Bernoulli

  • Bernoulli
  • Volume 24, Number 4A (2018), 2991-3012.

Simultaneous quantile inference for non-stationary long-memory time series

Weichi Wu and Zhou Zhou

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We consider the simultaneous or functional inference of time-varying quantile curves for a class of non-stationary long-memory time series. New uniform Bahadur representations and Gaussian approximation schemes are established for a broad class of non-stationary long-memory linear processes. Furthermore, an asymptotic distribution theory is developed for the maxima of a class of non-stationary long-memory Gaussian processes. Using the latter theoretical results, simultaneous confidence bands for the aforementioned quantile curves with asymptotically correct coverage probabilities are constructed.

Article information

Source
Bernoulli, Volume 24, Number 4A (2018), 2991-3012.

Dates
Received: April 2016
Revised: April 2017
First available in Project Euclid: 26 March 2018

Permanent link to this document
https://projecteuclid.org/euclid.bj/1522051231

Digital Object Identifier
doi:10.3150/17-BEJ951

Mathematical Reviews number (MathSciNet)
MR3779708

Zentralblatt MATH identifier
06853271

Keywords
heterogeneity local linear quantile estimation long memory simultaneous confidence bands

Citation

Wu, Weichi; Zhou, Zhou. Simultaneous quantile inference for non-stationary long-memory time series. Bernoulli 24 (2018), no. 4A, 2991--3012. doi:10.3150/17-BEJ951. https://projecteuclid.org/euclid.bj/1522051231


