Dierential Geometry of Autoregressive. Fractionally Integrated Moving Average. Models. Nalini Ravishanker. University of Connecticut

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1 Dierential Geometry of Autoregressive Fractionally Integrated Moving Average Models Nalini Ravishanker University of Connecticut Department of Statistics Storrs, CT 6269 November, 994 SUMMARY The dierential geometry of autoregressive fractionally integrated moving average processes is developed. Properties of Toeplitz forms associated with the spectral density functions of these long memory processes are used to compute the geometric quantities. The role of these geometric quantities on the asymptotic bias of the maximum likelihood estimates of the model parameters and on the Bartlett corrections to the likelihood ratio test statistics for the fractional dierence parameter is discussed. Key Words: asymptotic bias, Bartlett correction factor, curvature, fractional dierencing, long memory models. INTRODUCTION Time series data occuring 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 average (ARF IM A) processes. Applications of these processes to model

2 time series include work by Diebold and Rudebusch (989), Geweke and Porter-Hudak (983), Hosking (984), Lo (989) and Sowell (992). A time series fz t g is generated by an autoregressive fractionally integrated moving average (ARFIMA) process (Granger and Joyeaux, 98 and Hosking, 98) if (B)( B) d z t = (B)" t ; () where (B) = B p B p and (B) = B q B q are polynomials in B of degrees p and q respectively, p and q are integers, B is the backshift operator, i.e. Bz t = z t ; d is a real number denoting the fractional degree of dierencing, the fractional dierence operator is dened by a binomial series! ( B) d = d 6 (B) j j= j and the " t are independent and identically distributed as normal random variables with mean and variance 2 : It is assumed that < d < and 2 2 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 ; 2 ; d) denote the vector of model parameters of dimension m = p + q + 2. Since the eect of the d parameter on distant observations decays hyperbolically as a function of increasing lags while the eect of the and parameters decays exponentially, (.) is useful in modeling time series that exhibit both short memory and long memory behavior. When p 6= ; q 6= and d = ; (.) reduces to the ARMA(p; q) model; when p = ; q = and d 6= ; (.) represents the fractional Gaussian noise ( B) d z t = " t : (2) Based on a sample of n observations z n = (z ; :::; z n ) generated by an ARF IMA(p; d; q) process (.), the exact likelihood function has a multivariate normal form: L(z n ; ) = (2) n=2 j 6 j 2 expf 2 z n 6 z n g (3) where 6 = Cov(z n ) and is a function of with elements given by z k ; the autocovariances of fz t g of lag k: Hosking (98) explicitly derived k z directly, while Sowell (992) computed the elements k z as the inverse Fourier 2

3 transforms of the spectral density function of fz t g which has the form: s (w) = 2 (e iw )(e iw ) 2 (e iw )(e iw ) f( eiw )( e iw )g d ; < w 2; (4) and i= p : As a consequence of the long memory behavior, a property of the ARF IMA(p; d; q) process that distinguishes it from the short memory ARMA(p; q) process is reected in the spectral density in (.4) being unbounded at w = whereas the spectral density of an ARMA(p; q) process is a bounded rational function. One approach used for the estimation of model parameters is based on maximizing the likelihood (.3) or approxi- mations to (.3) and has been discussed by Boes, Davis and Gupta (989), Brockwell and Davis (987), Fox and Taqqu (986), Hosking (98, 984), Luce~no (993) and Sowell (992). Some properties of the maximum likelihood estimators are discussed in Fox and Taqqu (986, 987), Dahlhaus (989) and Cheung and Deibold (994). The study of asymptotic inference for the ARF IM A(p; d; q) process is of considerable current research interest. The use of dierential geometry to characterize statistical inference has been widely studied in the last two decades and an excellent review was given by Barndor-Nielsen et.al. (986). The application of the geometrical approach for time series problems was addressed by Amari (987) who discussed the geometrical theory of manifolds for linear systems, Ravishanker et.al. (99) and Ravishanker (994), where the geometry of ARM A(p; q) models was discussed. In particular, Ravishanker et.al. (99) characterized the ARMA(p; q) models as members of the curved exponential family, derived their geometric properties and used them to obtain the higher-order asymptotic bias of the maximum likelihood estimators of model parameters, parameter transformations to satisfy predetermined statistical properties and the Bartlett correction to the likelihood ratio test statistic for some simple models. The computation of the geometrical quantities for the ARM A(p; q) models utilized the form of their rational spectral density function and a result due to Taniguchi (986). The aim of this paper is to present the dierential geometry for the ARF IM A process. The derivations of the geometrical quantities here is not merely a trivial extension of the ARMA case, due to the unboundedness of s at w =. In Section 2, we briey review some dierential geometrical (w) concepts and present the geometrical framework for the ARF IMA(p; d; q) 3

