A Powerful Portmanteau Test of Lack of Fit for Time Series

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1 A Powerful Portanteau Test of Lack of Fit for Tie Series Daniel Peña and Julio Rodríguez A new portanteau test for tie series, ore powerful than the tests of Ljung and Box and Monti, is proposed. The test is based on the th root of the deterinant of the th autocorrelation atrix. It is shown that the proposed statistic is a function of all of the squared ultiple correlation coef cients of the regressions of the residuals on their lags when the nuber of lags goes fro 1 to. It can also be written as a function of the rst partial autocorrelation coef cients. The asyptotic distribution of the test statistic is a linear cobination of chi-squared distributions and can be approxiated by a gaa distribution. It is shown, depending on the odel and saple size, that this test can be up to 50% ore powerful than the Ljung and Box and Monti tests. The test is applied to the detection of several types of nonlinearity by using the autocorrelation atrix of the squared residuals, and it is shown that, in general, the new test is ore powerful than the test of McLeod and Li. An exaple is presented in which this test nds nonlinearity in the residuals of the sunspot series. KEY WORDS: AR IMA odel; Autocorrelation coef cient; Autocorrelation atrix; Goodness of t; Heteroscedasticity odel; Nonlinearity test; Partial autocorrelation. 1. INTRODUCTION Suppose that a tie series 8X t 9 is generated by a stationary and invertible ARMA4p1q5 process of the for 4B5X t D ˆ4B5 t, where t N and 4B5 and ˆ4B5 are polynoials given by 4B5 D 1 ƒ 1 B ƒ ƒ p B p and ˆ4B5 D 1 ƒ ˆ1B ƒ ƒ ˆqB q, where B k X t D X tƒk 0 Usually X t is soe transforation of an observed tie series such as differencing. De ning Oˆ4B5 and 4B5 O as the estiated polynoials where the coef cients i and ˆj are replaced by the axiu likelihood estiators, i O and Oˆj, the residuals of this odel are given by Õ t D Oˆƒ1 4B5 4B5X O t. Several diagnostic goodness-of- t tests have been proposed based on the residual autocorrelation coef cients given by r k D P n tdkc1 Õ t Õ tƒk = P n td1 Õ t for k D 11 1 : : :. Box and Pierce (1970) introduced a portanteau test to check the adequacy of the tted odel, using the statistic Q D n r k 1 (1) and they showed that the asyptotic distribution of Q can be approxiated by a distribution with ƒ4p Cq5 degrees of freedo. Ljung and Box (1978) iproved this approxiation by replacing the autocorrelation coef cients r k in (1) with their standardized values leading to the statistic Qr k Q LB D n4n C 5 kd1 D 4n C 5 4n ƒ k5 r k 1 () 4n ƒ k5 ƒ1 r 0 (3) k Daniel Peña and Julio Rodríguez are with the Departent of Statistics and Econoetrics, University Carlos III of Madrid, Calle Madrid 16, CP 8903, Getafe, Madrid, Spain (E-ail: dpena@est-econ.uc3.es or puerta@estecon.uc3.es). This research has been sponsored by Ministerio de Ciecia y Techología (Spain) under project BEC and by the Cátedra Banco BVA de Calidad. The authors are very grateful to Andrés Alonso for any stiulating discussions, to Mike Wiper and Victor Yohai for their help with the nal draft, and to an associate editor and a referee for very helpful coents. kd1 Ljung (1986) showed that when is sall, the estiated size of Q LB can be iproved by using the scaled distribution and that coputing Q LB with too any residual autocorrelations can reduce the power of the test. Monti (1994) proposed a portanteau test of goodness-of- t based on partial autocorrelations. Let O k be the kth residual partial autocorrelation. If the odel is correctly speci ed, O k is approxiately distributed as a noral rando variable with ean zero and variance 4n ƒ k5=4n4n C 550 Thus, a portanteau test statistic, siilar to the Ljung Box statistic, can be de ned by 601 Q MT D n4n C 5 4n ƒ k5 ƒ1 O k 0 (4) Under the assuption that the tie series has been generated by an ARMA4p1q5, the asyptotic distribution of Q MT is with ƒ 4p C q5 degrees of freedo. Monti (1994) showed by siulation that when the tted odel underestiates the order of the oving average coponent, Q MT is ore powerful than Q LB. Kwan and Wu (1997) exained via Monte Carlo siulation the nite-saple properties of (3) and (4) for data generated with onthly seasonality, nding only sall differences between the powers of Q MT and Q LB. This article proposes a new portanteau goodness-of- t test based on a general easure of ultivariate dependence and is organized as follows. Section presents the test and its ain properties. Section 3 obtains its asyptotic distribution and shows that it can be approxiated by a gaa distribution. Section 4 includes a Monte Carlo study of the properties of the test and shows that it is ore powerful than the tests proposed by Ljung and Box (1978) and Monti (1994). Section 5 extends the test to check nonlinearity by using the autocorrelations of the squared residuals. It is shown that the proposed test is ore powerful than that proposed by McLeod and Li (1983). Section 6 discusses soe advantages and liitations of the proposed test. kd1 00 Aerican Statistical Association Journal of the Aerican Statistical Association June 00, Vol. 97, No. 458, Theory and Methods

