PORTMANTEAU TESTS FOR ARMA MODELS WITH INFINITE VARIANCE

doi:10.1111/j.1467-9892.2007.00572.x PORTMANTEAU TESTS FOR ARMA MODELS WITH INFINITE VARIANCE By J.-W. Lin and A. I. McLeod The University of Western Ontario First Version received Septeber 2006 Abstract. Autoregressive and oving-average (ARMA) odels with stable Paretian errors are soe of the ost studied odels for tie series with infinite variance. Estiation ethods for these odels have been studied by any researchers but the proble of diagnostic checking of fitted odels has not been addressed. In this article, we develop portanteau tests for checking the randoness of a tie series with infinite variance and for ARMA diagnostic checking when the innovations have infinite variance. It is assued that least squares or an asyptotically equivalent estiation ethod, such as Gaussian axiu likelihood, is used. It is also assued that the distribution of the innovations is identically and independently distributed (i.i.d.) stable Paretian. It is seen via siulation that the proposed portanteau tests do not converge well to the corresponding liiting distributions for practical series length so a Monte Carlo test is suggested. Siulation experients show that the proposed Monte Carlo test procedure works effectively. Two illustrative applications to actual data are provided to deonstrate that an incorrect conclusion ay result if the usual portanteau test based on the finite variance assuption is used. Keywords. ARMA odel diagnostic check; portanteau test; residual autocorrelation function; stable paretian distribution; testing for randoness. 1. INTRODUCTION Tie-series odels with stable Paretian errors have been studied by any researchers. Adler et al. (1998) discussed any aspects of how to apply standard Box Jenkins techniques to stable autoregressive oving-average (ARMA) processes. Adler et al. (1998) concluded that, in principle, the standard Box Jenkins techniques do carry over to the stable setting but a great deal of care needs to be exercised. In Section 2 we briefly review the stable Paretian distribution and in Section 3 we develop portanteau tests for whiteness or randoness for an identically and independently distributed (i.i.d.) series. The whiteness test is illustrated with a brief application to daily returns on the S&P 500 stock index. In Section 4 we develop portanteau diagnostic checks for residuals of an AR odel fitted by least squares assuing that the true innovations are i.i.d.-stable Paretiandistributed. This is extended to the ARMA odel using the equality of residuals in AR and ARMA odels. An illustrative exaple shows the differences in inferences that ay result between the finite-variance and infinite-variance portanteau tests. 0143-9782/08/03 600 617 Journal copilation Ó 2008 Blackwell Publishing Ltd., 9600 Garsington Road, Oxford OX4 2DQ, UK and 350 Main Street, Malden, MA 02148, USA.

PORTMANTEAU TESTS 601 2. THE STABLE PARETIAN DISTRIBUTION A stable distribution is usually defined through its characteristic function. A rando variable Z, or Z a (r,b,l), is said to have a stable distribution if its characteristic function has the following for: Eðe itz Þ¼ exp rjtja 1 ib sgnðtþ tan pa 2 þ ilt if a 6¼ 1 exp rjtj 1 þ ib 2 ð1þ p sgnðtþ log jtj þ ilt if a ¼ 1, where i 2 ¼ 1, t the paraeter of the characteristic function, a the index of stability or the characteristic exponent, satisfying 0 < a 2, r > 0 the scale paraeter, b the skewness satisfying 1 b 1, l 2 R 1 is the location paraeter, and ( 1 if t > 0 sgnðtþ ¼ 0 if t ¼ 0 1 if t < 0. 3. PORTMANTEAU TESTS FOR RANDOMNESS OF STABLE PARETIAN TIME SERIES In this section, we study portanteau tests for checking randoness of a sequence of stable Paretian rando variables. We consider the stable analogues of portanteau tests of Box and Pierce (1970) as well as Penˇ a and Rodriguez (2002), denoted by Q BP and ^D, respectively. For this, we require soe iportant properties of saple autocorrelation functions (ACF) and saple partial autocorrelation functions (PACF) of stable Paretian ARMA processes (Brockwell and Davis, 1991, Ch. 13; Saorodnitsky and Taqqu, 1994; Adler et al., 1998). 3.1. Asyptotic distribution of saple ACF Let fz t : t ¼ 0, ±1, ±2,...g be an (i.i.d.) sequence of stable Paretian rando variables and X t be a strictly stationary process defined by X t ¼ X1 j¼ 1 w j Z t j ; t ¼ 1;...; n; ð2þ where X 1 j¼ 1 jjw j d j < 1; for soe d 2ð0; aþ\½0; 1Š: ð3þ The stable analogue of the autocorrelation function at lag k is defined as Journal copilation Ó 2008 Blackwell Publishing Ltd.

