Bootstrap and approximation methods for long memory processes

Fractionally integrated processes ARFIMA(p,d,q), introduced by Granger (1980) and Hosking (1981) independently, offer a useful tool to model the second order dependence structure (autocovariance and autocorrelation functions) of an observed time series. The literature is rich of paper on identification of the data generating process (dgp, from now on) and estimation of the parameters: Yajima (1985), Taqqu (1986) and Dahlhaus (1988), Dahlhaus (1989) wrote papers on parametric estimate of the memory parameter d, whereas Hurst (1951), Geweke and Porter-Hudak (1983), Higuchi (1988), Robinson (1995) and Hurvich (1998) developed semi-parametric estimation methods. It is not possible to define the best method, according to the situation each method offers advantages and drawbacks. Parametric estimators are asymptotically Normal and they are the most efficient, however in the case of misspecification of the model the estimates might be dramatically biased. On the other hand, semi-parametric estimators offer the possibility of estimating the long memory parameter from the short memory part with the drawback of a slower convergence rate (o(n -1/2)) or less) than with parametric techniques (o(n-1)). Moreover, Agiakloglou (1993) showed that the GPH Geweke and Porter-Hudak (1983) is biased in presence of arma parameter near the non-stationary area. A generalisation of ARFIMA processes are the Gegenbauer processes, introduced by Hosking (1981) and then studied by Woodward (1989), Woodward (1998), Giraitis (1995), Smallwood (2004), Sadek (2004) and Caporale (2006). Also in this case parametric and semi-parametric technique are available in the literature. One the main problem is the maximization in a multidimensional space of the likelihood function because there is not a close form and the existing numerical procedures are quite burdensome. Semi-parametric procedures play and important role to compute good starting values to maximize the likelihood function and to identify the order of the dgp. In the last years many bootstrap methods for time series have been developed, such as the model-based resampling, the block bootstrap (Kunsch, 1989), the autoregressive-aided periodogram bootstrap (Kreiss, 2003), the local bootstrap (Paparoditis, 1999), the sieve bootstrap (Kreiss, 1992), the parametric bootstrap (Andrews, 2006), the kernel bootstrap (Franke, 1992,Dahlhaus, 1996) and the phase scrambling (Theiler, 1992b). Bootstrap methods for time series have been widely used to build confidence intervals especially when asymptotic theory does not provide satisfactory results (Efron, 1979, Efron, 1982, Efron, 1987, Efron, 1987a, Hall, 1988, Hall, 1992, Arteche, 2005). The problem is still open when we want to replicate the dependence structure of a long memory process such as ARFIMA(p,d,q). In this thesis we develop a new bootstrap method for time series, the ACF bootstrap, based on a result of Ramsey (1974), that generates the surrogate series from the observed autocorrelation function. The thesis is divided in five chapters: the first two chapters review some literature, the last three chapters are new contributions. The first chapter reviews the literature on long memory processes, the properties of their sample autocorrelation and autocovariance functions, the most common parametric and semi-parametric estimators and, shortly, Gegenbauer processes. In the second chapter, we introduce briefly some bootstrap methods for time series. In Chapter 3 we introduce the new bootstrap method. We apply the ACF bootstrap to improve the performance of semi-parametric estimators for the memory parameter d for ARFIMA(0,d,0) processes in terms of smaller standard error, smaller mean squared error and better coverage for confidence intervals. Since the condition of Gaussianity of the observed process is very restrictive, we show, by means of Monte Carlo simulation, that the method is consistent even relaxing this hypothesis. In particular the method seems to be robust against fat tails and itshape asymmetry}. Another application is building confidence intervals for the memory parameter d. For the parametric Whittle estimator, the confidence intervals based on the bootstrap distribution have a closer coverage to the theoretical level if the time series is relatively short (n=128,300). For semi-parametric estimators, applying bootstrap improves coverage of confidence intervals for d when the dgp is a ARFIMA(1,d,0) process. In Chapter 4 we study the asymptotic behaviour of sample autocovariance and autocorrelation functions of a long memory processes. This results are useful to give a theoretical support for the consistency of the method in replicating long memory. Last, in Chapter 5, we propose an algorithm to estimate non-parametrically the parameters of a Gegenbauer process with one and two peaks in the spectral density. The bootstrap method will be useful to give an estimate of the distribution of the frequency parameter eta. Its asymptotic distribution is given for the estimators proposed by Chung (1996) and Sadek (2004) but it is very complicate and difficult to handle. The main aim is proposing a method to identify seasonal persistences and provide starting values for maximize a (penalized) likelihood function.