2 60 Journal of the Aerican Statistical Association, June 00. THE PROPOSED TEST The estiated residuals can be considered a saple of ultivariate data fro soe distribution. We are interested in testing whether or not the covariance (correlation) atrix of their distribution is proportional (equal) to the identity. In ultivariate analysis the likelihood ratio test for checking if a set of noral rando variables has a scalar covariance atrix is proportional to the deterinant of the correlation atrix of the ultivariate variables. Thus, it is sensible to explore a test based on this statistic. For stationary tie series data the residual correlation atrix of order, b R, is given by 3 1 r 1 r r 1 1 r ƒ1 br D (5) r r ƒ1 1 We propose to test for autocorrelation in the estiated residuals by using a transforation of br that has a sipler distribution under the null hypothesis. The proposed portanteau statistic is bd D n61 ƒ b R 1= 70 (6) This statistic has two interesting interpretations. The rst can be obtained by using a recursive expression for the deterinant of the atrix (5). If we de ne Or 45 D 4r 1 1 : : : 1 r 5 0, we can write µ 1 Or 0 45 br D 1 Or 45 br ƒ1 and, using the properties of the deterinant of a partitioned atrix, we have b R D b Rƒ1 41 ƒ R 51 where R D Or0 b 45Rƒ1 ƒ1 Or 45 is the square of the ultiple correlation coef cient in the linear t Õ t D P jd1 b j Õ tƒj C u t. By recursive use of this expression, we have " # 1= Y b R 1= D 41 ƒ R i 5 0 (7) Note that 1 ƒ R i is a easure of dependence and b R 1= is the geoetric average of these dependence easures. Thus, 1 ƒ b R 1= can be interpreted as an average squared correlation coef cient, obtained when autoregressive odels of increasing order are tted to the residuals of the estiated tie series. The second interpretation is based on the partial autocorrelation coef cients. Note that 1 ƒ R i D RSS411 i5=tss, where RSS411 i5 is the residual or unexplained su of squares in the analysis of variance (ANOVA) decoposition of the residual regression Õ t D P i b jd1 j Õ tƒj C u t, and TSS D P Õ t is the total su of squares. In the sae way, 1ƒR D RSS411 i ƒ15=tss iƒ1 and 1 ƒ R i RSS411 i5 D 1 ƒ R iƒ1 RSS411 i ƒ 15 D 41 ƒ O i 51 (8) where O i D 6RSS411 iƒ15ƒrss411 i57=rss411 i ƒ15 is the ith squared partial autocorrelation coef cient. Thus, using (8) and (7), we obtain b R 1= D Y 41 ƒ O i 54C1ƒi5= 0 (9) This expression shows that b R 1= is a weighted function of the rst partial autocorrelation coef cients of the residuals. 3. DISTRIBUTION OF THE PROPOSED STATISTIC In this section we derive the asyptotic distribution of the proposed statistic (6) for all, where is the nuber of saple autocorrelations. As this distribution is coplicated, we follow Box and Pierce (1970) in obtaining an approxiation to this distribution when is oderately high. Then, we will show by Monte Carlo siulation that this approxiation works well in sall saples. 3.1 Asyptotic Distribution The statistic bd is a continuous function of the saple partial autocorrelations, O i, as shown in Equation (9). De ning O 45 D 4 O 1 1 : : : 1 O 5 0, and using a result of Monti (1994), we have that n 1= O 45 is asyptotically ultivariate noral with zero ean vector and covariance atrix 4 I ƒ Q 5, where Q D X V ƒ1 X 0, V is the inforation atrix for the paraeters and ˆ1 and X is an 4p C q5 atrix, with eleents 0 and ˆ0 de ned by 1= 4B5 D Pˆ id0 0 i Bi and 1=ˆ4B5 D Pˆ id0 ˆ0 i Bi (see Brockwell and Davis, 1991, pp ). The coef cients 0 i and ˆ0i are readily coputed using the recursive procedure of Box and Jenkins (1976, pp ). Theore 1. If the odel is correctly identi ed, bd is asyptotically distributed as P i11 i, where 11 i 4i D 11 : : : 1 5 are independent 1 rando variables and i 4i D 11 : : : 1 5 are the eigenvalues of 4 I ƒ Q 5W, where W is a diagonal atrix with eleents w i D 4 ƒ i C 15= 4i D 11 : : : 1 5. The proof of this theore is given in the Appendix. For a general ARMA odel the expression for the eigenvalues of 4 I ƒq 5W is coplicated. However, these