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QUANTILOGRAMS UNDER STRONG DEPENDENCE

Published online by Cambridge University Press:  30 August 2019

Ji Hyung Lee ,
Oliver Linton  and
Yoon-Jae Whang
Ji Hyung Lee*
Affiliation:
University of Illinois
Oliver Linton
Affiliation:
University of Cambridge
Yoon-Jae Whang
Affiliation:
Seoul National University
*
*Address correspondence to Ji Hyung Lee, Department of Economics, University of Illinois, 1407 W. Gregory Dr., 214 David Kinley Hall, Urbana, IL 61801, USA; e-mail: jihyung@illinois.edu.
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Abstract

We develop the limit theory of the quantilogram and cross-quantilogram under long memory. We establish the sub-root-n central limit theorems for quantilograms that depend on nuisance parameters. We propose a moving block bootstrap (MBB) procedure for inference and establish its consistency, thereby enabling a consistent confidence interval construction for the quantilograms. The newly developed reduction principles for the quantilograms serve as the main technical devices used to derive the asymptotics and establish the validity of MBB. We report some simulation evidence that our methods work satisfactorily. We apply our method to quantile predictive relations between financial returns and long-memory predictors.

Type
ARTICLES
Information
Econometric Theory , Volume 36 , Issue 3 , June 2020 , pp. 457 - 487
Copyright
Copyright © Cambridge University Press 2019 

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Footnotes

We acknowledge helpful comments from Hongqi Chen, Rui Fan, Roger Koenker, Boyuan Zhang and participants from the seminars at University of Cambridge, UIUC, 2017 Asian meeting of the Econometric Society and Korea University. We thank the Co-Editor, Anna Mikusheva, and three anonymous referees for very constructive comments. We are also grateful for the vast amount of editorial input by the Editor, Peter Phillips, on the final version of the manuscript. Any errors are the responsibility of the authors.