602 J.-W. LIN AND A. I. MCLEOD q k ¼ P j w jw jþk P ; j w2 j k ¼ 1; 2;...: ð4þ Equation (4) can be estiated by the saple autocorrelation function as follows: P n k t¼1 r k ¼ P X tx tþk n t¼1 X t 2 ; k ¼ 1; 2;...; ð5þ for a > 0. According to Davis and Resnick (1986), for any positive integer k, the liiting distribution of saple autocorrelation functions is given by 1 n a ð r1 q 1 ;...; r k q k Þ T! ðy 1 ;...; Y k Þ T ; ð6þ where Ô!Õ denotes convergence in distribution and Y h ¼ X1 j¼1 S j q kþj þ q k j 2q j q k ; h ¼ 1;...; k; ð7þ S 0 where S 0, S 1,... are independent stable variables; S 0 is positive with S 0 Z a=2 ðc 2=a a=2 ; 1; 0Þ; and S j are Z a ðca 1=a ; 0; 0Þ, where ð1 aþ=ðcð2 aþ cosðpa=2þþ if a 6¼ 1 C a ¼ 2=p if a ¼ 1. Under the null hypothesis that X t are a sequence of i.i.d. stable Paretian rando variables, we have q 0 ¼ 1 and q k ¼ 0 for k 1, so the liiting distribution of saple ACFs can be further siplified as follows: where W h are given by 1 n a ð r1 ;...; r k Þ T! ðw 1 ;...; W k Þ T ; ð8þ W h ¼ S h ; h ¼ 1;...; k: ð9þ S 0 Note that, for a > 1, we ay also use the ean-corrected saple ACF at lag k, denoted as ~r k, which is given by ~r k ¼ P n k t¼1 ðx t X ÞðX tþk X Þ P n t¼1 ðx t X Þ 2 ; k ¼ 1; 2;...: ð10þ Davis and Resnick (1986) indicated that the liiting distribution of ~r k is the sae as that of r k. Journal copilation Ó 2008 Blackwell Publishing Ltd.

PORTMANTEAU TESTS 603 3.2. Asyptotic distribution of saple PACF The saple PACF at lag k is defined as the saple estiate of the k-th eleent of the Yule Walker solution to an AR process. The saple PACF ay also be calculated using the Durbin Levison algorith. Let p k be the saple PACF at lag k, and p () ¼ (p 1,..., p ) T. By the Durbin Levison algorith, the vector p () can be expressed as a function of r (), p () ¼ w(r () ), with the k-th eleent given by p k ¼ wðr ðkþ Þ¼ r k r T ðk 1Þ R 1 ðk 1Þ r ðk 1Þ 1 r T ðk 1Þ R 1 ðk 1Þ r ; ð11þ ðk 1Þ where r (p) ¼ (r 1,..., r p ) T is the p 1 vector of saple ACFs, R (p) ¼ (r ji jj ) pp is the p p saple autocorrelation atrix and r ðkþ ¼ðr k;...; r 1 Þ T. Following the proof in Monti (1994), we can derive the asyptotic distribution of saple PACFs. Under the null hypothesis that X t are independent, the ACFs are all zero, and according to Brockwell and Davis (1991, Ch. 13),! n 1=a r h ¼ O p ; h ¼ 1; 2;...: Therefore, R ðkþ ¼ 1 k þ O p! n 1=a ; where 1 k is a k k identity atrix. By eqn (1),! n 2=a p ðþ ¼ r ðþ þ O p : ð12þ Using eqn (8), we have 1 n a ð p1 ;...; p Þ T! ðw 1 ;...; W Þ T : ð13þ 3.3. Asyptotic distributions of Q BP and ^D tests Under the assuption 1 < a < 2, Runde (1997) derived the liiting distribution of Q BP, based on the ean-corrected saple ACFs. His result is given by n 2=aX Q BP ðþ ¼ ~r j 2! W 1 2 þþw 2 ; ð14þ j¼1 where fw k : k ¼ 1,..., g are defined in eqn (9). Note that if 0 < a 1, the liiting distribution of eqn (14) reains the sae if ~r k are replaced by r k. Journal copilation Ó 2008 Blackwell Publishing Ltd.

604 J.-W. LIN AND A. I. MCLEOD Consider the ^D test of Penˇ a and Rodriguez (2002) next. In the stable case we ay define the test statistic, n 2=a ^DðÞ ¼ 1 jr ðþj 1= : ð15þ Using the results in Sections 3.1 and 3.2, and following the arguents of Penˇ a and Rodriguez (2002), we ay have the asyptotic distribution of eqn (15) in Theore 1 Theore 1. ^DðÞ in eqn (15) is asyptotically distributed as X i¼1 þ 1 i Wi 2 ; where fw i : i ¼ 1,..., g are as defined in eqn (9). The proof of this theore is given in Appendix A. Reark 1. The liiting distributions of the Q BP and ^D tests can be coputed by aking use of the change variable technique and soe nuerical algoriths of calculating the probability density function of stable rando variables, such as Mittnik et al. (1999). This approach requires, however, intensive nuerical coputations. Reark 2. Another approach to obtaining the asyptotic distributions of the Q BP and ^D tests is to siulate the aforeentioned tests based on their asyptotic distributions. For exaple, ^D is siulated as defined in Theore 1. This approach also requires lengthy coputations but it is uch less intensive coputationally than the approach entioned in Reark 1. This approach will be adopted in the subsequent analysis based on 10 4 siulations. Reark 3. Penˇ a and Rodriguez (2006) consider a slightly different noralization for the ^D statistic and an iproved finite-saple approxiation to its liiting distribution is provided. In the stable case, the new test statistic ay be written, ^D n 2=a ðþ ¼ jr ðþj 1= : ð16þ It ay be shown that the liiting distribution of ^D ðþ is as stated in Theore 1. Moreover, identical results are produced if ^D ðþ is used instead of ^DðÞ in the Monte Carlo test given in Appendix B. 3.4. Siulation experients Based on 250 siulations, the 5, 10, 30, 50, 70, 90, 95, 97.5, 99 (%) epirical quantiles of both Q BP () and ^DðÞ tests with lag ¼ 5 were Journal copilation Ó 2008 Blackwell Publishing Ltd.

