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QUANTILE DOUBLE AUTOREGRESSION

Published online by Cambridge University Press:  25 June 2021

Qianqian Zhu  and
Guodong Li
Qianqian Zhu*
Affiliation:
Shanghai University of Finance and Economics
Guodong Li
Affiliation:
University of Hong Kong
*
Address correspondence to Qianqian Zhu, School of Statistics and Management, Shanghai University of Finance and Economics, Shanghai, China; e-mail: zhu.qianqian@mail.shufe.edu.cn.
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Abstract

Many financial time series have varying structures at different quantile levels, and also exhibit the phenomenon of conditional heteroskedasticity at the same time. However, there is presently no time series model that accommodates both of these features. This paper fills the gap by proposing a novel conditional heteroskedastic model called “quantile double autoregression”. The strict stationarity of the new model is derived, and self-weighted conditional quantile estimation is suggested. Two promising properties of the original double autoregressive model are shown to be preserved. Based on the quantile autocorrelation function and self-weighting concept, three portmanteau tests are constructed to check the adequacy of the fitted conditional quantiles. The finite sample performance of the proposed inferential tools is examined by simulation studies, and the need for use of the new model is further demonstrated by analyzing the S&P500 Index.

Type
ARTICLES
Information
Econometric Theory , Volume 38 , Issue 4 , August 2022 , pp. 793 - 839
Copyright
© The Author(s), 2021. Published by Cambridge University Press

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Footnotes

We are grateful to the Editor, Co-Editor, Associate Editor, and three anonymous referees for their valuable comments and suggestions, which led to substantial improvement of this article. Zhu’s research was supported by an NSFC grant 12001355, Shanghai Pujiang Program 2019PJC051, and Shanghai Chenguang Program 19CG44. Li’s research was partially supported by a Hong Kong RGC grant 17304617 and an NSFC grant 72033002.