4 process. We describe the derivation of the geometrical quantities by utilizing properties of Toeplitz forms and a result due to Dahlhaus (989). The complete set of quantities is presented in the Appendix. In Section 3, we discuss asymptotic inference for the ARF IM A(p; d; q) processes, indicating the role of geometry. 2 GEOMETRY OF ARFIMA PROCESSES This section describes the geometry of the ARF IMA(p; d; q) process. A brief review of relevant geometrical ideas is given in Section 2., a detailed reference for which is Amari (985). In Section 2.2, we provide details of the derivations of these geometrical quantities for the ARF IM A process. 2. Review of Dierential 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 ; :::; x n ) from this family, the log-likelihood function is `(x ; ) = t (x ) ( ) + c(x ); (5) where t (x ) is the r-dimensional vector of minimal sucient statistics, c(x ) is a function dened on X ; ( ) = log R expf t (x ) + c(x )gd and is a measure over X : An m-dimensional curved exponential family M parametrized by is a subfamily of S and is specied by expressing the canonical parameter as a smooth function ( ) where dim( ) = m < r. M is then referred to as an (r; m)curved exponential family; a curved exponential family is an exponential family in which the dimension of the minimal sucient statistic exceeds the dimension of the parameter space (Efron, 975). The families S and M may be considered as dierentiable manifolds with respective coordinate systems and, and M is a submanifold of S (Amari, 985). The Riemannian metric tensor and the -connections in S are dened respectively by g ij ( ) and ijk ( ) where g ij ( ) = E(`i`j) i; j = ; :::; r; (6) 4

5 ijk ( ) = E(`ij`k + ( ) `i`j`k) i; j; k = ; :::; r; (7) 2 where i (x ; ); j (x ; ) and is a real number. Note that g ij ( ) corresponds to the (i; j) th element of the expected Fisher information matrix for and the -connections ijk () are ane connections that allow the comparison of vectors in two tangent spaces of X at neighboring points and + d : Let 3 denote an alternate coordinate system in S; the corresponding Riemannian metric tensor g IJ ( 3 ) and -connections IJK ( 3 ) may be expressed in terms of the original coordinate system by means of the coordinate transformation laws g IJ ( 3 ) = B i I Bj J g ij( ) (8) and IJK ( 3 ) = B i I Bj J Bk K ijk ( ) + g jk ( )B k IB j J ; (9) where BI i 3 I B j J 3 ; i; j; k = ; :::; r; I; J; K = ; :::; r: Note I I J Einstein's summation convention is used in (2.4) - (2.5) and in the rest of the paper. i.e. summation is assumed over indices which appear both as a subscript and a superscript in the expression. The -Riemann-Christoel curvature tensor is fundamental to characterize the geometry of S and has elements Rijk` ( ) s s = (@ i j ik)g s` + ( is` s jk js` s ik) i; j; k; ` = ; :::; r; () where s jk= jkt g st ; g st denotes the (s; t) th element of the inverse of the expected Fisher information matrix i : The scalar curvature of S is dened by C= Rijk` g i`g jk ; () and the manifold S is said to be -at if Rijk` ( ) = for all ; i; j; k; ` = ; :::; r or equivalently if C= for all : 5