PORTMANTEAU TESTS 605 calculated and plotted against the corresponding asyptotic distributions as stated in Section 3.3. It is seen in Figures 1 and 2 that the epirical and asyptotic quantiles do not agree very well unless n is very large. The practical applications of both tests would be severely ipeded by the slow convergence. A solution to this proble is to use the Monte Carlo test or paraetric bootstrap. Consider the siulation experients: i.i.d. rando sequence of Z a (1,0,0) with series length n ¼ 250 and a ¼ 1.9,1.7,1.5,1.3,1.1 was siulated. The epirical sizes of both tests were calculated based on N ¼ 10 4 siulations and each Monte Carlo test was siulated based on B ¼ 10 3 siulations see Appendix B for algorith details. The results are tabulated in Table I. It is seen that the epirical sizes of both tests are very close to the 5% noinal level even with n ¼ 250. n = 1000 epirical asyptotic n = 2000 epirical asyptotic 0 10 20 30 40 50 60 70 D^ 0 20 40 60 80 D^ n = 5000 epirical asyptotic n = 10,000 epirical asyptotic 0 50 100 150 200 250 D^ 0 50 100 150 D^ 200 250 Figure 1. plot coparing the epirical and asyptotic distributions of ^DðÞ; ¼ 5 defined in eqn (15). Rando sequences of series length n ¼ 1000,2000,5000,10,000 were siulated fro S a (1,0,0), a ¼ 1.2 and epirical quantiles corresponding to 5, 10, 30, 50, 70, 90, 95, 97.5, 99 (%) are shown. Journal copilation Ó 2008 Blackwell Publishing Ltd.

606 J.-W. LIN AND A. I. MCLEOD n = 1000 epirical asyptotic n = 2000 epirical asyptotic 0 50 Q BP 100 150 0 50 100 Q BP 150 n = 5000 epirical asyptotic n = 10,000 epirical asyptotic 0 20 40 60 80 100 Q BP 120 0 100 200 300 400 500 600 Q BP Figure 2. plot coparing the epirical and asyptotic distributions of Q BP (), ¼ 5 defined in eqn (14). Rando sequences of series length n ¼ 1000,2000,5000,10,000 were siulated fro S a (1,0,0),a ¼ 1.2 and epirical quantiles corresponding to 5, 10, 30, 50, 70, 90, 95, 97.5, 99 (%) are shown. TABLE I Epirical Sizes (%) of Monte Carlo Tests for a Noinal 5% Test for Randoness of Stable White Noise Innovations with Index a ^D a ¼ 5 ¼ 10 ¼ 15 ¼ 5 ¼ 10 ¼ 15 1.9 5.30 4.66 4.78 4.96 4.71 4.87 1.7 5.18 4.44 4.44 4.82 4.43 4.41 1.5 4.82 4.99 5.13 5.07 5.27 5.30 1.3 4.80 5.03 5.18 5.04 5.00 5.27 1.1 5.26 5.33 5.12 5.33 5.25 5.15 The epirical size for each test was calculated based on N ¼ 10 4 siulations. Each Monte Carlo test used B ¼ 10 3 siulations. Series length n ¼ 250 and lags ¼ 5,10,15 were investigated. 3.5. Illustrative exaple The daily S & P 500 stock index fro 2 January, 1999 to 29 Deceber, 2006 was obtained fro Wharton Data Research Services. This results in a series with Journal copilation Ó 2008 Blackwell Publishing Ltd. Q BP