References

REFERENCES

An, H. Z. & Chen, Z. G. (1982) On convergence of LAD estimates in autoregresion with infinite variance. Journal of Multivariate Analysis 12, 335–345.CrossRefGoogle Scholar
Artzner, P., Delbaen, F., Eber, J. M., & Heath, D. (1999) Coherent measures of risk. Mathematical Finance 9, 203–228.CrossRefGoogle Scholar
Bassett, G., Koenker, R., & Kordas, G. (2004) Pessimistic portfolio allocation and Choquet expected utility. Journal of Financial Econometrics 4, 477–492.CrossRefGoogle Scholar
Bofinger, E. (1975) Estimation of a density function using order statistics. Australian Journal of Statistics 17, 1–7.CrossRefGoogle Scholar
Bollerslev, T. (1986) Generalized autoregression conditional heteroscedasticity. Journal of Econometrics 31, 307–327.CrossRefGoogle Scholar
Box, G. E. P. & Jenkins, G. M. (1970) Time Series Analysis, Forecasting and Control. Holden-Day.Google Scholar
Cai, Y., Montes-Rojas, G., & Olmo, J. (2013) Quantile double AR time series models for financial returns. Journal of Forecasting 32, 551–560.CrossRefGoogle Scholar
Chan, N. H. & Peng, L. (2005) Weighted least absolute deviations estimation for an AR(1) process with ARCH(1) errors. Biometrika 92, 477–484.CrossRefGoogle Scholar
Chernozhukov, V., Fernández-Val, I., & Galichon, A. (2010) Quantile and probability curves without crossing. Econometrica 78, 1093–1125.Google Scholar
Christoffersen, P. F. (1998) Evaluating interval forecasts. International Economic Review 39, 841–862.CrossRefGoogle Scholar
Davis, R. A., Knight, K., & Liu, J. (1992) M-estimation for autoregressions with infinite variances. Stochastic Processes and Their Applications 40, 145–180.CrossRefGoogle Scholar
Engle, R. F. (1982) Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica 50, 987–1007.CrossRefGoogle Scholar
Engle, R. F. & Manganelli, S. (2004) CAViaR: Conditional autoregressive value at risk by regression quantiles. Journal of Business and Economic Statistics 22, 367–381.CrossRefGoogle Scholar
Feigin, P. D. & Tweedie, R. L. (1985) Random coefficient autoregressive processes: A Markov chain analysis of stationarity and finiteness of moments. Journal of Time Series Analysis 6, 1–14.CrossRefGoogle Scholar
Francq, C. & Zakoian, J.-M. (2004) Maximum likelihood estimation of pure GARCH and ARMA–GARCH processes. Bernoulli 10, 605–637.CrossRefGoogle Scholar
Francq, C. & Zakoian, J.-M. (2010) GARCH Models: Structure, Statistical Inference and Financial Applications. Wiley.CrossRefGoogle Scholar
Francq, C. & Zakoian, J.-M. (2015) Risk-parameter estimation in volatility models. Journal of Econometrics 184, 158–174.CrossRefGoogle Scholar
Galvao, A. F. Jr., Montes-Rojas, G., & Olmo, J. (2011) Threshold quantile autoregressive models. Journal of Time Series Analysis 32, 253–267.CrossRefGoogle Scholar
Gross, S. & Steiger, W. L. (1979) Least absolute deviation estimates in autoregression with infinite variance. Journal of Applied Probability 16, 104–116.CrossRefGoogle Scholar
Hall, P. & Sheather, S. (1988) On the distribution of a studentized quantile. Journal of the Royal Statistical Society, Series B 50, 381–391.Google Scholar
Huber, P. (1967) The behavior of maximum likelihood estimates under nonstandard conditions. Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability 1, 221–233.Google Scholar
Huber, P. (1973) Robust regression: Asymptotics, conjectures and Monte Carlo. Annals of Statistics 1, 799–821.CrossRefGoogle Scholar
Jiang, F., Li, D., & Zhu, K. (2020) Non-standard inference for augmented double autoregressive models with null volatility coefficients. Journal of Econometrics 215, 165–183.CrossRefGoogle Scholar
Knight, K. (1998) Limiting distributions for ${L}_1$ regression estimators under general conditions. Annals of Statistics 26, 755–770.CrossRefGoogle Scholar
Koenker, R. (2005) Quantile Regression. Cambridge University Press.CrossRefGoogle Scholar
Koenker, R. & Xiao, Z. (2006) Quantile autoregression. Journal of the American Statistical Association 101, 980–990.CrossRefGoogle Scholar
Koenker, R. & Zhao, Q. (1996) Conditional quantile estimation and inference for ARCH models. Econometric Theory 12, 793–813.CrossRefGoogle Scholar
Lee, S. & Noh, J. (2013) Quantile regression estimator for GARCH models. Scandinavian Journal of Statistics 40, 2–20.CrossRefGoogle Scholar
