Local Bootstrap Approach for the Estimation of the Memory Parameter J. Arteche and J. Orbe * Abstract— The log periodogram regression is widely used in empirical applications because of its simplicity to estimate the memory parameter, d, its good asymp- totic 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 sam- ple size is small. Here the finite sample performance of different nonparametric residual bootstrap proce- dures is analyzed when applied to construct confi- dence intervals. In particular, in addition to the basic residual bootstrap the local bootstrap that might ad- equately replicate the structure that may arise in the errors of the regression is considered when the series shows weak dependence in addition to the long mem- ory component. Bias correcting bootstrap to adjust the bias caused by that structure is also considered. Keywords: bootstrap, confidence interval, log peri- odogram regression, long memory 1 Introduction Long memory processes are characterized by a strong de- pendence such that the lag-j autocovariances γ j decrease hyperbolically as j →∞ γ j ∼ Gj 2d-1 for some finite constant G, d is the memory parameter and a ∼ b means that a/b tends in the limit to 1. For d> 0, ∑ |γ j | = ∞ but stationarity is guaranteed as long as d< 1/2 and mean reversion holds for d< 1. It is also usually assumed that d> -1/2, which warrants invert- ibility. Long memory can alternatively be defined in the frequency domain. A stationary time series process has long memory if its spectral density function f (·) satisfies f (λ) ∼ C|λ| -2d as λ → 0, (1) for some positive finite constant C. Under positive long memory, which is the most common case in economic and * Authors are with Dpt. Econometr´ ıa y Estad´ ıstica, University of the Basque Country, Avda. Lehendakari Agirre 83, 48015 Bil- bao, Spain (email: josu.arteche@ehu.es, jesus.orbe@ehu.es). The authors wish to acknowledge financial support from the Spanish Ministerio de Ciencia y Tecnologa and FEDER grants MTM2006- 06550 and SEJ2007-61362/ECON and from the Department of Education of the Basque Government through grant IT-334-07 (UPV/EHU Econometrics Research Group). financial series, the spectral density diverges at the origin at a rate governed by d. If d> 1/2 the process is not sta- tionary and, by definition, the spectral density does not exist. However pseudo spectral density functions can be similarly defined (e.g. [1]) with a behavior as in (1). One issue of main interest in these processes is the estimation of d. Perhaps the most popular is the log periodogram regression estimator (LPE hereafter) originally proposed by [2] and analyzed in detail in [3] and [4]. The LPE is widely used in empirical applications because of its sim- plicity, since only a least squares regression is required, its good asymptotic and finite samples properties and its robustness to misspecification of the short term behavior of the series. Taking logarithms of the local specification of the spectral density in (1), the LPE ( ˆ d) is obtained by least squares in the regression log I j = a + dX j + u j , j =1, ..., m, (2) where X j = -2 log λ j , a = log C + c, c =0.577216 is Euler’s constant, I j = (2πn) -1 | ∑ n t=1 x t exp(-itλ j )| 2 is the periodogram of the series x t , t = 1, .., n, at Fourier frequency λ j =2πj/n, n is the sample size, u j = log(I j f (λ j ) -1 ) - c and m represents the bandwidth, that is the number of frequencies used in the estimation. For the asymptotics, this bandwidth has to increase with n but at a slower rate such that the band of frequencies used in the estimation degenerates to zero and the local specification in (1) remains valid. [3] and [4] proved the consistency of ˆ d in the stationary and invertible region -0.5 <d< 0.5, and obtained its limit distribution √ m( ˆ d - d) d → N  0, π 2 24  . (3) Reference [1] showed that the consistency holds even in the nonstationary region [0.5, 1) and the same limit dis- tribution remains valid for d ∈ [0.5, 0.75). In practice the choice of the bandwidth is crucial, a large m decreases the variance at the cost of a higher bias which can be extremely large in some situations, for example in the presence of some short term component such as those analyzed below. The choice of an optimal bandwidth is not a simple task. Some attempts have been made in [5], [6] and [7]. However, the performance of all these proce- dures is not very satisfactory and the results for a grid of bandwidths are usually shown in empirical applications. Proceedings of the World Congress on Engineering 2009 Vol II WCE 2009, July 1 - 3, 2009, London, U.K. ISBN:978-988-18210-1-0 WCE 2009