PORTMANTEAU TESTS 607 TABLE II P-values Using Monte Carlo Portanteau Tests and Ljung Box v 2 Test, Q LB, for Testing Randoness of Returns of S&P 500 Stock Index Test Statistic Method ¼ 5 ¼ 10 ¼ 20 ¼ 50 ^D Monte Carlo/stable 0.104 0.087 0.068 0.049 ^D Monte Carlo/noral 0.151 0.117 0.050 0.006 Q BP Monte Carlo/stable 0.078 0.136 0.053 0.048 Q BP Monte Carlo/noral 0.107 0.237 0.019 0.005 Q LB v 2 0.112 0.251 0.018 0.003 length n ¼ 2011. The returns, log(z tþ1 /z t ), were coputed and tested for randoness. The consistent estiators of McCulloch (1986) were used to estiate a and b for the returns. We obtained ^a M ¼ 1:587 and ^b M ¼ 0:081. It is seen that ^b M is close to zero so the series is not highly skewed. Since ^a M is uch less than 2, the Monte Carlo test using the stable distribution should be used (Appendix B). For coparison, the Monte Carlo tests were also perfored using noral rando variables as well. The Ljung Box tests based on the v 2 approxiation are shown in Table II. The Ljung Box test (Ljung and Box, 1978) rejects at level a when Q LB > v 2 ð1 aþ, where Q LB ðþ ¼nðn þ 2Þ X k¼1 r 2 k ðn kþ ; ð17þ and v 2 ð1 aþ denotes the 1 a quantile of a v2 distribution with degrees of freedo. As expected the Ljung Box test agrees very well with the Monte Carlo test using the Box Pierce statistic Q BP for the Monte Carlo test using noral rando variables. There is a striking difference in the stable/noral Monte Carlo tests when ¼ 50. Under the norality assuption both ^D and Q BP indicate that the randoness assuption is strongly rejected whereas the P-value is only about 5% when the stable assuption is ade. Reark 4. Portanteau tests based on the nonparaetric bootstrap procedure could also be used but it is expected that they would be less powerful since less inforation is used. In addition, the proposed paraetric bootstrap test procedure can be used to test nonlinear tie series driven by stable Paretian innovations. 4. DIAGNOSTIC CHECK FOR MODEL ADEQUACY OF AR(p) MODELS WITH STABLE PARETIAN ERRORS 4.1. Soe asyptotic results In this section, we consider Q BP and ^D tests for diagnostic checks in odel adequacy of AR(p) odels with stable Paretian errors. Journal copilation Ó 2008 Blackwell Publishing Ltd.

608 J.-W. LIN AND A. I. MCLEOD Define the general AR(p) process as follows: /ðbþx t ¼ Z t ; ð18þ where fz t : t ¼ 0, ±1, ±2,...g is an i.i.d. sequence of stable Paretian rando variables, B denotes the backward operator, and /(B) ¼ 1 / 1 B / p B p. Let ^/ ðpþ ¼ð^/ 1 ;...; ^/ p Þ T denote the estiates of autoregressive coefficients. The residuals of the fitted odel are given as follows: ^Z t ¼ Z t ð^/ ðpþ Þ¼X t ^/ 1 X t 1 ^/ p X t p ¼ ^/ðbþx t ; ð19þ and the corresponding residual autocorrelation at lag k is given by P ^Z t^z t k ^r k ¼ P : ^Z t 2 Consider the estiators of ^/ ðpþ satisfying! n 1=a ^/ ðpþ ¼ / ðpþ þ O p : Fro Lin and McLeod (2007, Appendix B) the residual autocorrelation at lag k, ^r k, can be approxiated by the first-order Taylor expansion about error ACFs, r k. Specifically, the approxiation is ^r k ¼ r k þ Xp j¼1 ð/ j ^/ j Þw k j þ O p n where w j is the ipulse response coefficient at lag j and P Zt Z t k r k ¼ P Z 2 t! 2=a ; ð20þ is the error autocorrelation at lag k. Equation (20) can also be written in atrix for, to order O p ([n/ log(n)] 2/a ), ^r ðpþ ¼ r ðpþ þ Xð/ ðpþ ^/ ðpþ Þ; ð21þ where 2 3 1 0 0. w 1 1.. 0... X ¼.... 0 : ð22þ 6.... 7 4.. 0 5 w 1 w 2 w p By aking use of eqn (20) or (21) as well as following the proof in Penˇ a and Rodriguez (2002), we ay derive the asyptotic distributions of the Journal copilation Ó 2008 Blackwell Publishing Ltd.