Export citation

References

  • [1] Baillie, R.T. (1996). Long memory processes and fractional integration in econometrics. J. Econometrics 73 5–59.
  • [2] Bekaert, G. and Harvey, C.R. (1995). Time-varying world market integration. J. Finance 50 403–444.
  • [3] Beran, J. (2009). On parameter estimation for locally stationary long-memory processes. J. Statist. Plann. Inference 139 900–915.
  • [4] Berman, S.M. (1972). Maximum and high level excursion of a Gaussian process with stationary increments. Ann. Math. Stat. 43 1247–1266.
  • [5] Bickel, P.J. and Rosenblatt, M. (1973). On some global measures of the deviations of density function estimates. Ann. Statist. 1 1071–1095.
  • [6] Chaudhuri, P. (1991). Nonparametric estimates of regression quantiles and their local Bahadur representation. Ann. Statist. 19 760–777.
  • [7] Clarke, R.T. (2007). Hydrological prediction in a non-stationary world. Hydrol. Earth Syst. Sci. 11 408–414.
  • [8] Coeurjolly, J.-F. (2008). Bahadur representation of sample quantiles for functional of Gaussian dependent sequences under a minimal assumption. Statist. Probab. Lett. 78 2485–2489.
  • [9] Cooley, T.F. and Prescott, E.C. (1976). Estimation in the presence of stochastic parameter variation. Econometrica 44 167–184.
  • [10] Csörgő, M. and Kulik, R. (2008). Reduction principles for quantile and Bahadur–Kiefer processes of long-range dependent linear sequences. Probab. Theory Related Fields 142 339–366.
  • [11] Csörgő, M. and Révész, P. (1981). Strong Approximations in Probability and Statistics. Probability and Mathematical Statistics. Academic Press: New York.
  • [12] Dahlhaus, R. and Polonik, W. (2009). Empirical spectral processes for locally stationary time series. Bernoulli 15 1–39.
  • [13] Dahlhaus, R. and Subba Rao, S. (2006). Statistical inference for time-varying ARCH processes. Ann. Statist. 34 1075–1114.
  • [14] Dehling, H. and Taqqu, M.S. (1989). The empirical process of some long-range dependent sequences with an application to $U$-statistics. Ann. Statist. 17 1767–1783.
  • [15] Eichner, J.F., Koscielny-Bunde, E., Bunde, A., Havlin, S. and Schellnhuber, H.-J. (2003). Power-law persistence and trends in the atmosphere: A detailed study of long temperature records. Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics 68 046133.
  • [16] Einmahl, U. (1987). A useful estimate in the multidimensional invariance principle. Probab. Theory Related Fields 76 81–101.
  • [17] Einmahl, U. (1987). Strong invariance principles for partial sums of independent random vectors. Ann. Probab. 15 1419–1440.
  • [18] Einmahl, U. (1989). Extensions of results of Komlós, Major, and Tusnády to the multivariate case. J. Multivariate Anal. 28 20–68.
  • [19] Fryzlewicz, P., Sapatinas, T. and Subba Rao, S. (2008). Normalized least-squares estimation in time-varying ARCH models. Ann. Statist. 36 742–786.
  • [20] Gray, H.L., Zhang, N.-F. and Woodward, W.A. (1989). On generalized fractional processes. J. Time Ser. Anal. 10 233–257.
  • [21] Gray, H.L., Zhang, N.-F. and Woodward, W.A. (1994). A correction: “On generalized fractional processes” [J. Time Ser. Anal. 10 (1989), no. 3, 233–257; MR1028940 (90m:62208)]. J. Time Series Anal. 15 561–562.
  • [22] Härdle, W. (1989). Asymptotic maximal deviation of $M$-smoothers. J. Multivariate Anal. 29 163–179.
  • [23] Harvey, C.R. (1989). Time-varying conditional covariances in tests of asset pricing models. J. Financ. Econ. 24 289–317.
  • [24] Haslett, J. and Raftery, A. (1989). Space-time modelling with long-memory dependence: Assessing Ireland’s wind power resource. Appl. Stat. 38 1–50.
  • [25] He, X. and Shao, Q.-M. (1996). A general Bahadur representation of $M$-estimators and its application to linear regression with nonstochastic designs. Ann. Statist. 24 2608–2630.
  • [26] Henry, M. and Zaffaroni, P. (2003). The long-range dependence paradigm for macroeconomics and finance. In Theory and Applications of Long-Range Dependence (P. Doukhan, G. Oppenheim and M.S. TurTaqqukman, eds.) 417–438. Boston, MA: Birkhäuser.
  • [27] Ho, H.-C. and Hsing, T. (1997). Limit theorems for functionals of moving averages. Ann. Probab. 25 1636–1669.
  • [28] Holton, G.A. (2003). Value-at-Risk: Theory and Practice. San Diego: Academic Press.
  • [29] Hurst, H.E. (1951). Long term storage capacity of reservoirs. Trans. Am. Soc. Civ. Eng. 116 770–799.
  • [30] Jorion, P. (2006). Value at risk. In The New Benchmark for Managing Financial Risk, Vol. 9, 3rd ed. New York: McGraw Hill Professional.
  • [31] Kärner, O. (2002). On nonstationarity and antipersistency in global temperature series. Journal of Geophysical Research D: Atmospheres 107.
  • [32] Koenker, R. (2005). Quantile Regression. Econometric Society Monographs 38. Cambridge: Cambridge Univ. Press.
  • [33] Kokoszka, P.S. and Taqqu, M.S. (1994). Infinite variance stable ARMA processes. J. Time Series Anal. 15 203–220.
  • [34] Kokoszka, P.S. and Taqqu, M.S. (1995). Fractional ARIMA with stable innovations. Stochastic Process. Appl. 60 19–47.
