Asymptotic spectral theory for nonlinear time series. (English) Zbl 1147.62076

Shao, Xiaofeng; Wu, Wei Biao

Ann. Stat. 35, No. 4, 1773-1801 (2007); correction ibid. 50, No. 5, 3088-3089 (2022).

Summary: We consider asymptotic problems in spectral analysis of stationary causal processes. Limiting distributions of periodograms and smoothed periodogram spectral density estimates are obtained and applications to the spectral domain bootstrap are given. Instead of the commonly used strong mixing conditions, in our asymptotic spectral theory we impose conditions only involving (conditional) moments, which are easily verifiable for a variety of nonlinear time series.

MSC:

62M15	Inference from stochastic processes and spectral analysis
62E20	Asymptotic distribution theory in statistics
62M10	Time series, auto-correlation, regression, etc. in statistics (GARCH)

Keywords:

cumulants; Fourier transform; frequency domain bootstrap; geometric moment contraction; lag window estimator; periodogram; spectral density estimates

Software:

FinTS

References:

[1]	Anderson, T. W. (1971). The Statistical Analysis of Time Series . Wiley, New York. · Zbl 0225.62108
[2]	Andrews, D. W. K. (1984). Nonstrong mixing autoregressive processes. J. Appl. Probab. 21 930–934. JSTOR: · Zbl 0552.60049
[3]	Andrews, D. W. K. (1991). Heteroskedasticity and autocorrelation consistent covariance matrix estimation. Econometrica 59 817–858. JSTOR: · Zbl 0732.62052
[4]	Bickel, P. J. and Freedman, D. A. (1981). Some asymptotic theory for the bootstrap. Ann. Statist. 9 1196–1217. · Zbl 0449.62034
[5]	Bougerol, P. and Picard, N. (1992). Strict stationarity of generalized autoregressive processes. Ann. Probab. 20 1714–1730. · Zbl 0763.60015
[6]	Brillinger, D. R. (1969). Asymptotic properties of spectral estimates of second order. Biometrika 56 375–390. JSTOR: · Zbl 0179.23902
[7]	Brillinger, D. R. (1975). Time Series : Data Analysis and Theory . Holden-Day, San Francisco. · Zbl 0321.62004
[8]	Brockwell, P. J. and Davis, R. A. (1991). Time Series : Theory and Methods , 2nd ed. Springer, New York. · Zbl 0709.62080
[9]	Carrasco, M. and Chen, X. (2002). Mixing and moment properties of various GARCH and stochastic volatility models. Econometric Theory 18 17–39. JSTOR: · Zbl 1181.62125
[10]	Chan, K. S. and Tong, H. (1985). On the use of the deterministic Lyapunov function for the ergodicity of stochastic difference equations. Adv. in Appl. Probab. 17 666–678. JSTOR: · Zbl 0573.60056
[11]	Chanda, K. C. (2005). Large sample properties of spectral estimators for a class of stationary nonlinear processes. J. Time Ser. Anal. 26 1–16. · Zbl 1091.62093
[12]	Chen, Z. G. and Hannan, E. J. (1980). The distribution of periodogram ordinates. J. Time Ser. Anal. 1 73–82. · Zbl 0499.62085
[13]	Chow, Y. S. and Teicher, H. (1988). Probability Theory . Independence , Interchangeability , Martingales , 2nd ed. Springer, New York. · Zbl 0652.60001
[14]	Davidson, J. (1994). Stochastic Limit Theory. Oxford Univ. Press.
[15]	Davidson, J. (2002). Establishing conditions for the functional central limit theorem in nonlinear and semiparametric time series processes. J. Econometrics 106 243–269. · Zbl 1041.60032
[16]	Diaconis, P. and Freedman, D. (1999). Iterated random functions. SIAM Rev. 41 45–76. JSTOR: · Zbl 0926.60056
[17]	Ding, Z., Granger, C. and Engle, R. (1993). A long memory property of stock market returns and a new model. J. Empirical Finance 1 83–106.
[18]	Duflo, M. (1997). Random Iterative Models . Springer, Berlin. · Zbl 0868.62069
[19]	Elton, J. H. (1990). A multiplicative ergodic theorem for Lipschitz maps. Stochastic Process. Appl. 34 39–47. · Zbl 0686.60028
[20]	Engle, R. F. (1982). Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica 50 987–1007. JSTOR: · Zbl 0491.62099
[21]	Fan, J. and Yao, Q. (2003). Nonlinear Time Series : Nonparametric and Parametric Methods . Springer, New York. · Zbl 1014.62103
[22]	Fay, G. and Soulier, P. (2001). The periodogram of an i.i.d. sequence. Stochastic Process. Appl. 92 315–343. · Zbl 1046.62097
[23]	Feigin, P. D. and Tweedie, R. L. (1985). Random coefficient autoregressive processes: A Markov chain analysis of stationarity and finiteness of moments. J. Time Ser. Anal. 6 1–14. · Zbl 0572.62069