PORTMANTEAU TESTS 609 aforeentioned portanteau tests for diagnostic checks in AR(p) odels. This distribution, however, is usually very coplicated and ay not be traceable unless the AR(p) odels of interest are fitted by least squares (LS). For siplicity, therefore, we only consider the case that eqn (18) is estiated using least squares in the subsequent analysis. According to Section 4 in Davis (1996), if the ARMA paraeters, b, are estiated using least squares, we have ½n=Š 1=a ð^b LS bþ converge in distribution, where ^b LS denotes the LS estiates of b. Hence, in ters of our notation, we have ^/ ðpþ / ðpþ ¼ O p ð½n=š 1=a Þ. Then, by Box and Pierce (1970), p f^z t g in eqn (19) satisfy the orthogonality conditions and, to order O p ð1= ffiffi n ½n=Š 1=a Þ, ^r T ðpþ X ¼ 0: ð23þ If we now ultiply eqn (21) on both sizes by then using eqn (23) we have Q ¼ XðX T XÞ 1 X T ; ^r ðpþ ¼ð1 QÞr ðpþ ð24þ approxiately, where 1 is an identity atrix and Q ¼ X(X T X) 1 X T.It was shown by Box and Pierce (1970) that 1 Q is idepotent of rank p. Hence, the asyptotic distribution of the Q BP test is given by n 2=aX ^r k 2 log n! WT ð1 QÞW ; ð25þ 1 where W ¼ (W 1,..., W ) T and fw i : i ¼ 1,..., g are defined in eqn (9). Consider the asyptotic distributions of residual partial autocorrelations next. Let ^p ðþ be the vector of the first residual partial autocorrelations and p () the vector of error partial autocorrelations. The Taylor expansion of wð^r ðþ Þ around r () yields ^p ðþ ¼ p ðþ þ @p ðþ @r ðþ ð^r ðþ r ðþ ÞþO p! n 2=a : ð26þ log n By eqns (11) and (12), eqn (26) becoes! n 2=a ^p ðþ ¼ ^r ðþ þ O p : ð27þ log n Consider the Penˇ a Rodriguez test as the for of n 2=a ^D ¼ ð1 j^r ðþj 1= Þ; ð28þ log n Journal copilation Ó 2008 Blackwell Publishing Ltd.

610 J.-W. LIN AND A. I. MCLEOD where ^R ðþ ¼ð^r ji jj Þ ; is the residual autocorrelation atrix. By eqn (27) and following the proof in Penˇ a and Rodriguez (2002), the liiting distribution of eqn (28) is W T A W, where A ¼ (1 Q) T W, (1 Q) and W, is an diagonal atrix with (i, i)-th eleent equal to ( i þ 1)/ for i ¼ 1,...,. Reark 5. It is shown in Lin and McLeod (2007, Appendix B.4) that the residuals in a fitted ARMA odel are asyptotically equivalent to those in a particular AR odel. Hence the asyptotic results for the AR ay be extended to the ARMA case. 4.2. Soe size and power calculations As in Section 3.4, the slow convergence of Q BP and ^D tests to their asyptotic distributions is also present at the residual autocorrelations. The first-order autoregressive process X t ¼ 0.5X t 1 þ Z t with Z t Z 1.2 (1,0,0) was siulated and AR(1) odels were fitted to the data. Then the 5, 10, 30, 50, 70, 90, 95, 97.5, 99 (%) epirical quantiles of ^r 1 were copared with the corresponding asyptotic quantiles. It is seen fro Figure 3 that the epirical quantiles of ^r 1 get closer to the asyptotic quantiles as n increases. The slow convergence of residual autocorrelations to their asyptotic distribution ay cause difficulties in using portanteau tests in practice. Therefore, as in Section 3.4, we suggested using the Monte Carlo test to iprove the effectiveness of portanteau tests. We now investigate the effectiveness of Q BP and ^D tests for diagnostic check in fitted AR odels with stable Paretian errors. The epirical sizes of ^D and Q BP tests for a 5% significance test were first calculated via siulation. In this experient, AR(1) odels, X t ¼ / 1 X t 1 þ Z t, were siulated, where Z t Z 1.5 (1,0,0) and / 1 ¼ 0, ±0.1, ±0.3, ±0.5, ±0.7, ±0.9 and AR(1) odels were fitted to the siulated data by the Burg algorith. The epirical size for each test was calculated based on N ¼ 10 4 siulations and each Monte Carlo test used 10 3 siulations. Series length n ¼ 100 and lags ¼ 5,10,20 were investigated. It is seen in Table III that the epirical sizes of both tests are very close to their noinal level. The epirical powers of ^D and Q BP tests as diagnostic tools were also investigated via siulation. Twelve ARMA (2,2) odels of series length n ¼ 100 in Table IV of Penˇ a and Rodriguez (2002) were siulated and AR(1) odels were fitted to the siulated data using the Burg algorith. Both tests with lags ¼ 5,10,20 were calculated using the paraetric bootstrap procedure. The epirical powers were calculated based on N ¼ 10 3 siulations and each Monte Carlo test used 10 3 siulations. It is seen in Table IV that the epirical powers of both tests are reasonably good for ost odels. Soe of the are even better than the powers listed in Penˇ a and Rodriguez (2002). In addition, increasing the series length can also iprove the effectiveness of the proposed test procedure. For exaple, with odel 3 in Table II, if the series length was increased to n ¼ 250, Journal copilation Ó 2008 Blackwell Publishing Ltd.