References

REFERENCES

Baillie, R.T. (1996) Long memory processes and fractional integration in econometrics. Journal of Econometrics 73(1), 5–59.CrossRefGoogle Scholar
Beutner, E., Wu, W.B., & Zahle, H. (2012) Asymptotics for statistical functionals of long-memory sequences. Stochastic Processes and their Applications 122(3), 910–929.CrossRefGoogle Scholar
Birr, S., Volgushev, S., Kley, T., Dette, H., & Hallin, M. (2017) Quantile spectral analysis for locally stationary time series. Journal of the Royal Statistical Society: Series B (Statistical Methodology) 79(5), 1619–1643.CrossRefGoogle Scholar
Bollerslev, T., Osterrieder, D., Sizova, N., & Tauchen, G. (2013) Risk and return: Long-run relations, fractional cointegration, and return predictability. Journal of Financial Economics 108(2), 409–424.CrossRefGoogle Scholar
Dehling, H. & Taqqu, M.S. (1989) The empirical process of some long-range dependent sequences with an application to U-statistics. The Annals of Statistics 17(4), 1767–1783.CrossRefGoogle Scholar
Dette, H., Hallin, M., Kley, T., & Volgushev, S. (2015) Of copulas, quantiles, ranks and spectra: An L 1-approach to spectral analysis. Bernoulli 21(2), 781–831.CrossRefGoogle Scholar
Doukhan, P., Oppenheim, G., & Taqqu, M. (eds.) (2002) Theory and Applications of Long-Range Dependence. Springer Science & Business Media.Google Scholar
Efron, B. (1979) Bootstrap methods: Another look at the jackknife. The Annals of Statistics 7, 1–26.CrossRefGoogle Scholar
Fan, R. & Lee, J.H. (2019) Predictive quantile regressions under persistence and conditional heteroskedasticity. Journal of Econometrics, forthcoming. Available at SSRN: https://ssrn.com/abstract=3016449.Google Scholar
Geweke, J. & Porter-Hudak, S. (1983) The estimation and application of long memory time series models. Journal of Time Series Analysis 4(4), 221–238.CrossRefGoogle Scholar
Giraitis, L., Koul, H.L., & Surgailis, D. (2012) Large Sample Inference for Long Memory Processes, Vol. 10. AMC. p. 12.CrossRefGoogle Scholar
Granger, C.W. (1980) Long memory relationships and the aggregation of dynamic models. Journal of Econometrics 14(2), 227–238.CrossRefGoogle Scholar
Hagemann, A. (2011) Robust Spectral Analysis. Working paper.CrossRefGoogle Scholar
Han, H., Linton, O., Oka, T., & Whang, Y.J. (2016) The cross-quantilogram: Measuring quantile dependence and testing directional predictability between time series. Journal of Econometrics 193(1), 251–270.CrossRefGoogle Scholar
Hjort, N.L. & Pollard, D. (2011) Asymptotics for minimisers of convex processes. arXiv preprint arXiv:1107.3806.Google Scholar
Ho, H.C. & Hsing, T. (1996) On the asymptotic expansion of the empirical process of long-memory moving averages. The Annals of Statistics 24(3), 992–1024.Google Scholar
Ho, H.C. & Hsing, T. (1997) Limit theorems for functionals of moving averages. The Annals of Probability 25(4), 1636–1669.CrossRefGoogle Scholar
Honda, T. (2009) A limit theorem for sums of bounded functionals of linear processes without finite mean. Probability and Mathematical Statistics 29(2), 337.Google Scholar
Ibragimov, I.A. & Linnik, Yu.V. (1971) Independent and Stationary Sequences of Random Variables. Wolters-Noordoff.Google Scholar
Kim, Y.M. & Nordman, D.J. (2011) Properties of a block bootstrap under long-range dependence. Sankhya A 73(1), 79–109.CrossRefGoogle Scholar
Knight, K. (1998) Limiting distributions for L 1-regression estimators under general conditions. The Annals of Statistics 26(2), 755–770.Google Scholar
Koenker, R. (2005) Quantile Regression, vol. 38. Cambridge University Press.CrossRefGoogle Scholar
Koenker, R. (2017) Quantile regression: 40 years on. Annual Review of Economics 9(1), 155–176.CrossRefGoogle Scholar
Koul, H.L. & Surgailis, D. (2002) Asymptotic expansion of the empirical process of long memory moving averages. In Dehling, H., Mikosch, T., Sørensen, M. (eds.), Empirical Process Techniques for Dependent Data, pp. 213–239. Birkhauser Boston.CrossRefGoogle Scholar
Kreiss, J.P. & Paparoditis, E. (2011) Bootstrap methods for dependent data: A review. Journal of the Korean Statistical Society 40(4), 357–378.CrossRefGoogle Scholar
Künsch, H.R. (1989) The jackknife and the bootstrap for general stationary observations. The Annals of Statistics 17(3), 1217–1241.CrossRefGoogle Scholar
Lahiri, S.N. (2003) Resampling Methods for Dependent Data. Springer.CrossRefGoogle Scholar
Lee, J.H. (2016) Predictive quantile regression with persistent covariates: IVX-QR approach. Journal of Econometrics 192(1), 105–118.CrossRefGoogle Scholar
Li, T.H. (2008) Laplace periodogram for time series analysis. Journal of the American Statistical Association 103(482), 757–768.CrossRefGoogle Scholar
Li, T.H. (2012) Quantile periodograms. Journal of the American Statistical Association 107(498), 765–776.CrossRefGoogle Scholar
Linton, O. & Whang, Y.J. (2007) The quantilogram: With an application to evaluating directional predictability. Journal of Econometrics 141(1), 250–282.CrossRefGoogle Scholar
Liu, R.Y. & Singh, K. (1992) Moving blocks jackknife and bootstrap capture weak dependence. Exploring the Limits of Bootstrap 225, 248.Google Scholar
Maynard, A., Shimotsu, K., & Wang, Y. (2011) Inference in predictive quantile regressions. Unpublished manuscript.Google Scholar
Mikusheva, A. (2007) Uniform inference in autoregressive models. Econometrica 75(5), 1411–1452.CrossRefGoogle Scholar
Pollard, D. (1991) Asymptotics for least absolute deviation regression estimators. Econometric Theory 7(02), 186–199.CrossRefGoogle Scholar
Robinson, P.M. (1995) Gaussian semiparametric estimation of long range dependence. The Annals of Statistics 23(5), 1630–1661.CrossRefGoogle Scholar
Shao, X. (2015) Self-normalization for time series: A review of recent developments. Journal of the American Statistical Association 110(512), 1797–1817.CrossRefGoogle Scholar
Shimotsu, K. & Phillips, P.C. (2005) Exact local Whittle estimation of fractional integration. The Annals of Statistics 33(4), 1890–1933.CrossRefGoogle Scholar
Sun, S. & Lahiri, S.N. (2006) Bootstrapping the sample quantile of a weakly dependent sequence. Sankhyā: The Indian Journal of Statistics 68(1), 130–166.Google Scholar
Tewes, J. (2016) Block bootstrap for the empirical process of long-range dependent data. arXiv preprint arXiv:1601.01122.Google Scholar
Tsay, W.J. & Chung, C.F. (2000) The spurious regression of fractionally integrated processes. Journal of Econometrics 96(1), 155–182.CrossRefGoogle Scholar
Van der Vaart, A.W. (2000) Asymptotic Statistics, vol. 3. Cambridge University Press.Google Scholar
Welch, I. & Goyal, A. (2008) A comprehensive look at the empirical performance of equity premium prediction. Review of Financial Studies 21(4), 1455–1508.CrossRefGoogle Scholar
Zhang, T., Ho, H.C., Wendler, M., & Wu, W.B. (2013) Block sampling under strong dependence. Stochastic Processes and their Applications 123(6), 2323–2339.CrossRefGoogle Scholar