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

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. | Turk J Math 35 (2011) , 151 – 158. ¨ ITAK ˙ c TUB doi:10.3906/mat-0906-31 Jackknife and bootstrap with cycling blocks for the estimation of fractional parameter in ARFIMA model Lorenc Ekonomi and Argjir Butka Abstract 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. Key Words: Jackknife, bootstrap, fractionally parameter, ARFIMA model, moving blocks, non overlapping blocks. 1. Introduction Over the last couple decades, there has been much special interest in the application of long memory time series to various fields such as economy, finance, hydrology and geology. Of particular focus has been estimation of ARFIMA model parameters, especially that of fractional parameter d . For this, Granger and Joueux [10] approximated the ARFIMA model by a high-order autoregressive process and estimated d by comparing variances for different choices of d . Geweke and Porter-Hudak [9] and Kashyap and Eom [12] used a regression procedure for the logarithm of the periodogram to estimate d . Hassler [11] considered tests based on the asymptotic results of Kashyap and Eom [12], of the periodogram regression estimator of d . He showed that this test is valid only if the series are generated from a fractional white noise process. In Section 2 we give the classical method [9] for the estimation of unknown parameter d , and in Section 3 given is a short expose of the contribution to jackknife and bootstrap for time series. In Section 4 we propose two jackknife .