PORTMANTEAU TESTS 611 n = 100 epirical asyptotic n = 500 epirical asyptotic 0.05 0.00 0.05 0.10 ^ r 1 0.02 0.00 0.02 0.04 0.06 ^ r 1 n = 1000 epirical asyptotic n = 10,000 epirical asyptotic 0.01 0.00 0.01 0.02 0.03 ^ r 1 0.000 0.005 0.010 ^ r 1 Figure 3. plot ^r 1 and its asyptotic approxiation. The AR(1) process, X t ¼ 0.5X t 1 þ Z t, of series length n ¼ 100,500,10,000 was siulated with stable innovations, fz t gs 1.2 (1,0,0). For each n,10 4 siulations were perfored and AR(1) odels were then fitted to siulated series. The 5, 10, 30, 50, 70, 90, 95, 97.5, 99 (%) epirical quantiles of ^r 1 are plotted and the corresponding asyptotic distribution is shown. the epirical powers of the ^D test at lags ¼ 5,10,20 were increased significantly fro 23.37%, 20.10% and 17.61% to 58.27%, 43.71% and 35.52%, respectively. Siilar iproveent was also found in the Q BP test. Finally, as in Penˇ aand Rodriguez (2002), our siulation experients show that ^D is ore powerful than Q BP as a diagnostic tool. Reark 6. It is well known that the Burg estiate of / 1 is close to the LS estiate. The advantage of using Burg estiate is that it is always in the stationary region and this is needed for the Monte Carlo test. 4.3. Illustrative application Consider the onthly siple returns of CRSP value-weighted index fro January 1926 to Deceber 1997, n ¼ 864 (Tsay, 2002, Ch. 2). After fitting an AR(5), we Journal copilation Ó 2008 Blackwell Publishing Ltd.

612 J.-W. LIN AND A. I. MCLEOD TABLE III Epirical Sizes (%) of^d and Q BP for a 5% Significance Test / 1 ¼ 5 ¼ 10 ¼ 20 ^D Q BP ^D Q BP ^D Q BP 0.9 4.90 4.60 4.75 4.71 4.88 4.96 0.7 4.97 4.95 5.20 4.94 5.16 5.42 0.5 5.37 5.55 5.32 5.12 5.14 5.16 0.3 5.11 5.13 4.90 4.80 4.82 5.26 0.1 4.92 5.14 5.01 4.75 5.20 4.86 0.1 5.30 5.25 5.45 5.08 5.29 4.90 0.3 5.00 4.79 5.20 5.30 5.33 5.45 0.5 5.00 5.00 4.93 4.93 5.10 5.26 0.7 5.62 5.20 5.73 5.45 5.65 5.41 0.9 5.21 5.01 5.02 5.00 5.07 5.30 ^D and Q BP tests for checking odel adequacy of AR(1) odels fitted by the Burg algorith. Both tests were ipleented by the paraetric bootstrap procedure. The epirical size for each test was calculated based on N ¼ 10 4 siulations. Each Monte Carlo test also used B ¼ 10 3 siulations. Series length n ¼ 100 and lags ¼ 5,10,20 were investigated. TABLE IV Epirical Powers (%) of^d and Q BP for a 5% Significance Test ¼ 5 ¼ 10 ¼ 20 Model ^D Q BP ^D Q BP ^D Q BP 1 53.32 29.59 38.31 21.76 32.77 19.25 2 99.01 94.53 98.56 70.46 98.01 59.61 3 23.37 21.62 20.10 16.71 17.61 15.17 4 77.13 60.82 59.38 40.29 48.12 35.15 5 93.22 84.66 87.62 66.68 79.84 58.46 6 13.74 10.68 11.17 9.13 10.05 8.61 7 26.51 17.56 26.25 13.80 24.92 13.05 8 33.92 27.36 26.68 20.60 23.57 19.25 9 99.44 98.71 99.27 93.17 99.16 78.88 10 76.71 40.62 58.06 28.39 48.50 25.94 11 99.01 94.02 98.46 67.04 97.87 57.11 12 99.89 99.86 99.87 99.63 99.48 99.48 ^D and Q BP tests for checking odel adequacy of twelve ARMA(2,2) odels in Table 3 of Peňa and Rodriguez (2002) fitted by AR (1) using the Burg algorith. Both tests were ipleented based on the paraetric bootstrap procedure. The epirical power for each test was calculated based on N ¼ 10 4 siulations. Each Monte Carlo test also used B ¼ 10 3 siulations. Series length n ¼ 100 and lags ¼ 5,10,20 were investigated. obtained ^a ¼ 1:635 using the ethod of McCulloch (1986). So the infinite variance hypothesis is plausible for this data. In Table V we copare the P-values obtained for the Monte Carlo tests with ^D and Q BP for both the stable and noral cases with ¼ 10,20,30. The P-value for the Q LB test using the v 2 approxiation is also shown. As expected the Monte Carlo test using Q BP agrees well with the Q LB test using the v 2 approxiation since n is quite large. It is interesting that when ¼ 10 all tests have siilar P-values but when ¼ 20,30 the P-values of Journal copilation Ó 2008 Blackwell Publishing Ltd.