  • [35] Komlós, J., Major, P. and Tusnády, G. (1975). An approximation of partial sums of independent $\mathrm{RV}$’s and the sample $\mathrm{DF}$. I. Z. Wahrsch. Verw. Gebiete 32 111–131.
  • [36] Komlós, J., Major, P. and Tusnády, G. (1976). An approximation of partial sums of independent RV’s, and the sample DF. II. Z. Wahrsch. Verw. Gebiete 34 33–58.
  • [37] Leipus, R. and Surgailis, D. (2013). Asymptotics of partial sums of linear processes with changing memory parameter. Lith. Math. J. 53 196–219.
  • [38] Mann, M.E. (2010). On long range dependence in global surface temperature series. Clim. Change 107 267–276.
  • [39] Mercurio, D. and Spokoiny, V. (2004). Statistical inference for time-inhomogeneous volatility models. Ann. Statist. 32 577–602.
  • [40] Mills, T.C. (2007). Time series modelling of two millennia of northern hemisphere temperatures: Long memory or shifting trends? J. Roy. Statist. Soc. Ser. A 170 83–94.
  • [41] Orbe, S., Ferreira, E. and Rodriguez-Poo, J. (2005). Nonparametric estimation of time varying parameters under shape restrictions. J. Econometrics 126 53–77.
  • [42] Palma, W. (2010). On the sample mean of locally stationary long-memory processes. J. Statist. Plann. Inference 140 3764–3774.
  • [43] Palma, W. and Olea, R. (2010). An efficient estimator for locally stationary Gaussian long-memory processes. Ann. Statist. 38 2958–2997.
  • [44] Pollard, D. (1990). Empirical Processes: Theory and Applications. NSF-CBMS Regional Conference Series in Probability and Statistics 2. Hayward, CA: IMS.
  • [45] Porter-Hudak, S. (1990). An application of the seasonal fractionally differenced model to the monetary aggregates. J. Amer. Statist. Assoc. 85 338–344.
  • [46] Ravn, M.O., Schmitt-Grohé, S. and Uribe, M. (2008). Macroeconomics of subsistence points. Macroecon. Dyn. 12 136–147.
  • [47] Rea, W., Reale, M., Brown, J. and Oxley, L. (2011). Long memory or shifting means in geophysical time series? Math. Comput. Simulation 81 1441–1453.
  • [48] Roueff, F. and von Sachs, R. (2011). Locally stationary long memory estimation. Stochastic Process. Appl. 121 813–844.
  • [49] Shao, Q.M. (1995). Strong approximation theorems for independent random variables and their applications. J. Multivariate Anal. 52 107–130.
  • [50] Smith, V. (1993). Long Range Dependence and Global Warming. In Statistics for the Environment (V. Barnett and F. Turkman, eds.) Wiley: New York.
  • [51] Stock, J.H. and Watson, M.W. (1996). Evidence on structural instability in macroeconomic time series relations. J. Bus. Econom. Statist. 14 11–30.
  • [52] Sun, J. (1993). Tail probabilities of the maxima of Gaussian random fields. Ann. Probab. 21 34–71.
  • [53] Sun, J. and Loader, C.R. (1994). Simultaneous confidence bands for linear regression and smoothing. Ann. Statist. 22 1328–1345.
  • [54] Tsay, R.S. (2010). Analysis of Financial Time Series, 3rd ed. Wiley Series in Probability and Statistics. Hoboken, NJ: Wiley.
  • [55] Wang, Q., Lin, Y.-X. and Gulati, C.M. (2003). Strong approximation for long memory processes with applications. J. Theoret. Probab. 16 377–389.
  • [56] Wu, W. and Zhou, Z. (2017). Supplement to “Simultaneous Quantile Inference For Non-Stationary Long-Memory Time Series.” DOI:10.3150/17-BEJ951SUPP.
  • [57] Wu, W.B. (2007). Strong invariance principles for dependent random variables. Ann. Probab. 35 2294–2320.
  • [58] Wu, W.B. and Zhou, Z. (2011). Gaussian approximations for non-stationary multiple time series. Statist. Sinica 21 1397–1413.
  • [59] Xia, Y. (1998). Bias-corrected confidence bands in nonparametric regression. J. R. Stat. Soc. Ser. B. Stat. Methodol. 60 797–811.
  • [60] Zaitsev, A.Yu. (2000). Multidimensional version of a result of Sakhanenko in the invariance principle for vectors with finite exponential moments. I. Theory Probab. Appl. 45 624–641.
  • [61] Zaitsev, A.Yu. (2001). Multidimensional version of a result of Sakhanenko in the invariance principle for vectors with finite exponential moments. II. Theory Probab. Appl. 46 490–514.
  • [62] Zaitsev, A.Yu. (2001). Multidimensional version of a result of Sakhanenko in the invariance principle for vectors with finite exponential moments. III. Theory Probab. Appl. 46 676–698.
  • [63] Zhang, T. and Wu, W.B. (2012). Inference of time-varying regression models. Ann. Statist. 40 1376–1402.
  • [64] Zhou, Z. (2010). Nonparametric inference of quantile curves for nonstationary time series. Ann. Statist. 38 2187–2217.
  • [65] Zhou, Z. (2013). Heteroscedasticity and autocorrelation robust structural change detection. J. Amer. Statist. Assoc. 108 726–740.
  • [66] Zhou, Z. and Wu, W.B. (2009). Local linear quantile estimation for nonstationary time series. Ann. Statist. 37 2696–2729.
  • [67] Zhou, Z. and Wu, W.B. (2011). On linear models with long memory and heavy-tailed errors. J. Multivariate Anal. 102 349–362.

Supplemental materials

  • Supplement to “Simultaneous quantile inference for non-stationary long-memory time series”. In the supplementary material, we provide complete proofs for lemmas, corollaries, propositions and theorems.