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Efficiency improvements in inference on stationary and nonstationary fractional time series. (English) Zbl 1078.62096
Summary: We consider a time series model involving a fractional stochastic component, whose integration order can lie in the stationary/invertible or nonstationaly regions and be unknown, and an additive deterministic component consisting of a generalized polynomial. The model can thus incorporate competing descriptions of trending behavior. The stationary input to the stochastic component has parametric autocorrelation, but innovation with distribution of unknown form. The model is thus semiparametric, and we develop estimates of the parametric component which are asymptotically normal and achieve an \(M\)-estimation efficiency bound, equal to that found in work using an adaptive LAM/LAN approach.
A major technical feature which we treat is the effect of truncating the autoregressive representation in order to form innovation proxies. This is relevant also when the innovation density is parameterized, and we provide a result for that case also. Our semiparametric estimates employ nonparametric series estimation, which avoids some complications and conditions in kernel approaches featured in much work on adaptive estimation of time series models; our work thus also contributes to methods and theory for nonfractional time series models, such as autoregressive moving averages. A Monte Carlo study of finite sample performance of the semiparametric estimates is included.

MSC:
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62J05 Linear regression; mixed models
62G08 Nonparametric regression and quantile regression
65C05 Monte Carlo methods
62G10 Nonparametric hypothesis testing
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References:
[1] Beran, J. (1995). Maximum likelihood estimation of the differencing parameter for invertible short and long memory autoregressive integrated moving average models. J. Roy. Statist. Soc. Ser. B 57 659–672. · Zbl 0827.62088
[2] Beran, R. (1976). Adaptive estimates for autoregressive processes. Ann. Inst. Statist. Math. 28 77–89. · Zbl 0362.62093 · doi:10.1007/BF02504731
[3] Bickel, P. (1982). On adaptive estimation. Ann. Statist. 10 647–671. JSTOR: · Zbl 0489.62033 · doi:10.1214/aos/1176345863 · links.jstor.org
[4] Bloomfield, P. (1973). An exponential model for the spectrum of a scalar time series. Biometrika 60 217–226. · Zbl 0261.62074 · doi:10.1093/biomet/60.2.217
[5] Box, G. E. P. and Jenkins, G. M. (1970). Time Series Analysis , Forecasting and Control . Holden–Day, San Francisco. · Zbl 0249.62009
[6] Dahlhaus, R. (1989). Efficient parameter estimation for self-similar processes. Ann. Statist. 17 1749–1766. JSTOR: · Zbl 0703.62091 · doi:10.1214/aos/1176347393 · links.jstor.org
[7] Dahlhaus, R. (1995). Efficient location and regression estimation for long range dependent regression models. Ann. Statist. 23 1029–1047. JSTOR: · Zbl 0838.62084 · doi:10.1214/aos/1176324635 · links.jstor.org
[8] Drost, F. C., Klaassen, C. A. J. and Werker, B. J. M. (1997). Adaptive estimation in time series models. Ann. Statist. 25 786–817. · Zbl 0941.62093 · doi:10.1214/aos/1031833674
[9] Fox, R. and Taqqu, M. S. (1986). Large-sample properties of parameter estimates for strongly dependent stationary Gaussian time series. Ann. Statist. 14 517–532. JSTOR: · Zbl 0606.62096 · doi:10.1214/aos/1176349936 · links.jstor.org
[10] Freud, G. (1971). Orthogonal Polynomials . Pergamon, Oxford. · Zbl 0226.33014 · doi:10.1007/BF01094355
[11] Giraitis, L. and Surgailis, D. (1990). A central limit theorem for quadratic forms in strongly dependent linear variables and its application to asymptotic normality of Whittle’s estimate. Probab. Theory Related Fields 86 87–104. · Zbl 0717.62015 · doi:10.1007/BF01207515
[12] Hájek, J. (1972). Local asymptotic minimax and admissibility in estimation. Proc. Sixth Berkeley Symp. Math. Statist. Probab. 1 175–194. Univ. California Press, Berkeley. · Zbl 0281.62010
[13] Hall, P. and Heyde, C. C. (1980). Martingale Limit Theory and Its Application . Academic Press, New York. · Zbl 0462.60045
[14] Hallin, M. and Serroukh, A. (1998). Adaptive estimation of the lag of a long-memory process. Stat. Inference Stoch. Process . 1 111–129. · Zbl 1061.62555 · doi:10.1023/A:1009905527102
[15] Hallin, M., Taniguchi, M., Serroukh, A. and Choy, K. (1999). Local asymptotic normality for regression models with long-memory disturbance. Ann. Statist. 27 2054–2080. · Zbl 0957.62077 · doi:10.1214/aos/1017939250
[16] Hannan, E. J. (1973). The asymptotic theory of linear time series models. J. Appl. Probability 10 130–145. · Zbl 0261.62073 · doi:10.2307/3212501
[17] Knuth, D. E. (1968). The Art of Computer Programming 1 . Fundamental Algorithms . Addison–Wesley, Reading, MA. · Zbl 0191.17903
[18] Koul, H. L. and Schick, A. (1997). Efficient estimation in nonlinear autoregressive time series models. Bernoulli 3 247–277. · Zbl 1066.62537 · doi:10.2307/3318592
[19] Kreiss, J.-P. (1987). On adaptive estimation in stationary ARMA processes. Ann. Statist. 15 112–133. JSTOR: · Zbl 0616.62042 · doi:10.1214/aos/1176350256 · links.jstor.org
[20] Le Cam, L. (1960). Locally asymptotically normal families of distributions. Univ. California Publ. Statist. 3 37–98. · Zbl 0104.12701
[21] Li, W. K. and McLeod, A. I. (1986). Fractional time series modelling. Biometrika 73 217–221.
[22] Ling, S. (2003). Adaptive estimators and tests of stationary and nonstationary short- and long-memory ARFIMA–GARCH models. J. Amer. Statist. Assoc. 98 955–967. · Zbl 1045.62089 · doi:10.1198/016214503000000918
[23] Loeve, M. (1977). Probability Theory 1 , 4th ed. Springer, New York. · Zbl 0359.60001
[24] Martin, R. D. (1982). The Cramér–Rao bound and robust \(M\)-estimates for autoregressions. Biometrika 69 437–442. · Zbl 0499.62029
[25] Newey, W. K. (1988). Adaptive estimation of regression models via moment restrictions. J. Econometrics 38 301–339. · Zbl 0686.62045 · doi:10.1016/0304-4076(88)90048-6
[26] Robinson, P. M. (1994). Efficient tests of nonstationary hypotheses. J. Amer. Statist. Assoc. 89 1420–1437. · Zbl 0813.62016 · doi:10.2307/2291004
[27] Scott, D. J. (1973). Central limit theorems for martingales and for processes with stationary increments using a Skorokhod representation approach. Adv. in Appl. Probab. 5 119–137. · Zbl 0263.60011 · doi:10.2307/1425967
[28] Stone, C. J. (1975). Adaptive maximum-likelihood estimators of a location parameter. Ann. Statist. 3 267–284. · Zbl 0303.62026 · doi:10.1214/aos/1176343056
[29] Velasco, C. and Robinson, P. M. (2000). Whittle pseudo-maximum likelihood estimation for nonstationary time series. J. Amer. Statist. Assoc. 95 1229–1243. · Zbl 1008.62087 · doi:10.2307/2669763
[30] Whittaker, E. T. and Watson, G. N. (1940). A Course of Modern Analysis , 4th ed. Cambridge Univ. Press. · Zbl 0951.30002
[31] Yajima, Y. (1988). On estimation of a regression model with long-memory stationary errors. Ann. Statist. 16 791–807. JSTOR: · Zbl 0661.62090 · doi:10.1214/aos/1176350837 · links.jstor.org
[32] Yajima, Y. (1991). Asymptotic properties of the LSE in a regression model with long memory stationary errors. Ann. Statist. 19 158–177. · Zbl 0728.62085 · doi:10.1214/aos/1176347975
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