PORTMANTEAU TESTS 613 TABLE V An Exaple Using the Monthly Siple Return of CRSP Value-Weighted Index Data fro Tsay (2002) Test statistic Method ¼ 10 ¼ 20 ¼ 30 ^D Monte Carlo/stable 5.1 3.1 1.7 Q BP Monte Carlo/stable 5.0 2.5 2.6 ^D Monte Carlo/noral 6.6 1.4 0.1 Q BP Monte Carlo/noral 4.8 0.2 0.2 Q LB v 2 test 4.7 0.2 0.2 The data were fitted by an AR(5) odel. The entries in the first two coluns are the P-values, in percent, of ^D S and ^Q S in Section 4 based on the Monte Carlo test; those in the third and fourth coluns are the P-values, in percent, of the Monte Carlo test for ^D N and ^Q N based on the noral distribution and in the last colun the P-value for the standard asyptotic Ljung Box test based on the noral assuption is given. the noral-based tests are uch saller, by a factor of about 10, than the ore correct test based on the stable distribution. In general it appears that using tests based on the assuption of norally distributed innovations ay produce P-values which are too sall when the innovations are generated by a stable distribution. In other words, the test based on the noral distribution would err in the opposite direction as a conservative test and lead to an inflated type I error rate. We ay conclude that in fitting ARMA tie series with stable innovations, it is iportant to use the Monte Carlo diagnostic test presented in this article. 5. CONCLUDING REMARKS McLeod and Li (1983) considered using a portanteau diagnostic test based on the squared residuals. Li (2004, Ch. 5) shows that this test is also useful for testing for the presence of conditional heteroscedasticity in the residuals of fitted ARMA odels. The Monte Carlo test presented in this article is generalized to the squared residuals case in Appendix B and ipleented in our R package (Lin and McLeod, 2007). APPENDIX A. PROOF OF THEOREM 1 First, by decoposing the deterinant of the saple autocorrelation atrix R (), Pena and Rodriguez (2002) showed that jr () j 1/ is a weighted function of the first partial autocorrelations. Specifically, jr ðþ j 1= ¼ Y ð1 p 2 i Þðþ1 iþ= : i¼1 ð29þ Journal copilation Ó 2008 Blackwell Publishing Ltd.

614 J.-W. LIN AND A. I. MCLEOD Suppose that under the null hypothesis, ^D is asyptotic distributed as X. By applying the d-ethod to g(x) ¼ log(1 x), it follows that (n/log(n)) 2/a log (jr () j 1/ ) is asyptotically distributed as X. Fro eqn (29), we can have n 2=a logðjr j 1= n Þ¼ 2=aX i¼1 i þ 1 logð1 p 2 i Þ: Next suppose that n 2=a p 2 1 ; p2 2 ;...; T! p2 Y ; ð31þ and apply the ultivariate d-ethod to it follows that gðp 2 1 ; p2 2 ;...; p2 Þ¼ X i¼1 X i¼1 i þ 1 logð1 p 2 i Þ; i þ 1 logð1 p 2 1 i Þ! 1; ;...; 1 Y : Fro the Craer Wold theore, it follows that 1; 1 ;...; 1 n 2=a p 2 1 ;...; n 2=a p 2! 1; 1 ;...; 1 Y ð33þ Under the null hypothesis that X t are a sequence of i.i.d. stable Paretian rando variables, we have q 0 ¼ 1 and q k ¼ 0 for k 1 so the liiting distribution of saple ACFs can be further siplified as follows: 1 n a ð r1 ;...; r k Þ T! ðw 1 ;...; W k Þ T ; ð34þ where W h are given by W h ¼ S h ; h ¼ 1;...; k: ð35þ S 0 Note that, for a > 1, we ay also use the ean-corrected saple autocorrelation function at lag k, denoted as ~r k, which is given by P n k t¼1 ~r k ¼ ðx t X ÞðX tþk X Þ P n t¼1 ðx t X Þ 2 ; k ¼ 1; 2;...: ð36þ Using eqn (8), we have 1 n a ð p1 ;...; p Þ T! ðw 1 ;...; W Þ T : ð37þ it follows that! T ð30þ ð32þ Journal copilation Ó 2008 Blackwell Publishing Ltd.

PORTMANTEAU TESTS 615 1; 1 ;...; 1 n 2=a p 2 1 ;...; n 2=a p 2! W1 2 þ 1 W 2 2 þþ1 W 2 ; ð38þ Finally, fro eqns (33) and (38), and fro eqn (31), we have 1; 1 ;...; 1 Y! X ^D! X i¼1 i¼1 þ 1 i Wi 2 : þ 1 i Wi 2 ;! T B. MONTE CARLO TEST FOR RANDOMNESS AND GOODNESS-OF-FIT OF ARMA MODELS The Monte Carlo test procedure for diagnostic checking of AR and ARMA odels with stable Paretian errors can be suarized below. Step 1. Fit an AR odel to data using least squares or the Burg algorith or for ARMA, an approxiate Gaussian axiu likelihood algorith is used. Calculate residuals f^z t g and the portanteau test of interest, say ^D. Step 2. Estiate the paraeters of the stable distribution fro residuals f^z t g in step 1. The quantile estiator given by McCulloch (1986) ay be used or the MLE ipleented in the R package fbasics. Step 3. Select the nuber of Monte Carlo siulations, B. Typically 100 B 1000. Step 4. Siulate the fitted odel using the estiated AR or ARMA paraeters in step 1 and ^a in step 2. Obtain ^D after estiating the paraeters in the siulated series. Step 5. Repeat step 4 B ties counting the nuber of ties k that a value of ^D greater than or equal to that in step 1 has been obtained. Step 6. The P-value for the test is (k þ 1)/(B þ 1). Step 7. Reject the null hypothesis if the P-value is saller than a predeterined significance level. This algorith is easily odified to handle the proble of testing a tie series for randoness as well as for Monte Carlo testing when the norality assuption is ade. Siulation of ARMA odels with stable innovations The stable paraeters ay be estiated using the ethod of McCulloch (1986). Independent and identically distributed stable rando variables ay be siulated using the fbasics package. The ARMA(p,q) odel ay be approxiated as a high-order ovingaverage odel, Journal copilation Ó 2008 Blackwell Publishing Ltd.

