Jackknife and bootstrap with cycling blocks for the estimation of fractional parameter in ARFIMA model

One of most important problems concerning the ARFIMA time series model is the estimation of fractional parameter d. Various methods have been used to solve this problem, such as the log-periodogram regression of a process. In this article we propose two jackknife and bootstrap methods, which aid in the estimation of fractional parameter d. These methods involve non-overlapping blocks and moving blocks with random starting point and length. We have conducted several simulations and the results show that the estimations obtained are very close to the real parameter value.

Anahtar Kelimeler:

Jackknife, bootstrap, fractionally parameter, ARFIMA model, moving blocks, non overlapping blocks

Jackknife and bootstrap with cycling blocks for the estimation of fractional parameter in ARFIMA model

Keywords:

Jackknife, bootstrap, fractionally parameter, ARFIMA model, moving blocks, non overlapping blocks,

PDF

___

Carlstein, E.: The use of subseries values for estimating the variance of a general statistic from a stationary sequence, Ann. Statist. 14, 1171–1179, (1986).
Carlstein, E., Do, K., Hall, P., Hesterberg, T. and Kunsch, H.: Matched-block bootstrap for dependent data, Bernoulli 4, 305–328, (1998).
Davis, W. W.: Robust interval estimation of the innovation variance of an ARMA model, Ann. Statist. 5, 700–708, (1977).
Efron, B.: Bootstrap methods: another look at the jackknife, Annals of Statistics 7, 1–26, (1979).
Efron, B. and Tibshirani, R.J.: Bootstrap methods for standard errors, conŞdence intervals and other measures of statistical accuracy (with discussion), Statist. Sci. 1, 54–77, (1986).
Franco, G.C. and Reisen, V.A.: Bootstrap techniques in semiparametric estimation methods for ARFIMA models: a comparison study, Computational Statistics & Data Analysis 19, 243–259, (2004).
Freedman, D.A.: On bootstrapping two-stage least squares estimates in stationary linear models, Ann. Statist. 12, –842, (1984).
Geweke, J. and Porter-Hudak, S.: The estimation and application of long memory time series model, Journal of Time Series Analysis 4, 221–238, (1983).
Granger, C.W.J. and Joyeux, R.: An introduction to long memory time series models and fractional diﬀerencing, Journal of Time Series Analysis 1, 15–39, (1980).
Hassler, U.: Regression of spectral estimators with fractionally integrated time series, Journal of Time Series Analysis 14, 369–379, (1993).
Kashyap, R.L. and Eom, K.B.: Estimation in long-memory time series models, Journal of Time Series Analysis 9, –41, (1988).
Kunsch, H.: The jackknife and bootstrap for general stationary observations, Annals of Statistics 17, 1217–1241, (1989).
Kunsch, H. and Carlstein, E.: The linked blockwise bootstrap for serially dependent observations, Technical Report, Dept. of Statistics, Univ. of North Carolina, Chapel Hill. (1990).
Liu, R. and Singh, K.: Moving blocks jackknife and bootstrap capture weak dependence, Exploring the Limits of Bootstrap, Wiley, New York, 225–248, (1992).
Maharaj, A.E.: A test for the diﬀerence parameter of the ARFIMA model using the moving blocks bootstrap, Working Paper, Dept. of Economics and Business Statistics, Monash University Australia, (1999).
Phillips, P.C.B. and Yu, J.: JackkniŞng bond option prices, Review of Financial Studies 18, 707–742, (2005).
Phillips, P.C.B. and Yu, J.: Comment: A selective overview of nonparametric methods in Şnancial econometrics, Statistical science, Vol. 20, No.4, 338–343, (2005).
Politis, D. and Romano, J.: The stationary bootstrap, JASA, 89, 1303–1313, (1994).
Priestley, M.B.: Spectral analysis and time series 1, 2, Academic New York, (1981).
Silva, E.M., Franco, G.C., Reisen, V.A. and Cruz F.R.B.: Local bootstrap approaches for fractional diﬀerential parameter estimation in ARFIMA models, Computational Statistics & Data Analysis 51, 1002–1011, (2006).
Singh, K.: On the asymptotic accuracy of Efron’s bootstrap, Annals of Statistics 9, 1187–1195, (1981).
Tukey, J.: Bias and conŞdence in not quite large samples (abstract), Ann. Math. Statist. 29, 614, (1958).
Wu, J.: Jackknife, bootstrap and other resampling methods in regression analysis, Annals of Statistics 14, 1261– , (1986). Lorenc EKONOMI
Natural Sciences Department University of Korca, Korce, ALBANIA e-mail: lorencekonomi@yahoo.co.uk Argjir BUTKA Natural Sciences Department University of Korca, Korce, ALBANIA e-mail: argjirbutka@yahoo.com