6 For the submanifold M of dimension m with coordinate system ; the Riemannian metric tensor, -connections, the -Riemann-Christoel curvature tensor and the scalar curvature are dened analogous to equations (2.2), (2.3), (2.6) and (2.7) by replacing i; j; k; ` with a; b; c; d which subscript the components of the parameter vector : The -imbedding curvature of M in S is dened by (Amari, 985): H ab= (@ b Ba i + Bb k Bj a i jk)b`k g i`; a; b = ; :::; m; = ; :::; r m; i; j; k; ` = ; :::; r) (2) where B j a j =@ a b B i a 2 i =@ b ; B` and = ( ; :::; rm ) are the coordinates of the submanifold M? of S, which is orthogonal to M. If H ab= ; then M is -at in S. The quantity ( H) 2 = ab H ach bd g g cd ; (3) a; b; c; d = ; :::; m; ; = ; :::; rm represents the square of the -imbedding curvature of M, where g and g cd denote the (; ) th and (c; d) th elements respectively of the inverse of the expected Fisher information matrices with respect to and (Amari, 987). When m =, Efron (975) referred to the scalar quantity 2 = ( H) 2 ab gab (4) as the statistical curvature of the one-dimensional submanifold M. 2.2 Computation of Geometrical Quantities for the ARFIMA Processes This section provides the details of the derivation of the geometrical quantities dened in Section 2. for the ARF IMA(p; d; q) process. These computations involve the evaluation of expectations of the form E(`i`j); E(`ij`k) and E(`i`j`k) corresponding to the logarithm of the exact likelihood in (.3). To evaluate such expectations for the ARM A(p; q) process, Ravishanker et.al. (99) used an approach due to Taniguchi (986) that utilized the properties of Toeplitz forms corresponding to the rational spectral density function of the process. However, that approach cannot be extended for 6

7 ARF IM A(p; d; q) process since it requires that the spectral density function be bounded everywhere except possibly on a set of measure zero, which the ARF IMA spectral density s (w) (see (.4)) does not satisfy. Here, we apply a result (Dahlhaus, 989) that uses the asymptotic properties of Toeplitz forms corresponding to the spectral density functions of strongly dependent processes, which are unbounded at frequency w =. This enables us to evaluate the required expectations and hence derive the geometrical quantities. First, it is necessary to reexpress the ARF IMA(p; d; q) spectral density s (w) dened in (.4) in terms of an alternate parametrization. To do this, assume that j ; j = ; :::; p and `; ` = ; :::; q denote the roots of (z) = and (z) = respectively. We may reexpress (.4) as f 5 q ( j e iw )( j e iw )g s (w) = 2 j= 2 f 5 p ( j e iw )( j e iw )g f( e iw )( e iw )g d (5) j= where < w 2; = ( ; :::; p ; ; :::; q ; 2 ; d); j j j< ; j = ; :::; p and j ` j< ; ` = ; :::; q: Let s a s (w) and s a@ b s (w); a; b = ; :::; m: The geometrical quantities for the ARF IMA process are derived through the following steps: Step : Express the expectations in (2.2) and (2.3) as traces of the covariance matrix 6 and suitable derivatives of 6 (using Lemma, Taniguchi, 986). Specically, for our problem, we write, in terms of the new parametrization ) b 2 tr(6 6 a 6 6 b ); and c ) = tr(6 6 a 6 6 b 6 6 c E( ) c 2 tr(6 6 (a;b) 6 6 c ) tr(6 6 a 6 6 b 6 6 c ) 6 a ; 6 (a;b) b ; a; b = ; :::; m: 7

8 Step 2: Express the traces in Step, up to O(n ) as integrals of products of s (w) and suitable derivatives of s (w) (using Theorem 5., Dahlhaus, 989). Specically, for the expressions written down in Step, this gives tr(6 6 a 6 6 b ) = 2 Z2 (w) s (w) s (b) (w) s (w) dw + O(n ); s (a) and tr(6 6 a 6 6 b 6 6 c ) = 2 Z2 tr(6 6 (a;b) 6 6 c ) = 2 (w) s (w) s (b) (w) s (w) s (c) (w) s (w) dw + O(n ) s (a) Z2 s (a;b) (w) s (w) s (c) (w) s (w) dw + O(n ): Step 3: Evaluate the integrals obtained in Step 2. Since the spectral density function is a rational function for ARM A(p; q) process, the integrands are also rational expressions; hence the integrals could be directly evaluated by the application of Hille's (959) residue theorem to yield the required geometrical quantities in terms of the parametrization (see Ravishanker et.al., 99). For the ARF IMA(p; d; q) process however, the integrals do not always involve rational integrands and the resulting integrations are more complicated. For terms that do not involve the fractional dierencing parameter d, the integrals may still be evaluated using Hille's residue theorem. When the derivatives of s (w) with respect to d are involved, Hille's residue theorem does not apply and we resort to special complex integration techniques. For some simple expressions, formulas may be obtained from Gradshteyn and Ryzhik (965). In general, numerical integration may be alternately used if analytical expressions are not necessary for the intended application of the geometrical terms. Step 4: Step 3 yields analytic expressions for the geometric quantities in terms of : Based on the relationship ( z p z p ) = ( z)( 2 z) ( p z) 8

9 we may obtain the elements of the Jacobian matrix of the transformation from ( ; :::; p ) to ( ; :::; p ) k = pk j p 5 (` j ) `= `6=j : (6) A similar result holds for the transformation from ( ; :::; q ) to ( ; :::; q ). The geometric quantities in terms of the coordinate system are then computed using (2.2) and the coordinate transformation laws (2.4) and (2.5). Although for simplicity, we have focused here on real roots of (z) = and (z) =, this is not binding. We discuss briey the situation where, for instance, k of the roots of (z) = and k 2 of the roots of (z) = are complex (Ravishanker et.al., 99). Let the complex conjugate roots of (z) = be j exp(6i j ); j = ; :::; k =2; the complex conjugate roots of (z) = be ` exp(6i `); ` = ; :::; k 2 =2 and let the real roots of (z) = and (z) = be respectively k +; :::; p and k2 +; :::; q : The transformation of to 3 where 3 = ( ; ; :::; k =2; k =2; k +; :::; p ; ; ; :::; k2 =2; k2 =2; k2+ ; :::; q ; 2 ; d) is well dened and dierentiable. The geometrical quantities may be computed in terms of 3 using steps -4 and obtained in terms of by applying the coordinate transformation laws. We only consider real roots in the rest of the article. The elements of the Riemannian metric tensor and -connections for the ARF IMA(p; d; q) process under the parametrization = ( ; :::; p ; ; :::; q ; 2 ; d) are presented in the Appendix. 3 ASYMPTOTIC INFERENCE BASED ON GEOMETRY First consider the ARF IM A(; d; ) or fractional Gaussian noise process: ( B) d z t = " t (7) 9

10 whose spectral density function may be expressed as s (w) = 2 2 (2 sin w=2)2d ; < w ; (8) where = (d; 2 ): The non-zero elements of the Riemannian metric tensor, -connections and -Riemann-Christoel curvature tensor for this process are presented in Table. Note that C is nonvanishing for any real and hence M is not -at for any. If we assume that 2 is known (let, in this case, 2 without loss of generality); then M reduces to a one-dimensional manifold parametrized by d: For this one-dimensional manifold, C is trivially zero and Efron's statistical curvature (see (2.)) is computed as: 2 d = Q3 6 36( 6 =k 3 ) 2 ; (9) n 6 =26 where Q 3 6 = [ =k =`(k + `) 3 ] and approximately, `= d 2 ' :249 : The statistical curvature is independent of d: Based on Efron's n (975) suggestion that a curvature greater than =8 is `large', we see that the statistical curvature for the one-parameter ARF IM A(; d; ) process is `large' for sample size n < 89: The magnitude of d 2 plays a role in the Bartlett correction to the likelihood ratio test statistic for d; discussed in Section

11 3. Parameter Transformations Amari (985) showed that an -ane coordinate system for M exists if and only if C vanishes identically for all parameter values. Under such -ane parametrizations, should they exist, it may be shown that either (a) the asymptotic bias vanishes (for = ) or (b) the Fisher information matrix is a constant matrix not depending on the parameters (for = ), or (c) the asymptotic skewness of the distribution of the transformed parameter is zero (for = ), or (d) the distribution of the transformed parameter estimator 3 is close to a normal distribution (for = ) (Kass, 989). 3 Clearly, for the ARF IMA(; d; ) process with unknown 2, no -ane parametrization exists since C does not vanish for any. If M is a one-dimensional manifold, then, as mentioned earlier, the - Riemann-Christoel curvature tensor and hence C vanish trivially for any real and hence there always exists an -ane parametrization in M for any. To investigate the nature of such parametrizations, let us set 2 in the ARF IMA(; d; ) process so that now where = (d): Then s (w) = 2 (2 sin w 2 )2d ; < w ; (2) g = 2 6 ; = 6 6 =k 3 and C= : (2) Let 3 denote the -ane parameter in M; then 3 can be determined from by solving the following dierential equation (Amari, 985): yielding the transformation d 2 3 d 2 3 = () d d ; (22) 8 3 >< = >: 2 36( 6 =k 3 ) exp( (=k 3 )g; 6= : = : (23)

12 3.2 Asymptotic Bias in MLE Following Amari (985), the asymptotic bias (of order O(n )) of the MLE of in the manifold M may be approximated by b ( b a ) ' 2n a cd gcd ; a = ; :::; m (24) where b = ( b ; :::; b m ) is obtained on the basis of a sample of size n. The bias in (3.8) is the component derived from the --connections in M and is common to all rst-order ecient estimators of. Here, we have ignored a second component (Amari, 985) that is a function of the -- curvature of the ancillary family A associated with the particular estimate b, as it is unclear at this point how this quantity may be derived in this time series problem. For the process (3.), use of (3.8) yields and b ( b d) ' 8( 6 =k 3 ) n 4 ' :33=n (25) b (b 2 ) ' 2 =n: (26) The asymptotic bias in b d is constant and negative, i.e. d tends to be underestimated. Further, as expected, the bias decreases as n increases. For n = 5, n = ; n = 3 and n = 5, we see that b ( b d) assumes values :266; :33; :44 and :26 respectively. Our results compare in magnitude with the simulation results obtained by Cheung and Newbold (994) on the bias in the MLE of d from the ARF IMA(; d; ) process for time-domain maximum likelihood with the true mean removed, although they report an inverse relation between the bias and the d value. Further, it is of interest to contrast the asymptotic bias in b d from the ARF IMA(; d; ) process to the asymptotic bias in the estimate of d from the ARF IMA(; d; ) or the ARF IMA(; d; ) processes. For the ARF IMA(; d; ) process ( B)( B) d z t = " t ; 2

13 where j j< and = (d; 2 ; ), the asymptotic bias in b d, b 2 and b are computed from (3.8) and the geometrical quantities in the Appendix as b ( b d) ' 3B 3 ( )=nb 3 2( ); b (b 2 ) ' 2 2 =n; (27) b ( b ) ' 6B 3 3 ( )=nb 3 2 ( ): Similar calculations for the ARF IMA(; d; ) process ( B) d z t = ( B)" t ; where j j< and = (d; 2 ; ) yield b ( b d) ' 3B 3 ( )=nb 3 2( ); b (b 2 ) ' 2 2 =n; (28) b ( b ) ' 6B 3 3( )=nb 3 2( ): where B 3 ( ) = 2Q 3 2 ( ) 2 (log( )) 2 8Q 3 ( ) 3 log( ) +( 2 Q 3 ( ) =k 3 + q 2 log( ) + 24Q 3 2( )(log( )) 2 ) 4 +8Q 3 ( ) 5 log( ) ( 2 Q 3 ( ) + 2Q 3 2 ( )(log( )) 2 ) 6 ; B 3 2 ( ) = ( 2 2 6( 2 )(log( )) 2 ) 2 ; B 3 3( ) = q 2 6Q 3 ( ) 2 (log( )) 2 +(8 6 =k 3 2 Q 3 2 ( ) Q3 ( )) 3 log( ) +2Q 3 ( ) 4 (log( )) 2 +(2 2 Q 3 2 ( ) 2 Q 3 ( ) 8 6 =k 3 ) 5 log( ) 6Q 3 ( ) 6 (log( )) 2 +( 2 Q3 ( ) Q 3 2( )) 2 7 log( ); 3

14 and where ( Q Q ( ) = ( ) for the ARF IMA(; d; ) process, Q 3 5( ) for the ARF IMA(; d; ) process, ( 4 for the ARF IMA(; d; ) process q = 2 for the ARF IMA(; d; ) process ; ( 3 6 for the ARF IMA(; d; ) process q 2 = for the ARF IMA(; d; ) process, and the other Q 3 j ( ) are dened in the Appendix. B 3 i ( ); Q 3 j ( ) are obtained by substituting for in the relevant formulas while B 3 i ( ) and Q 3 j( ) are obtained likewise by substituting for in those formulas. 3.3 Bartlett Correction to the Likelihood Ratio Test Statistic The likelihood ratio test statistic LR for testing H : = is given by LR = 2f`( b ; z n ) `( ; z n )g (29) where ` denotes the log-likelihood function of the ARF IMA(p; d; q) process and b denotes the maximum likelihood estimator of : Under the null hypotheses, LR has an asymptotic 2 distribution with m degrees of freedom where m is the number of restricted parameters (Dahlhaus, 989). The rate of convergence to the 2 distribution may be improved from O(n ) to O(n 3=2 ) by using a scaled likelihood ratio test statistic LR 3 (Barndor- Nielsen and Cox, 984); the scaling factor, called the Bartlett correction to the likelihood ratio test statistic is : LR 3 = LR + R=m (3) wherer has the form (Barndor-Nielsen and Blaesild, 986). R = 2 (K ijk`g ij g k` + K ijk`mn g ij g k`g mn ); K ijk` = 3 i;j;k;` + 6 ik;j` 3 ij;k` 6 ij;k;`; K ijk`mn = 3 i;j;k `;m;n +6 i;j;k `;mn +3 ij;k `;mn +2 i;k;m j;`;n 6 i;km j;`n ; ij = E(`ij ); i;j;k = E(`i`j`k); i;jk = E(`i`jk ); i;j;k;` = E(`i`j`k``); ij;k` = E(`ij`k`) ij k`; ij;k;` = E(`ij`k``)+ ij k`; i ; `ij j :

15 Lawley (956) provided an alternate form to compute R while McCullagh and Cox (986) presented an invariant expression of LR 3 and discussed its relation to the geometry of the process. In particular, they showed that the correction factor R increased as the statistical curvature increased. For the ARF IMA(p; d; q) process, the quantity R, which is derived from the exptectations of the derivatives of the log-likelihood function, may be computed using the procedure in Section 2.2. We present here the Bartlett correction for the ARF IMA(; d; ) model when testing H : d = d ; 2 unrestricted. The most common form of this hypothesis is that d is zero. We derive R = 36 6 (2 Q =k 3 ): In future work we will present the general form of the Bartlett correction factor for the ARF IMA(p; d; q) models and study by simulation the degree of improvement achieved. ACKNOWLEDGEMENTS This work was supported in part by a U.S. Army Grant DAAL3-9- G-4. The author is grateful to Zuqiang Qiou for his help with some computations. REFERENCES Amari, S. (985) Dierential-Geometrical methods in Statistics. Lecture notes in Statistics. New York: Springer Verlag. Amari, S. (987) Dierential-geometrical theory in Statistics - towards new developments. In Dierential Geometry in Statistical Inference, IMS Monograph Series. Hayward: Institute of Mathematical Statistics. Barndor-Nielsen, O.E. and Cox, D.R. (984) Bartlett adjustments to the likelihood ratio statistic and the distribution of the maximum likelihood estimator. J.R. Statist. Soc., Ser. B, 46, Barndor-Nielsen, O.E. and Blaesild, P. (986) A note on the calculation of Bartlett adjustments. J.R. Statist. Soc. B., 48, Barndor-Nielsen, O.E., Cox, D.R. and Reid, N. (986) The role of differential geometry in statistical theory. Int. Statist. Rev., 54,

16 Boes, D.C., Davis, R.A. and Gupta, S.N. (989) Parameter estimation in low order fractionally dierenced ARMA processes. Stochastic Hydrology and Hydraulics, 3, 97-. Brockwell, P.J. and Davis, R.A. (987) Time Series: Theory and Methods. Springer-Verlag, New York. Cheung, Y.W. and Diebold, F.X. (994) On maximum likelihood estimation of the dierencing parameter of fractionally-integrated noise with unknown mean. J. Econometrics, 62, Dahlhaus, R. (989) Ecient parameter estimation for self-similar processes. Ann. Statist., 7, Diebold, F.X. and Rudebusch, G.D. (989) Long memory and persistence in aggregate output. J. Monetary Economics, 24, Efron, B. (975) Dening the curvature of a statistical problem (with application to second order eciency) (with discussion). Ann. Statist., 3, Fox, R. and Taqqu, M.S. (986) Large-sample properties of parameter estimates for strongly dependent stationary Gaussian time series. Ann. Statist., 4, Fox, R. and Taqqu, M.S. (987) Central limit theorems for quadratic forms in random variables having long-range dependence. Probab. Theory Related Fields, 74, Geweke, J.F. and Porter-Hudak, S. (983) The estimation and application of long memory time series models. J. of Time Series Anal., 4(4), Gradshteyn, I.S. and Ryzhik, I.M. (965) Tables of Integrals Series and Products. Academic Press, New York. Granger, C.W.J. and Joyeaux, R. (98) An introduction to long-memory time series models and fractional dierencing. J. of Time Series Anal.,, Hille, E. (95) Analytical Function Theory, Vol., Lexington, MA: Lexington. Hosking. J.R.M. (98) Fractional dierencing. Biometrika, 68, Hosking, J.R.M. (984) Modeling persistence in hydrological time series using fractional dierencing. Water Resources Research, 2(2), Kashyap, R.L. and Eom, K.B. (988) Estimation in long-memory time series model. J. of Time Series Anal., 9(), Kass, R.E. (989) Canonical parametrization and zero parameter eects curvature. J.R. Statist. Soc., Ser. B., 46,

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18 TABLE Geometrical Quantities for the ARF IMA(; d; ) Model Riemannian Metric Tensor -Connections g = 2 =6 3 = 6 6 g 22 = =2 4 2 = 2 =6 2 k 3 2= ( + ) 2 =6 2 22= ( + )=2 8 C= (4= 4 + 2= 2 ) + (8= 4 + 6= 2 + 6) + 2 (4= 4 + 4= ( 6 =k 3 ) 2 ) 3 : This result has been noted previously by Kashyap and Eom (988) APPENDIX For the ARF IMA(p; d; q) process, the non-zero elements of the Riemannian metric tensor (2.2) under the parametrization = ( ; :::; p ; ; :::; ; 2 ; d) 8

19 are: g j ;` = ( j `) j; ` = ; :::; p g j ;` = ( j `) j; ` = ; :::; q g j ;` = ( j `) j = ; :::; p; ` = ; :::; q g d;d = 2 6 g j;d = j ln( j ) j = ; :::; p g `;d = ` ln( `) ` = ; :::; q g 2 ; 2 = 2 4 ; where `n(x) denotes the natural logarithm of x. In the same parametrization, the non-zero elements of the -connections in (2.3) are: = 6 6 d;d;d = 2 ; d;d; ; k 3 = = Q 3 ( j ); j = ; :::; p; d; j ;d d;d; j = = Q 3 d; j ;d d;d;j ( j ); j = ; :::; q; = (+)2 ; d; 2 6 ;d 2 d; 2 ; j = (+) ln( j) 2 j ; j = ; :::; p; d; 2 ; j = (+) ln( j) 2 j ; j = ; :::; q; = = ln( j ) d; j ; 2 2 ; j ;d 2 j ; j = ; :::; p; = n Q3 2 ( j ); j=k Q d; j ; 3 3 ( j; k ); j6=k; j;;:::;p; k 9