616 J.-W. LIN AND A. I. MCLEOD Y t ¼ : Z t þ w 1 Z t 1 þþw Q Z t Q : ð39þ Initial values Y 1,..., Y r,r ax(p, q) are obtained fro eqn (39) using the fast Fourier transfor to copute the convolution. The reaining portion of the tie series ay be coputed recursively. For speed, these recursions are ipleented in C and interfaced to our R functions. The function siulateara ipleents these ideas in our R package. Nonlinear test In steps 1 and 4 of the algorith in Appendix B, square the residuals, copute the eancorrected saple autocorrelations and use these autocorrelations to copute the test statistic ^D. R package The R package prtest that ipleents all ethods described above is available fro our online suppleent (Lin and McLeod, 2007). Scripts are also given (Lin and McLeod, 2007) for generating all figures and tables which were given in our paper. ACKNOWLEDGEMENTS A. I. McLeod acknowledges with thanks a Discovery Grant Award fro NSERC. The authors would also like to thank the referee for helpful suggestions. NOTE Corresponding author: A.I. McLeod, Departent of Statistical and Actuarial Sciences, The University of Western Ontario, London, Ontario N6A 5B7, Canada. E-ail: aicleod@uwo.ca REFERENCES Adler, R. J., Feldan, R. E. and Gallagher, C. (1998) Analysing stable tie series. In A Practical Guide to Heavy Tails: Statistical Techniques and Applications (eds R. J. Adler, R. E. Feldan and M. S. Taqqu). Boston, MA: Birkhäuser, 133 58. Box, G. E. P. and Pierce, D. A. (1970) Distribution of residual autocorrelation in autoregressiveintegrated oving average tie series odels. Journal of Aerican Statistical Association 65, 1509 26. Brockwell, P. J. and Davis, R. A. (1991) Tie Series: Theory and Methods. New York: Springer. Journal copilation Ó 2008 Blackwell Publishing Ltd.

PORTMANTEAU TESTS 617 Davis, R. A. (1996) Gauss Newton and M-estiation for ARMA processes. Stochastic Processes and their Applications 63, 75 95. Davis, R. A. and Resnick, S. (1986) Liit theory for the saple covariance and correlation functions of oving averages. The Annals of Statistics 14, 533 58. Li, W. K. (2004) Diagnostic Checks in Tie Series. New York: Chapan and Hall/CRC. Lin, J.-W. and McLeod, A. I. (2007) Online suppleents to Portanteau Tests for ARMA odels with Infinite Variance (available at http://www.stats.uwo.ca/faculty/ai/2007/linmcleod/). Ljung, G. M. and Box, G. E. P. (1978) On a easure of lack of fit in tie series odels. Bioetrika 65, 297 303. McCulloch, J. H. (1986) Siple consistent estiators of stable distribution paraeters. Counication in Statistics Coputation and Siulation 15, 1109 36. McLeod, A. I. and Li, W. K. (1983) Diagnostic checking ARMA tie series odels using squaredresidual autocorrelations. Journal of Tie Series Analysis 4, 269 73. Mittnik, S., Rachev, S. T., Doganoglu, T. and Chenyao, D. (1999) Maxiu likelihood estiation of stable Paretian odels. Matheatical and Coputer Modelling 29, 275 93. Monti, A. C. (1994) A Proposal for residual autocorrelation test in linear odels. Bioetrika 81, 776 80. Peňa, D. and Rodriguez, J. (2002) A powerful portanteau test of lack of fit for tie series. Journal of Aerican Statistical Association 97, 601 10. Peňa, D. and Rodriguez, J. (2006) The log of the autocorrelation atrix for testing goodness of fit in tie series. Journal of Statistical Inference and Planning 136, 2706 18. Runde, R. (1997) The asyptotic null distribution of the Box Pierce Q-statistic for rando variable with infinite variance: an application to Geran stock returns. Journal of Econoetrics 78, 205 16. Saorodnitsky, G. and Taqqu, M. (1994) Stable-Non-Gaussian Rando Processes. New York: Chapan-Hall. Tsay, R. S. (2002) Analysis of Financial Tie Series. New York: Wiley. Journal copilation Ó 2008 Blackwell Publishing Ltd.