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Small Sample Bootstrap Confidence Intervals for Long-Memory Parameter

Authors: Josu Arteche, Jesus Orbe

Abstract:

The log periodogram regression is widely used in empirical applications because of its simplicity, since only a least squares regression is required to estimate the memory parameter, d, its good asymptotic properties and its robustness to misspecification of the short term behavior of the series. However, the asymptotic distribution is a poor approximation of the (unknown) finite sample distribution if the sample size is small. Here the finite sample performance of different nonparametric residual bootstrap procedures is analyzed when applied to construct confidence intervals. In particular, in addition to the basic residual bootstrap, the local and block bootstrap that might adequately replicate the structure that may arise in the errors of the regression are considered when the series shows weak dependence in addition to the long memory component. Bias correcting bootstrap to adjust the bias caused by that structure is also considered. Finally, the performance of the bootstrap in log periodogram regression based confidence intervals is assessed in different type of models and how its performance changes as sample size increases.

Keywords: bootstrap, confidence interval, log periodogram regression, long memory.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1083581

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References:


[1] Velasco, C., 1999. Non stationary log-periodogram regression. J. Econometrics 91, 325-371.
[2] Geweke, J. and Porter-Hudak, S., 1983. The estimation and application of long-memory time series models. J. Time Ser. Anal. 4, 221-238.
[3] Robinson, P.M., 1995. Log-periodogram regression of time series with long-range dependence. Ann. Statist. 23, 1048-1072.
[4] Hurvich, C.M., Deo, R. and Brodsky, J., 1998. The mean squared error of Geweke and Porter-Hudak-s estimator of the memory parameter in a long-memory time series. J. Time Ser. Anal. 19, 19-46.
[5] Phillips, P.C.B., 2007. Unit root log periodogram regression. J. Econometrics 138, 104-124.
[6] Kim, C.S. and Phillips, P.C.B., 2006. Log periodogram regression: The nonstationary case. Cowles Foundation Discussion Paper No. 1587.
[7] Phillips, P.C.B. and Shimotsu, K., 2004. Local Whittle estimation in nonstationary and unit root cases. Ann. Statist. 32, 656-692.
[8] Arteche, J., 2004, Gaussian Semiparametric Estimation in Long Memory in Stochastic Volatility and Signal Plus Noise Models. J. Econometrics 119, 131-154.
[9] Giraitis, L., Robinson, P.M. and Samarov, A., 2000. Adaptive semiparametric estimation of the memory parameter. J. Multiv. Anal. 72, 183-207.
[10] Hurvich, C.M., and Deo, R.S., 1999. Plug-in selection of the number of frequencies in regression estimates of the memory parameter of a long-memory time series. J. Time Ser. Anal. 20, 331-341.
[11] Hassler, U. and Wolters, J., 1995. Long memory in inflation rates: International evidence. J. Business Econ. Stat. 13, 37-45.
[12] Diebold, F.X. and Rudebush, G., 1989. Long memory and persistence in aggregate output. J. Monet. Econ. 24, 189-209.
[13] Diebold, F.X. and Rudebush, G., 1991. Is consumption too smooth: Long memory and the Deaton paradox. Rev. Econ. Statist. 73, 1-9.
[14] Sowell, F., 1992a. Maximum likelihood estimation of stationary univariate fractionally integrated time series models. J. Econometrics 53, 165-188.
[15] Arteche, J. and Robinson, P.M., 2000. Semiparametric inference in seasonal and cyclical long memory processes. J. Time Ser. Anal. 21, 1-27.
[16] Sowell, F., 1992b. Modeling long-run behaviour with the fractional ARIMA model. J. Monet. Econ. 29, 277-302.
[17] Andrews, D.W.K., Lieberman, O. and Marmer, V., 2006. Higher-order improvements of the parametric bootstrap for long-memory time series. J. Econometrics 133, 673-702.
[18] Andersson, M.K. and Gredenhoff, M.P., 1998. Robust testing for fractional integration using the bootstrap. Working Paper Series in Economics and Finance 218, Stockholm School of Economics, Sweden.
[19] Hidalgo, J., 2003. An alternative bootstrap to moving blocks for time series regression models. J. Econometrics 117, 369-399.
[20] Silva, E.M., Franco, G.C., Reisen, V.A. and Cruz, F.R.B., 2006. Local bootstrap approaches for fractional differential parameter estimation in ARFIMA models. Comput. Statist. Data Anal. 51, 1002-1011.
[21] Arteche, J. and Orbe, J., 2005. Bootstrapping the log-periodogram regression. Econ. Letters 86, 70-85.
[22] Agiakloglou, C., Newbold, P. and Wohar, M., 1993. Bias in an estimator of the fractional difference parameter. J. Time Ser. Anal. 14, 235-246.
[23] Paparoditis, E. and Politis, D., 1999. The local bootstrap for periodogram statistics. J. Time Ser. Anal. 20, 193-222.
[24] Efron, B., 1982. The jackknife, the bootstrap, and other resampling plans. Volume 38 of CBMS-NSF Regional Conference Series in Applied Mathematics. SIAM.
[25] Efron, B., 1987. Better bootstrap confidence intervals. J. Amer. Statistical Assoc. 82, 171-200.
[26] Mackinnon, J.G. and Smith, A.A., 1998. Approximate bias correction in econometrics. J. Econometrics 85, 205-230.
[27] Efron, B., 1979. Bootstrap methods: Another look at the jackknife. Ann. Statist. 7, 1-26.
[28] Davidson, R. and MacKinnon, J.G., 2006. Bootstrap methods in econometrics. In: Mills, T.C. and Patterson, K.D. (eds.), Palgrave Handbooks of Econometrics: Volume 1, Econometric Theory. Palgrave Macmillan, 812-838.
[29] K¨unsch, H.R., 1989. The jackknife and the bootstrap for general stationary observations. Ann. Statist. 17, 1217-1261.
[30] Liu, R.Y. and Singh, K., 1992. Moving blocks jackknife and bootstrap capture weak dependence. In: LePage, R. and Billard, L.(Eds.) Exploring the limits of bootstrap. Wiley, New York, 225-248.
[31] Lahiri, S.N., 1999. Theoretical comparisons of block bootstrap methods. Ann. Statist. 27, 386-404.
[32] Kilian, L., 1998. Small sample confidence intervals for impulse response functions. Rev. Econ. Statist. 80, 218-230.
[33] Shao, J. and Tu, D., 1995. The jackknife and bootstrap. Springer Verlag: New York.
[34] Efron, B. and Tibshirani, R.J., 1993. An introduction to the bootstrap. Chapman and Hall: New York.
[35] Davison, A.C. and Hinkley, D.V., 1997. Bootstrap methods and their application. Cambridge University Press: Cambridge.
[36] Arteche, J., 2006. Semiparametric estimation in perturbed long memory series. Comput. Statist. Data Anal. 51, 2118-2141.
[37] Hurvich, C.M., Moulines, E. and Soulier, P., 2005. Estimating long memory in volatility. Econometrica 73, 1283-1328.
[38] Sun, Y. and Phillips, P.C.B., 2003. Nonlinear log-periodogram regression for perturbed fractional processes. J. Econometrics 115, 355-389.
[39] Andrews, D.W.K. and Guggenberger, P., 2003. A bias-reduced logperiodogram regression estimator for the long-memory parameter. Econometrica 71, 675-712.