Skip to main content Accessibility help
×
Home

QUANTILOGRAMS UNDER STRONG DEPENDENCE

  • Ji Hyung Lee (a1), Oliver Linton (a2) and Yoon-Jae Whang (a3)

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.

Copyright

Corresponding author

*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.

Footnotes

Hide All

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.

Footnotes

References

Hide All
Baillie, R.T. (1996) Long memory processes and fractional integration in econometrics. Journal of Econometrics 73(1), 559.
Beutner, E., Wu, W.B., & Zahle, H. (2012) Asymptotics for statistical functionals of long-memory sequences. Stochastic Processes and their Applications 122(3), 910929.
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), 16191643.
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), 409424.
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), 17671783.
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), 781831.
Doukhan, P., Oppenheim, G., & Taqqu, M. (eds.) (2002) Theory and Applications of Long-Range Dependence. Springer Science & Business Media.
Efron, B. (1979) Bootstrap methods: Another look at the jackknife. The Annals of Statistics 7, 126.
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.
Geweke, J. & Porter-Hudak, S. (1983) The estimation and application of long memory time series models. Journal of Time Series Analysis 4(4), 221238.
Giraitis, L., Koul, H.L., & Surgailis, D. (2012) Large Sample Inference for Long Memory Processes, Vol. 10. AMC. p. 12.
Granger, C.W. (1980) Long memory relationships and the aggregation of dynamic models. Journal of Econometrics 14(2), 227238.
Hagemann, A. (2011) Robust Spectral Analysis. Working paper.
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), 251270.
Hjort, N.L. & Pollard, D. (2011) Asymptotics for minimisers of convex processes. arXiv preprint arXiv:1107.3806.
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), 9921024.
Ho, H.C. & Hsing, T. (1997) Limit theorems for functionals of moving averages. The Annals of Probability 25(4), 16361669.
Honda, T. (2009) A limit theorem for sums of bounded functionals of linear processes without finite mean. Probability and Mathematical Statistics 29(2), 337.
Ibragimov, I.A. & Linnik, Yu.V. (1971) Independent and Stationary Sequences of Random Variables. Wolters-Noordoff.
Kim, Y.M. & Nordman, D.J. (2011) Properties of a block bootstrap under long-range dependence. Sankhya A 73(1), 79109.
Knight, K. (1998) Limiting distributions for L 1-regression estimators under general conditions. The Annals of Statistics 26(2), 755770.
Koenker, R. (2005) Quantile Regression, vol. 38. Cambridge University Press.
Koenker, R. (2017) Quantile regression: 40 years on. Annual Review of Economics 9(1), 155176.
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. 213239. Birkhauser Boston.
Kreiss, J.P. & Paparoditis, E. (2011) Bootstrap methods for dependent data: A review. Journal of the Korean Statistical Society 40(4), 357378.
Künsch, H.R. (1989) The jackknife and the bootstrap for general stationary observations. The Annals of Statistics 17(3), 12171241.
Lahiri, S.N. (2003) Resampling Methods for Dependent Data. Springer.
Lee, J.H. (2016) Predictive quantile regression with persistent covariates: IVX-QR approach. Journal of Econometrics 192(1), 105118.
Li, T.H. (2008) Laplace periodogram for time series analysis. Journal of the American Statistical Association 103(482), 757768.
Li, T.H. (2012) Quantile periodograms. Journal of the American Statistical Association 107(498), 765776.
Linton, O. & Whang, Y.J. (2007) The quantilogram: With an application to evaluating directional predictability. Journal of Econometrics 141(1), 250282.
Liu, R.Y. & Singh, K. (1992) Moving blocks jackknife and bootstrap capture weak dependence. Exploring the Limits of Bootstrap 225, 248.
Maynard, A., Shimotsu, K., & Wang, Y. (2011) Inference in predictive quantile regressions. Unpublished manuscript.
Mikusheva, A. (2007) Uniform inference in autoregressive models. Econometrica 75(5), 14111452.
Pollard, D. (1991) Asymptotics for least absolute deviation regression estimators. Econometric Theory 7(02), 186199.
Robinson, P.M. (1995) Gaussian semiparametric estimation of long range dependence. The Annals of Statistics 23(5), 16301661.
Shao, X. (2015) Self-normalization for time series: A review of recent developments. Journal of the American Statistical Association 110(512), 17971817.
Shimotsu, K. & Phillips, P.C. (2005) Exact local Whittle estimation of fractional integration. The Annals of Statistics 33(4), 18901933.
Sun, S. & Lahiri, S.N. (2006) Bootstrapping the sample quantile of a weakly dependent sequence. Sankhyā: The Indian Journal of Statistics 68(1), 130166.
Tewes, J. (2016) Block bootstrap for the empirical process of long-range dependent data. arXiv preprint arXiv:1601.01122.
Tsay, W.J. & Chung, C.F. (2000) The spurious regression of fractionally integrated processes. Journal of Econometrics 96(1), 155182.
Van der Vaart, A.W. (2000) Asymptotic Statistics, vol. 3. Cambridge University Press.
Welch, I. & Goyal, A. (2008) A comprehensive look at the empirical performance of equity premium prediction. Review of Financial Studies 21(4), 14551508.
Zhang, T., Ho, H.C., Wendler, M., & Wu, W.B. (2013) Block sampling under strong dependence. Stochastic Processes and their Applications 123(6), 23232339.

QUANTILOGRAMS UNDER STRONG DEPENDENCE

  • Ji Hyung Lee (a1), Oliver Linton (a2) and Yoon-Jae Whang (a3)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed