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Sieve-based inference for infinite-variance linear processes. (English) Zbl 1459.62168

Summary: We extend the available asymptotic theory for autoregressive sieve estimators to cover the case of stationary and invertible linear processes driven by independent identically distributed (i.i.d.) infinite variance (IV) innovations. We show that the ordinary least squares sieve estimates, together with estimates of the impulse responses derived from these, obtained from an autoregression whose order is an increasing function of the sample size, are consistent and exhibit asymptotic properties analogous to those which obtain for a finite-order autoregressive process driven by i.i.d. IV errors. As these limit distributions cannot be directly employed for inference because they either may not exist or, where they do, depend on unknown parameters, a second contribution of the paper is to investigate the usefulness of bootstrap methods in this setting. Focusing on three sieve bootstraps: the wild and permutation bootstraps, and a hybrid of the two, we show that, in contrast to the case of finite variance innovations, the wild bootstrap requires an infeasible correction to be consistent, whereas the other two bootstrap schemes are shown to be consistent (the hybrid for symmetrically distributed innovations) under general conditions.

MSC:

62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62M15 Inference from stochastic processes and spectral analysis
62G09 Nonparametric statistical resampling methods

Software:

robustbase
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Full Text: DOI Euclid

References:

[1] Arcones, M. A. and Giné, E. (1989). The bootstrap of the mean with arbitrary bootstrap sample size. Ann. Inst. Henri Poincaré Probab. Stat. 25 457-481. · Zbl 0712.62015
[2] Athreya, K. B. (1987). Bootstrap of the mean in the infinite variance case. Ann. Statist. 15 724-731. · Zbl 0628.62042 · doi:10.1214/aos/1176350371
[3] Berk, K. N. (1974). Consistent autoregressive spectral estimates. Ann. Statist. 2 489-502. · Zbl 0317.62064 · doi:10.1214/aos/1176342709
[4] Böttcher, A. and Silbermann, B. (1999). Introduction to Large Truncated Toeplitz Matrices . Springer, New York. · Zbl 0916.15012
[5] Brillinger, D. R. (2001). Time Series : Data Analysis and Theory. Classics in Applied Mathematics 36 . SIAM, Philadelphia, PA. · Zbl 0983.62056 · doi:10.1137/1.9780898719246
[6] Calder, M. and Davis, R. A. (1998). Inference for linear processes with stable noise. In A Practical Guide to Heavy Tails : Statistical Techniques and Applications (R. Adler, R. Feldman and M. Taqqu, eds.) 159-176. Birkhäuser, Basel. · Zbl 0922.62086
[7] Cavaliere, G., Georgiev, I. and Taylor, A. M. R. (2013). Wild bootstrap of the sample mean in the infinite variance case. Econometric Rev. 32 204-219. · doi:10.1080/07474938.2012.690660
[8] Cavaliere, G., Georgiev, I. and Taylor, A. (2016). Supplement to “Sieve-based inference for infinite-variance linear processes.” . · Zbl 1459.62168 · doi:10.1214/15-AOS1419
[9] Cornea-Madeira, A. and Davidson, R. (2015). A parametric bootstrap for heavy-tailed distributions. Econometric Theory 31 449-470. · Zbl 1441.62062 · doi:10.1017/S0266466614000395
[10] Davis, R. A. (2010). Heavy tails in financial time series. In Encyclopedia Quantitative Finance (R. Cont, ed.). Wiley, New York.
[11] Davis, R. A., Knight, K. and Liu, J. (1992). \(M\)-estimation for autoregressions with infinite variance. Stochastic Process. Appl. 40 145-180. · Zbl 0801.62081 · doi:10.1016/0304-4149(92)90142-D
[12] Davis, R. and Resnick, S. (1985a). Limit theory for moving averages of random variables with regularly varying tail probabilities. Ann. Probab. 13 179-195. · Zbl 0562.60026 · doi:10.1214/aop/1176993074
[13] Davis, R. and Resnick, S. (1985b). More limit theory for the sample correlation function of moving averages. Stochastic Process. Appl. 20 257-279. · Zbl 0572.62075 · doi:10.1016/0304-4149(85)90214-5
[14] Davis, R. and Resnick, S. (1986). Limit theory for the sample covariance and correlation functions of moving averages. Ann. Statist. 14 533-558. · Zbl 0605.62092 · doi:10.1214/aos/1176349937
[15] Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events : For Insurance and Finance. Applications of Mathematics ( New York ) 33 . Springer, Berlin. · Zbl 0873.62116
[16] Feller, W. (1971). An Introduction to Probability Theory and Its Applications. Vol. II , 2nd ed. Wiley, New York. · Zbl 0219.60003
[17] Finkenstädt, B. and Rootzén, H. (2003). Extreme Values in Finance , Telecommunication and the Environment . Chapman & Hall, London. · Zbl 1020.00022
[18] Gabaix, X. (2009). Power laws in economics and finance. Annual Review of Economics 1 255-293.
[19] Gonçalves, S. and Kilian, L. (2007). Asymptotic and bootstrap inference for \(\mathrm{AR}(\infty)\) processes with conditional heteroskedasticity. Econometric Rev. 26 609-641. · Zbl 1126.62079 · doi:10.1080/07474930701624462
[20] Hannan, E. J. and Kanter, M. (1977). Autoregressive processes with infinite variance. J. Appl. Probab. 14 411-415. · Zbl 0366.60033 · doi:10.2307/3213015
[21] Hill, J. B. (2013). Least tail-trimmed squares for infinite variance autoregressions. J. Time Series Anal. 34 168-186. · Zbl 1273.62064 · doi:10.1111/jtsa.12005
[22] Jaffard, S. (1990). Propriétés des matrices “bien localisées” près de leur diagonale et quelques applications. Ann. Inst. H. Poincaré Anal. Non Linéaire 7 461-476. · Zbl 0722.15004
[23] Knight, K. (1987). Rate of convergence of centred estimates of autoregressive parameters for infinite variance autoregressions. J. Time Series Anal. 8 51-60. · Zbl 0659.62107 · doi:10.1111/j.1467-9892.1987.tb00420.x
[24] Knight, K. (1989). On the bootstrap of the sample mean in the infinite variance case. Ann. Statist. 17 1168-1175. · Zbl 0687.62017 · doi:10.1214/aos/1176347262
[25] Kreiss, J.-P. (1997). Asymptotic properties of residual bootstrap for autoregression. Technical report, TU Braunschweig. Available at .
[26] Kreiss, J.-P., Paparoditis, E. and Politis, D. N. (2011). On the range of validity of the autoregressive sieve bootstrap. Ann. Statist. 39 2103-2130. · Zbl 1227.62067 · doi:10.1214/11-AOS900
[27] LePage, R. (1992). Bootstrapping signs. In Exploring the Limits of the Bootstrap (R. Lepage and L. Billard, eds.) 215-224. Wiley, New York. · Zbl 0842.62030
[28] LePage, R. and Podgórski, K. (1996). Resampling permutations in regression without second moments. J. Multivariate Anal. 57 119-141. · Zbl 0863.62039 · doi:10.1006/jmva.1996.0025
[29] LePage, R., Woodroofe, M. and Zinn, J. (1981). Convergence to a stable distribution via order statistics. Ann. Probab. 9 624-632. · Zbl 0465.60031 · doi:10.1214/aop/1176994367
[30] Lewis, R. and Reinsel, G. C. (1985). Prediction of multivariate time series by autoregressive model fitting. J. Multivariate Anal. 16 393-411. · Zbl 0579.62085 · doi:10.1016/0047-259X(85)90027-2
[31] Maronna, R. A., Martin, R. D. and Yohai, V. J. (2006). Robust Statistics : Theory and Methods . Wiley, Chichester. · Zbl 1094.62040 · doi:10.1002/0470010940
[32] McCulloch, J. H. (1997). Measuring tail thickness to estimate the stable index \(\alpha\): A critique. J. Bus. Econom. Statist. 15 74-81.
[33] Paparoditis, E. (1996). Bootstrapping autoregressive and moving average parameter estimates of infinite order vector autoregressive processes. J. Multivariate Anal. 57 277-296. · Zbl 0863.62078 · doi:10.1006/jmva.1996.0034
[34] Pötscher, B. M. and Leeb, H. (2009). On the distribution of penalized maximum likelihood estimators: The LASSO, SCAD, and thresholding. J. Multivariate Anal. 100 2065-2082. · Zbl 1170.62046 · doi:10.1016/j.jmva.2009.06.010
[35] Resnick, S. I. (1997). Heavy tail modeling and teletraffic data. Ann. Statist. 25 1805-1869. · Zbl 0942.62097 · doi:10.1214/aos/1069362376
[36] Samworth, R. (2003). A note on methods of restoring consistency to the bootstrap. Biometrika 90 985-990. · Zbl 1436.62114 · doi:10.1093/biomet/90.4.985
[37] Silverberg, G. and Verspagen, B. (2007). The size distribution of innovations revisited: An application of extreme value statistics to citation and value measures of patent significance. J. Econometrics 139 318-339. · Zbl 1418.62534 · doi:10.1016/j.jeconom.2006.10.017
[38] Strohmer, T. (2002). Four short stories about Toeplitz matrix calculations. Linear Algebra Appl. 343/344 321-344. · Zbl 0999.65026 · doi:10.1016/S0024-3795(01)00243-9
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