[24]	Franke, J. and Härdle, W. (1992). On bootstrapping kernel spectral estimates. Ann. Statist. 20 121–145. · Zbl 0757.62048
[25]	Freedman, D. and Lane, D. (1980). The empirical distribution of Fourier coefficients. Ann. Statist. 8 1244–1251. · Zbl 0449.62036
[26]	Freedman, D. and Lane, D. (1981). The empirical distribution of the Fourier coefficients of a sequence of independent, identically distributed long-tailed random variables. Z. Wahrsch. Verw. Gebiete 58 21–39. · Zbl 0449.62035
[27]	Götze, F. and Hipp, C. (1994). Asymptotic distribution of statistics in time series. Ann. Statist. 22 2062–2088. · Zbl 0827.62015
[28]	Granger, C. W. J. and Andersen, A. P. (1978). An Introduction to Bilinear Time Series Models. Vandenhoeck and Ruprecht, Göttingen. · Zbl 0379.62074
[29]	Grenander, U. and Rosenblatt, M. (1957). Statistical Analysis of Stationary Time Series . Wiley, New York. · Zbl 0080.12904
[30]	Hannan, E. J. (1973). Central limit theorems for time series regression. Z. Wahrsch. Verw. Gebiete 26 157–170. · Zbl 0246.62086
[31]	He, C. and Teräsvirta, T. (1999). Properties of moments of a family of GARCH processes. J. Econometrics 92 173–192. · Zbl 0929.62093
[32]	Hsing, T. and Wu, W. B. (2004). On weighted \(U\)-statistics for stationary processes. Ann. Probab. 32 1600–1631. · Zbl 1049.62099
[33]	Hurvich, C. M. and Zeger, S. L. (1987). Frequency domain bootstrap methods for time series. Technical Report 87–115, Graduate School of Business Administration, New York Univ.
[34]	Jarner, S. and Tweedie, R. (2001). Local contracting iterated functions and stability of Markov chains. J. Appl. Probab. 38 494–507. · Zbl 0996.60081
[35]	Jones, D. A. (1976). Non-linear autoregressive processes. Ph.D. dissertation, Univ. London.
[36]	Kokoszka, P. and Mikosch, T. (2000). The periodogram at the Fourier frequencies. Stochastic Process. Appl. 86 49–79. · Zbl 1025.62030
[37]	Kreiss, J.-P. and Paparoditis, E. (2003). Autoregressive-aided periodogram bootstrap for time series. Ann. Statist. 31 1923–1955. · Zbl 1042.62081
[38]	Lahiri, S. N. (2003). A necessary and sufficient condition for asymptotic independence of discrete Fourier transforms under short- and long-range dependence. Ann. Statist. 31 613–641. · Zbl 1039.62087
[39]	Li, W. K., Ling, S. and McAleer, M. (2002). Recent theoretical results for time series models with GARCH errors. In Contributions to Financial Econometrics . Theoretical and Practical Issues (M. McAleer and L. Oxley, eds.) 9–33. Blackwell, Oxford.
[40]	Ling, S. and McAleer, M. (2002). Necessary and sufficient moment conditions for the GARCH\((r,s)\) and asymmetric power GARCH\((r,s)\) models. Econometric Theory 18 722–729. JSTOR: · Zbl 1110.62332
[41]	Ling, S. and McAleer, M. (2002). Stationarity and the existence of moments of a family of GARCH processes. J. Econometrics. 106 109–117. · Zbl 1043.62073
[42]	Meyn, S. P. and Tweedie, R. L. (1993). Markov Chains and Stochastic Stability . Springer, London. · Zbl 0925.60001
[43]	Min, W. (2004). Inference on time series driven by dependent innovations. Ph.D. dissertation, Univ. Chicago.
[44]	Mokkadem, A. (1990). Propriétés de mélange des processus autorégressifs polynomiaux. Ann. Inst. H. Poincaré Probab. Statist. 26 219–260. · Zbl 0706.60040
[45]	Nordgaard, A. (1992). Resampling stochastic processes using a bootstrap approach. Bootstrapping and Related Techniques . Lecture Notes in Econom. and Math. Systems 376 181–185. Springer, Berlin.
[46]	Olshen, R. A. (1967). Asymptotic properties of the periodogram of a discrete stationary process. J. Appl. Probab. 4 508–528. JSTOR: · Zbl 0167.46403
[47]	Paparoditis, E. and Politis, D. N. (1999). The local bootstrap for periodogram statistics. J. Time Ser. Anal. 20 193–222. · Zbl 0938.62051
[48]	Parzen, E. (1957). On consistent estimates of the spectrum of a stationary time series. Ann. Math. Statist. 28 329–348. · Zbl 0081.14102
[49]	Pham, D. T. (1985). Bilinear Markovian representation and bilinear models. Stochastic Process. Appl. 20 295–306. · Zbl 0588.62162
[50]	Pham, D. T. (1986). The mixing property of bilinear and generalised random coefficient autoregressive models. Stochastic Process. Appl. 23 291–300. · Zbl 0614.60062
[51]	Pham, D. T. (1993). Bilinear time series models. In Dimension Estimation and Models (H. Tong, ed.) 191–223. World Scientific, Singapore. · Zbl 0820.62073