20 = = Q 3 3( j ; k ); j = ; :::; p; k = ; :::; q; d; j ; k j ; k ;d = = ln( j) d; j ; 2 2 ; j ;d 2 j ; j = ; :::; q; = n Q3 2 ( j ); j=k Q 3 d; j ; 3 ( j; k ); j6=k; j;;:::;q; k d; j ; k = Q 3 3( j ; k ); k = ; :::; p; j = ; :::; q; = (+) ; 2 ; 2 ; ; j ; k = j ; k ; 2 = 2 ; j ; k = j ; k ; 2 = 2 ; k ; j = 2 ; j ; k = j ; k ; 2 = 2 ( j k ; j; k = ; :::; p; ) 2 ( j k ; j = ; :::; p; k = ; :::; q; ) 2 ( j ; j; k k = ; :::; q; ) = n Q 3 4 ( j); j=k Q 3 j ; k ;d 3 ( j; k ); j6=k; j;;:::;p; = j; k ;` k ;`; j = 8 ( >< j + k +`3 j k `) ( j k )( j `)( k `) ; k6=j 2 >: j +()` +(3) 2 j ` ( j `) 2 ( 2 ; j ) k=j; j;k;`=;:::;p; 8 ( >< k +`+ j 3 k ` j ) ; k6=` ( j k )( j `)( k`) >: 2 k () j +(3) 2 k j ( 2 k )( k j )2 ; k=` = n ( j+ k+`3 jk `) ( `; j ; j k )( j `)( k `) k k; ` = ; :::; p; j = ; :::; q; ; j; k = ; :::; p; ` = ; :::; q; = (`+ j+ k 3` j k) (` `; j ; j )(` k )( j k ; ` = ; :::; p; j; k = ; :::; q; ) k = n Q 3 5 ( j); j=k Q 3 j ; k ;d 3 ( j; k ); j6=k; j;;:::;q; k ;`; j = 8 ( >< j + k +`3 j k ` ) ; k6=` ( j k )( j `)( k `) >: (+) j 2`+(+3) j 2` ( j `) 2 ( 2` ; ) k=`; j=;:::;p;k;`=;:::;q; 2

21 = j k ;` 8 ( >< j + k +`3 j k `) ( j k )( j `)( k ` ) ; k6=j >: (+)`+2 j (+3) 2 j ` ( j `) 2 ( 2 j ) ; k=j; j;k;`=;:::;q; where Q 3 ( j ) = fc (c ) c (ln 2) 2 + 4c ( 2 =48 + S ) 2c (c 3 c 4 + S 2 ) 2(S 3 + 2S 4 )=( + 2 j )g; Q 3 2( j ) = j ( j ) + (36 j +54 j +2 j ) 2 j (2 j )(+2 j )2 ln( j ); Q 3 3( j ; k ) = (+2 j 2 j k ) ( j k )( j k ) j ln( j ) + (+2 k 2 j k ) ( j k )( k j ) k ln( k ); Q 3 4( j ) = 2[(+42 j +54 j 26 j ) ln( j)+ j (+ j 3 2 j 23 j +4 j +5 j )] 2 j (2 j )3 + ( + )Q 3 2( j ); Q 3 5( j ) = 2 ln( j) ( 2 j ) ( + )Q 3 2( j ); c = j =( + 2 j ); c 2 = ln(( j ) 2 = j ); c 3 = 2 ln( j ) ln(( j )= j ); c 4 = ln j ln(( j )= j ); S = 6 S 2 = 6 2k 6 `= k 6 `= () k+`+ ( ) ' :8565; k` 2k` ( k` ) f()`[( k` j ) k + 2j k ] [`j( j ) k` + 2 k` j ( j )`]g S 3 = 6 k j =k 2 ; S 4 = 6 k 6 k= `= ( k` ) ()k`+ k j (k`+) 2 S 5 = 6 k= k j =(k + )2 : Q 3 i ( j ) and Q 3 i ( j ) are obtained by substituting j and j respectively for j in the formulas for Q 3 i ( j ) and their components, i = ; 2; while Q 3 3( j ; k ); Q 3 3( j ; k ) and Q 3 3( j ; k ) are obtained likewise by suitable substitutions in Q 3 3 ( j; k ): Note that we have retained the parameter symbols as subscripts on the Riemannian metric tensor and -connections for ease of reading. Correspondence with the notation of Section 2. is achieved by setting (d; 2 ; ; :::; p ; ; :::; q ) (; 2; 3; :::; p + 2; p + 3; :::; p + q + 2): 2