 307–327.CrossRefGoogle Scholar
Bollerslev, T. (2001) Financial econometrics: Past developments and future challenges. Journal of Econometrics 100, 41–51.CrossRefGoogle Scholar
Buchinsky, M. (1995) Quantile regression, Box-Cox transformation model, and the U.S. wage structure. 1963–1987. Journal of Econometrics 65, 65–154.10.1016/0304-4076(94)01599-UCrossRefGoogle Scholar
Campbell, J. & Cochrane, J. (1999) By force of habit: A consumption-based explanation of aggregate stock market behaviour. Journal of Political Economy 107, 205–251.CrossRefGoogle Scholar
Day, J.C. & Newburger, E.C. (2002) The big payoff: Educational attainment and synthetic estimates of work-life earnings. Special studies. Current population reports. Statistical report p23–210, U.S. Department of Commerce, U.S. Census Bureau.Google Scholar
Diebolt, J. & Guégan, D. (1993) Tail behavior of the stationary density of general nonlinear autoregressive processes of order 1. Journal of Applied Probability 30, 315–329.CrossRefGoogle Scholar
Engle, R. (1982) Autoregressive conditional heteroskedasticity with estimates of the variance of United Kingdom inflation. Econometrica 50, 987–1007.10.2307/1912773CrossRefGoogle Scholar
Fan, Y. & Li, Q. (1996) Consistent model specification tests: Omitted variables, parametric and semiparametric functional forms. Econometrica 64, 865–890.CrossRefGoogle Scholar
Fan, Y. & Li, Q. (1999) Central limit theorem for degenerate U-statistics of absolutely regular processes with applications to model specification tests. Journal of Nonparametric Statistics 10, 245–271.CrossRefGoogle Scholar
Franke, J. & Mwita, P. (2003) Nonparametric Estimates for Conditional Quantiles of Time Series. Wirtschaftsmathematik 87, University of Kaiserslautern.Google Scholar
Fuller, W. (1976) Introduction to Statistical Time Series. Wiley.Google Scholar
Granger, C. (1969) Investigating causal relations by econometric models and cross-spectral methods. Econometrica 37, 424–438.10.2307/1912791CrossRefGoogle Scholar
Granger, C. (1988) Some recent developments in a concept of causality. Journal of Econometrics 39, 199–211.CrossRefGoogle Scholar
Györfi, L., Härdle, W., Sarda, P., & Vieu, P. (1989) Nonparametric Curve Estimation from Time Series. Springer-Verlag.CrossRefGoogle Scholar
Hall, P. (1984) Central limit theorem for integrated square error of multivariate nonparametric density estimators. Journal of Multivariate Analysis 14, 1–16.CrossRefGoogle Scholar
Härdle, W., Ritov, Y., & Song, S. (2009) Bootstrap Partial Linear Quantile Regression and Confidence Bands. SFB649 Discussion paper 2010-002, Humboldt Universität zu Berlin.CrossRefGoogle Scholar
Härdle, W. & Stoker, T. (1989) Investigating smooth multiple regression by the method of average derivatives. Journal of the American Statistical Association 84, 986–995.Google Scholar
Hong, Y., Liu, Y., & Wang, S. (2009) Granger causality in risk and detection of extreme risk spillover between financial markets. Journal of Econometrics 150, 271–287.CrossRefGoogle Scholar
Hsiao, C. & Li, Q. (2001) A consistent test for conditional heteroskedasticity in time-series regression models. Econometric Theory 17, 188–221.CrossRefGoogle Scholar
Irle, A. (1997) On the consistency in nonparametric estimation under mixing assumptions. Multivariate Analysis 60, 123–147.10.1006/jmva.1996.1647CrossRefGoogle Scholar
Kwiatkowski, D., Phillips, P.C.B., Schmidt, P., & Shin, Y. (1992) Testing the null hypothesis of stationarity against the alternative of a unit root. Journal of Econometrics 54, 159–178.10.1016/0304-4076(92)90104-YCrossRefGoogle Scholar
Lee, T. & Yang, W. (2007) Money-Income Granger-Causality in Quantiles. Manuscript, University of California, Riverside.Google Scholar
Li, Q. (1999) Consistent model specification tests for time series econometric models. Journal of Econometrics 92, 101–147.10.1016/S0304-4076(98)00087-6CrossRefGoogle Scholar
Li, Q. & Wang, S. (1998) A simple consistent bootstrap test for a parametric regression functional form. Journal of Econometrics 87, 145–165.10.1016/S0304-4076(98)00011-6CrossRefGoogle Scholar
Masry, E. & Tjøstheim, D. (1995) Nonparametric estimation and identification of nonlinear ARCH time series: Strong convergence and asymptotic normality. Econometric Theory 11, 258–289.CrossRefGoogle Scholar
Masry, E. & Tjøstheim, D. (1997) Additive nonlinear ARX time series and projection estimates. Econometric Theory 13, 214–252.CrossRefGoogle Scholar
Meir, R. (2000) Nonparametric time series prediction through adaptive model selection. Machine Learning 39, 5–34.CrossRefGoogle Scholar
Modha, D. & Masry, E. (1998) Memory-universal prediction of stationary random processes. IEEE Transactions on Information Theory 44, 117–133.CrossRefGoogle Scholar
Phillips, P.C.B. & Perron, P. (1988) Testing for a unit root in time series regression. Biometricka 75, 335–346.10.1093/biomet/75.2.335CrossRefGoogle Scholar
Powell, J., Stock, J., & Stoker, T. (1989) Semiparametric estimation of index coefficients. Econometrica 57, 1403–1430.CrossRefGoogle Scholar
Robinson, P.M. (1988) Root- n-consistent semiparametric regression. Econometrica 56, 931–954.10.2307/1912705CrossRefGoogle Scholar
Roussas, G.G. (1988) Nonparametric estimation in mixing sequences of random variables. Journal of Statistical Planning and Inference 18, 135–149.CrossRefGoogle Scholar
Sherman, R. (1994) Maximal inequalities for degenerate U-processes with applications to optimization estimators. Annals of Statistics 22, 439–459.CrossRefGoogle Scholar
Yoshihara, K. (1976) Limiting behavior of u-statistics for stationary absolutely regular processes. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 35, 237–252.CrossRefGoogle Scholar
Yu, B. (1993) Density estimation in the l1 norm for dependent data with applications. Annals of Statistics 21, 711–735.10.1214/aos/1176349146CrossRefGoogle Scholar
Zheng, J. (1998) A consistent nonparametric test of parametric regression models under conditional quantile restrictions. Econometric Theory 14, 123–138.CrossRefGoogle Scholar