[52]	Priestley, M. B. (1981). Spectral Analysis and Time Series 1 . Univariate Series . Academic Press, London. · Zbl 0537.62075
[53]	Priestley, M. B. (1988). Nonlinear and Nonstationary Time Series Analysis. Academic Press, London. · Zbl 0667.62068
[54]	Robinson, P. M., ed. (2003). Time Series with Long Memory . Oxford Univ. Press. · Zbl 1113.62106
[55]	Rootzén, H. (1976). Gordin’s theorem and the periodogram. J. Appl. Probab. 13 365–370. JSTOR: · Zbl 0346.60025
[56]	Rosenblatt, M. (1984). Asymptotic normality, strong mixing, and spectral density estimates. Ann. Probab. 12 1167–1180. · Zbl 0545.62058
[57]	Rosenblatt, M. (1985). Stationary Sequences and Random Fields. Birkhäuser, Boston. · Zbl 0597.62095
[58]	Rudzkis, R. (1985). Distribution of the maximum deviation of an estimate of the spectral density of a Gaussian stationary sequence. Litovsk. Mat. Sb. 25 118–130. (In Russian.) · Zbl 0606.62104
[59]	Shao, X. and Wu, W. B. (2007). Local Whittle estimation of fractional integration for nonlinear processes. Econometric Theory . · Zbl 1274.62566
[60]	Slutsky, E. (1929). Sur l’extension de la théorie de périodogrammes aux suites des quantités dépendantes. C. R. Acad. Sci. Paris 189 722–725. · JFM 55.0939.04
[61]	Slutsky, E. (1934). Alcuni applicazioni di coefficienti di Fourier al analizo di sequenze eventuali coherenti stazionarii. Giorn. d. Instituto degli Atuari. 5 435–482. · Zbl 0010.31203
[62]	Subba Rao, T. and Gabr, M. M. (1984). An Introduction to Bispectral Analysis and Bilinear Time Series Models . Lecture Notes in Statist. 24 . Springer, New York. · Zbl 0543.62074
[63]	Terrin, N. and Hurvich, C. M. (1994). An asymptotic Wiener–Itô representation for the low frequency ordinates of the periodogram of a long memory time series. Stochastic Process. Appl. 54 297–307. · Zbl 0810.62023
[64]	Theiler, J., Linsay, P. S. and Rubin, D. M. (1994). Detecting nonlinearity in data with long coherence times. In Time Series Prediction . Forecasting the Future and Understanding the Past (A. Weigend and N. Gershenfeld, eds.) 429–455. Addison-Wesley, Reading, MA.
[65]	Tong, H. (1990). Nonlinear Time Series : A Dynamical System Approach. Oxford Univ. Press. · Zbl 0716.62085
[66]	Tsay, R. S. (2005). Analysis of Financial Time Series , 2nd ed. Wiley, New York. · Zbl 1086.91054
[67]	Tweedie, R. L. (1975). Sufficient conditions for ergodicity and recurrence of Markov chains on a general state space. Stochastic Process. Appl. 3 385–403. · Zbl 0315.60035
[68]	Tweedie, R. L. (1976). Criteria for classifying general Markov chains. Adv. in Appl. Probab. 8 737–771. JSTOR: · Zbl 0361.60014
[69]	Tweedie, R. L. (1988). Invariant measures for Markov chains with no irreducibility assumptions. J. Appl. Probab. 25A 275–285. · Zbl 0663.60054
[70]	Walker, A. M. (1965). Some asymptotic results for the periodogram of a stationary time series. J. Austral. Math. Soc. 5 107–128. · Zbl 0128.38701
[71]	Walker, A. M. (2000). Some results concerning the asymptotic distribution of sample Fourier transforms and periodograms for a discrete-time stationary process with a continuous spectrum. J. Time Ser. Anal. 21 95–109. · Zbl 0974.62082
[72]	Woodroofe, M. and Van Ness, J. W. (1967). The maximum deviation of sample spectral densities. Ann. Math. Statist. 38 1558–1569. · Zbl 0201.51801
[73]	Wu, W. B. (2005). Fourier transforms of stationary processes. Proc. Amer. Math. Soc. 133 285–293. · Zbl 1055.60016
[74]	Wu, W. B. (2005). Nonlinear system theory: Another look at dependence. Proc. Natl. Acad. Sci. USA 102 14,150–14,154. · Zbl 1135.62075
[75]	Wu, W. B. and Min, W. (2005). On linear processes with dependent innovations. Stochastic Process. Appl. 115 939–958. · Zbl 1081.62071
[76]	Wu, W. B. and Shao, X. (2004). Limit theorems for iterated random functions. J. Appl. Probab. 41 425–436. · Zbl 1046.60024
[77]	Yajima, Y. (1989). A central limit theorem of Fourier transforms of strongly dependent stationary processes. J. Time Ser. Anal. 10 375–383. · Zbl 0692.60024
[78]	Yao, J.-F. (2004). Stationarity, tails and moments of univariate GARCH processes.
[79]	Zhurbenko, I. G. and Zuev, N. M. (1975). On higher spectral densities of stationary processes with mixing. Ukrainian Math. J. 27 364–373. · Zbl 0333.60034