Differential Geometry of Arfima Processes

Autoregressive fractionally integrated moving average ARFIMA pro cesses are widely used for modeling time series exhibiting both long memory and short memory behavior Properties of Toeplitz matrices associated with the spectral density functions of Gaussian ARFIMA processes are used to compute di erential geometric quantities INTRODUCTION Time series data occurring in several areas such as geology hydrology and economics exhibit both short memory and long memory behavior which may be modeled by the class of autoregressive fractionally integrated moving aver age ARFIMA processes Beran A time series fXtg is generated by an autoregressive fractionally integrated moving average ARFIMA process if B B Xt B t where B B pB and B B qB are polynomials in B of degrees p and q respectively p and q are integers B is the backshift operator i e BXt Xt d is a real number denoting the fractional degree of di erencing the fractional di erence operator is de ned by a binomial series B d j d j B j and the t are independent and identically distributed as normal random vari ables with mean and variance It is assumed that d and that the roots of z and z lie outside the unit circle ensuring the stationarity and invertibility of the process It is further assumed that z and z do not have common roots Let p q d denote the vector of model parameters of dimension m p q Since the e ect of the parameter d on distant observations decays hyperbolically as a function of increasing lags while the e ect of the and parameters decays exponen tially is useful in modeling time series that exhibit both short memory and long memory behavior When p q and d reduces to the ARMA p q process when p q and d represents the fractional Gaussian noise B Xt t Based on a sample of n observations Xn X Xn generated by an ARFIMA p d q process the exact likelihood function has the form L Xn n j j expf X n Xng where Cov Xn and is a function of with elements given by x k the autocovariances of fXtg of lag k The spectral density function of fXtg has the form s w e e iw eiw e iw f e e iw g d w and i p While the spectral density of the short memory ARMA p q process is a bounded rational function s w is unbounded at w Properties such as consistency rst order asymptotic e ciency and asymp totic normality of the approximate and exact MLEs of the model parameters have been discussed in Fox and Taqqu and Dahlhaus The study of higher order asymptotic inference for the ARFIMA p d q process is of considerable interest and can be related to the di erential geometry of the process The use of di erential geometry to characterize statistical in ference has been widely studied in the last two decades Barndor Nielsen et al Murray and Rice The relationship between the geometry and asymptotic inference in statistical models was pointed out by Rao Application of the geometrical approach for time series prob lems was addressed by Amari who discussed the geometrical theory of manifolds for linear systems and by Ravishanker et al where the ge ometry of ARMA p q processes was discussed The latter characterized the ARMA p q processes as members of the curved exponential family derived their geometric properties and used them to obtain for some simple processes a approximate higher order asymptotic bias of the maximum likelihood esti mators of parameters b parameter transformations to satisfy predetermined statistical properties and c the Bartlett correction to the likelihood ratio test statistic In this paper analytical expressions for various geometric properties in ARFIMA p d q processes are computed by utilizing results on Toeplitz ma trices associated with their spectral density functions Dahlhaus Due to the unboundedness of s w at w these computations are consider ably di erent from those used with ARMA processes A brief review of the geometry is given in Section and the computational procedure is described in Section where the geometric quantities are presented in detail Section also shows how these quantities may be used to obtain a ne parametrizations for the fractional Gaussian noise parameter Section suggests application of these geometrical quantities for carrying out asymptotic inference A REVIEW OF DIFFERENTIAL GEOMETRY Let S denote distributions from an r parameter exponential family de ned on a sample space X and let denote the r dimensional canonical parameter of the family Based on a set of n observations x x xn from this family the log likelihood function is