Li, D., Ling, S., & Zhang, R. (2016) On a threshold double autoregressive model. Journal of Business & Economic Statistics 34, 68–80.CrossRefGoogle Scholar
Li, G. & Li, W. K. (2005) Diagnostic checking for time series models with conditional heteroscedasticity estimated by the least absolute deviation approach. Biometrika 92, 691–701.CrossRefGoogle Scholar
Li, G. & Li, W. K. (2008) Least absolute deviation estimation for fractionally integrated autoregressive moving average time series models with conditional heteroscedasticity. Biometrika 95, 399–414.CrossRefGoogle Scholar
Li, G. & Li, W. K. (2009) Least absolute deviation estimation for unit root processes with GARCH errors. Econometric Theory 25, 1208–1227.CrossRefGoogle Scholar
Li, G., Li, Y., & Tsai, C.-L. (2015) Quantile correlations and quantile autoregressive modeling. Journal of the American Statistical Association 110, 246–261.CrossRefGoogle Scholar
Li, G., Zhu, Q., Liu, Z., & Li, W. K. (2017) On mixture double autoregressive time series models. Journal of Business and Economic Statistics 35, 306–317.CrossRefGoogle Scholar
Li, W. K. (2004) Diagnostic Checks in Time Series. Chapman and Hall.Google Scholar
Ling, S. (2004) Estimation and testing stationarity for double-autoregressive models. Journal of the Royal Statistical Society, Series B 66, 63–78.CrossRefGoogle Scholar
Ling, S. (2005) Self-weighted least absolute deviation estimation for infinite variance autoregressive models. Journal of the Royal Statistical Society, Series B 67, 381–393.CrossRefGoogle Scholar
Ling, S. (2007) A double AR(p) model: Structure and estimation. Statistica Sinica 17, 161–175.Google Scholar
Ling, S. & Li, D. (2008) Asymptotic inference for a nonstationary double AR(1) model. Biometrika 95, 257–263.CrossRefGoogle Scholar
Ling, S. & McAleer, M. (2003) Asymptotic theory for a vector ARMA–GARCH model. Econometric Theory 19, 280–310.CrossRefGoogle Scholar
Ljung, G. M. & Box, G. E. P. (1978) On a measure of lack of fit in time series models. Biometrika 65, 297–303.CrossRefGoogle Scholar
Machado, J. A. (1993) Robust model selection and M-estimation. Econometric Theory 9, 478–493.CrossRefGoogle Scholar
Phillips, P. C. B. (2015) Halbert White Jr. memorial JFEC lecture: Pitfalls and possibilities in predictive regression. Journal of Financial Econometrics 13, 521–555.CrossRefGoogle Scholar
Pollard, D. (1985) New ways to prove central limit theorems. Econometric Theory 1, 295–314.CrossRefGoogle Scholar
Tsay, R. S. (2014) Multivariate Time Series Analysis: With R and Financial Applications. Wiley.Google Scholar
Tweedie, R. L. (1983) Criteria for rates of convergence of Markov chains, with application to queueing and storage theory. In Kingman, J. F. C. & Reuter, G. E. H. (eds.), Probability, Statistics and Analysis, pp. 260–276. Cambridge University Press.CrossRefGoogle Scholar
Weiss, A. (1986) Asymptotic theory for ARCH models: Estimation and testing. Econometric Theory 2, 107–131.CrossRefGoogle Scholar
Wu, G. & Xiao, Z. (2002) An analysis of risk measures. Journal of Risk 4, 53–75.CrossRefGoogle Scholar
Xiao, Z. & Koenker, R. (2009) Conditional quantile estimation for generalized autoregressive conditional heteroscedasticity models. Journal of the American Statistical Association 104, 1696–1712.CrossRefGoogle Scholar
Zheng, Y., Zhu, Q., Li, G., & Xiao, Z. (2018) Hybrid quantile regression estimation for time series models with conditional heteroscedasticity. Journal of the Royal Statistical Society, Series B 80, 975–993.CrossRefGoogle Scholar
Zhu, K. & Ling, S. (2011) Global self-weighted and local quasi-maximum exponential likelihood estimators for ARMA–GARCH/IGARCH models. Annals of Statistics 39, 2131–2163.CrossRefGoogle Scholar
Zhu, K. & Ling, S. (2013) Quasi-maximum exponential likelihood estimators for a double AR(p) model. Statistica Sinica 23, 251–270.Google Scholar
Zhu, Q., Li, G., & Xiao, Z. (2021) Quantile estimation of regression models with GARCH-X errors. Statistica Sinica, to appear. doi:10.5705/ss.202019.0003. CrossRefGoogle Scholar
Zhu, Q., Zheng, Y., & Li, G. (2018) Linear double autoregression. Journal of Econometrics 207, 162–174.CrossRefGoogle Scholar
Zhu, X., Pan, R., Li, G., Liu, Y., & Wang, H. (2017) Network vector autoregression. Annals of Statistics 45, 1096–1123.CrossRefGoogle Scholar
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