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Quantiles, Expectiles and Splines

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Author Info

  • DeRossi, G.
  • Harvey, A.

Abstract

A time-varying quantile can be fitted to a sequence of observations by formulating a time series model for the corresponding population quantile and iteratively applying a suitably modified state space signal extraction algorithm. It is shown that such time-varying quantiles satisfy the defining property of fixed quantiles in having the appropriate number of observations above and below. Expectiles are similar to quantiles except that they are defined by tail expectations. Like quantiles, time-varying expectiles can be estimated by a state space signal extraction algorithm and they satisfy properties that generalize the moment conditions associated with fixed expectiles. Time-varying quantiles and expectiles provide information on various aspects of a time series, such as dispersion and asymmetry, while estimates at the end of the series provide the basis for forecasting. Because the state space form can handle irregularly spaced observations, the proposed algorithms can be easily adapted to provide a viable means of computing spline-based non-parametric quantile and expectile regressions.

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File URL: http://www.econ.cam.ac.uk/research/repec/cam/pdf/cwpe0660.pdf
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Bibliographic Info

Paper provided by Faculty of Economics, University of Cambridge in its series Cambridge Working Papers in Economics with number 0702.

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Length: 29
Date of creation: Feb 2007
Date of revision:
Handle: RePEc:cam:camdae:0702

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Related research

Keywords: Asymmetric least squares; cubic splines; dispersion; non-parametric regression; quantile regression; signal extraction; state space smoother.;

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References

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  1. Siem Jan Koopman & Neil Shephard & Jurgen A. Doornik, 1999. "Statistical algorithms for models in state space using SsfPack 2.2," Econometrics Journal, Royal Economic Society, vol. 2(1), pages 107-160.
  2. Busettti, F. & Harvey, A., 2007. "Tests of time-invariance," Cambridge Working Papers in Economics 0701, Faculty of Economics, University of Cambridge.
  3. Newey, Whitney K & Powell, James L, 1987. "Asymmetric Least Squares Estimation and Testing," Econometrica, Econometric Society, vol. 55(4), pages 819-47, July.
  4. Engle, Robert F & Manganelli, Simone, 1999. "CAViaR: Conditional Autoregressive Value at Risk by Regression Quantiles," University of California at San Diego, Economics Working Paper Series qt06m3d6nv, Department of Economics, UC San Diego.
  5. Jondeau, Eric & Rockinger, Michael, 2003. "Conditional volatility, skewness, and kurtosis: existence, persistence, and comovements," Journal of Economic Dynamics and Control, Elsevier, vol. 27(10), pages 1699-1737, August.
  6. Koenker,Roger, 2005. "Quantile Regression," Cambridge Books, Cambridge University Press, number 9780521845731, October.
  7. Bosch, Ronald J. & Ye, Yinyu & Woodworth, George G., 1995. "A convergent algorithm for quantile regression with smoothing splines," Computational Statistics & Data Analysis, Elsevier, vol. 19(6), pages 613-630, June.
  8. Andrew Harvey & Siem Jan Koopman, 2000. "Signal extraction and the formulation of unobserved components models," Econometrics Journal, Royal Economic Society, vol. 3(1), pages 84-107.
  9. Durbin, James & Koopman, Siem Jan, 2001. "Time Series Analysis by State Space Methods," OUP Catalogue, Oxford University Press, number 9780198523543.
  10. DeRossi, G. & Harvey, A., 2006. "Time-Varying Quantiles," Cambridge Working Papers in Economics 0649, Faculty of Economics, University of Cambridge.
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Citations

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Cited by:
  1. Cathy Q. Ning & Loran Chollete, 2009. "The Dependence Structure of Macroeconomic Variables in the US," Working Papers 005, Ryerson University, Department of Economics.
  2. Bellini, Fabio & Klar, Bernhard & Müller, Alfred & Rosazza Gianin, Emanuela, 2014. "Generalized quantiles as risk measures," Insurance: Mathematics and Economics, Elsevier, vol. 54(C), pages 41-48.
  3. Zhijie Xiao & Roger Koenker, 2009. "Conditional Quantile Estimation for GARCH Models," Boston College Working Papers in Economics 725, Boston College Department of Economics.
  4. Tobias Adrian & Markus K. Brunnermeier, 2011. "CoVaR," NBER Working Papers 17454, National Bureau of Economic Research, Inc.
    • Tobias Adrian & Markus K. Brunnermeier, 2008. "CoVaR," Staff Reports 348, Federal Reserve Bank of New York.
  5. Fabio Busetti & Andrew Harvey, 2011. "When is a Copula Constant? A Test for Changing Relationships," Journal of Financial Econometrics, Society for Financial Econometrics, vol. 9(1), pages 106-131, Winter.
  6. Yuta Kurose & Yasuhiro Omori, 2012. "Bayesian Analysis of Time-Varying Quantiles Using a Smoothing Spline," CIRJE F-Series CIRJE-F-845, CIRJE, Faculty of Economics, University of Tokyo.
  7. Chollete, Lorán & de la Peña, Victor & Lu, Ching-Chih, 2011. "International diversification: A copula approach," Journal of Banking & Finance, Elsevier, vol. 35(2), pages 403-417, February.
  8. Chollete, Loran & Ning, Cathy, 2010. "Asymmetric Dependence in US Financial Risk Factors?," UiS Working Papers in Economics and Finance 2011/2, University of Stavanger.
  9. Harvey, Andrew, 2010. "Tracking a changing copula," Journal of Empirical Finance, Elsevier, vol. 17(3), pages 485-500, June.
  10. Huang, Xiaolin & Shi, Lei & Suykens, Johan A.K., 2014. "Asymmetric least squares support vector machine classifiers," Computational Statistics & Data Analysis, Elsevier, vol. 70(C), pages 395-405.
  11. Chollete, Loran & Pena, Victor de la & Lu, Ching-Chih, 2009. "International Diversification: A Copula Approach," UiS Working Papers in Economics and Finance 2009/27, University of Stavanger.
  12. Ng, Jason & Forbes, Catherine S. & Martin, Gael M. & McCabe, Brendan P.M., 2013. "Non-parametric estimation of forecast distributions in non-Gaussian, non-linear state space models," International Journal of Forecasting, Elsevier, vol. 29(3), pages 411-430.
  13. Chollete, Lorán & de la Peña, Victor & Lu, Ching-Chih, 2012. "International diversification: An extreme value approach," Journal of Banking & Finance, Elsevier, vol. 36(3), pages 871-885.
  14. Voudouris, Vlasios & Matsumoto, Ken'ichi & Sedgwick, John & Rigby, Robert & Stasinopoulos, Dimitrios & Jefferson, Michael, 2014. "Exploring the production of natural gas through the lenses of the ACEGES model," Energy Policy, Elsevier, vol. 64(C), pages 124-133.
  15. Rubia, Antonio & Sanchis-Marco, Lidia, 2013. "On downside risk predictability through liquidity and trading activity: A dynamic quantile approach," International Journal of Forecasting, Elsevier, vol. 29(1), pages 202-219.
  16. Mauro Bernardi & Ghislaine Gayraud & Lea Petrella, 2013. "Bayesian inference for CoVaR," Papers 1306.2834, arXiv.org, revised Nov 2013.
  17. Chollete, Loran & de la Pena , Victor & Lu, Ching-Chih, 2009. "International Diversification: An Extreme Value Approach," UiS Working Papers in Economics and Finance 2009/26, University of Stavanger.
  18. Harvey, A., 2008. "Dynamic distributions and changing copulas," Cambridge Working Papers in Economics 0839, Faculty of Economics, University of Cambridge.

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