eigenvalues are readily calculated for any given, ˆ, and. In the following subsection, we propose an approxiation to the distribution P i 11 i for oderately high. 3. An Approxiation to the Distribution of b D The probability Pr4bD > x5 can be evaluated by inverting the characteristic function of P i11 i ( Ihof, 1961). This procedure requires only one-diensional nuerical integration, but for siplicity, we prefer to use the approxiation proposed by Satterthwaite (1941, 1946) and Box (1954). They suggested approxiating a distribution of the for P i11 i by a distribution of the for ab, with ean and variance equal to those of the exact distribution, where the degrees of freedo, b, are usually fractional. This iplies a D P i = P i and b D 4 P i 5 = P i. Thus, following this suggestion, we approxiate the distribution of bd by a gaa

3 Peña and Rodríguez: A Powerful Portanteau Test of Lack of Fit for Tie Series 603 Relative frequency Relative frequency ^ D D Figure 1. The Solid Line is the Monte Carlo Distribution of b D (left) and D (right) Generated fro 10,000 Replications of an AR(1) Model With 1 D.5, When n D 100 and D 0. The dashed line is the (, ) approxiation With D 7.3 and D.77. distribution, 4 D b=1 D 1=a5, where the paraeters are de ned by and D D 364 C 15 ƒ 4p C q57 64 C 154 C 15 ƒ 14p C q C 15 ƒ 4p C q57 4 C 154 C 15ƒ 14p C q5 1 and the distribution has a ean of = D 4 C15= ƒ 4p Cq5 and a variance of = D 4C154C15=3ƒ4pCq5. The details of this approxiation are given in the Appendix. This approxiation has been checked by a Monte Carlo experient. As the gaa approxiation to the asyptotic distribution iproves when the standardized autocorrelation coef cients Qr k de ned by () are used, we will consider both bd, given by (6), and D de ned by D D n61 ƒ e R 1= 71 (10) where e R is the correlation atrix built by using the standardized autocorrelation coef cients Qr k. Under the null hypothesis bd and D are asyptotically equivalent, but obviously, soe differences ay be expected in their sall-saple behaviors. Figure 1 illustrates the accuracy of the approxiation of the epirical distribution of bd and D to the gaa distribution, using 10,000 replications of saple size 100 by an AR(1) process with 1 D 05 and D 0. We have found siilar results in a larger siulation study, and, as the approxiation by the standardized autocorrelations is better, we recoend the use of D, especially for sall saple sizes. Table 1 shows the percentiles corresponding to D 005 for different values of and the estiated paraeters 4p C q5 for the proposed approxiating distribution. Note that in the approxiation c4p1q51 where c4p1 q5 34p C q ƒ 1=5 is obtained fro the restriction that the variance ust be positive. 4. SMALL-SAMPLE SIGNIFICANCE LEVEL AND POWER OF D In this section we present a coparative study of the signi - cance level and power of the three statistics D, Q LB, and Q MT given by (10), (3), and (4). The signi cance levels of Q LB and Q MT have been obtained using the percentiles of the distribution, and those of D using the approxiation obtained in Section 3.. These signi cance levels have been evaluated under both low-order AR and MA odels, but here we report only the results for the AR(1). Siilar results were found in the other Table 1. 95% Points of the Recoended Approxiation to the Distribution of D, for Different Values of and (p + q) p C q

4 604 Journal of the Aerican Statistical Association, June 00 Table. Signi cance Levels of D, Q LB, and Q MT Under an AR(1) Model = 10 = 15 = 0 1 D Q LB Q MT D Q LB Q MT D Q LB Q MT D D cases, and they are available upon request fro the authors. In each case, 10,000 Gaussian series of saple size n D 100 were generated. Three values for, 10, 15, and 0, were considered. Table shows the signi cance levels of the three statistics for several values of the AR(1) paraeter when the noinal levels are.05 and.01. The observed signi cance level is close to the noinal level for the three statistics, and the results obtained for Q LB and Q MT are siilar to those presented by Monti (1994). In all 30 cases the observed signi - cance level of Q LB is larger than the noinal level, whereas the behavior of Q MT and D shows less variability. For D 0051 the signi cance levels of D belong to the interval ( ) and are always between those of Q LB (interval ) and Q MT (interval 0039 ƒ For D 001 the observed values for D and Q MT are in the interval ( ), whereas those of Q LB are in the range (.01.01). The signi cance level of D does not see to be affected by the value of. The power of the tests is analyzed for the odels proposed by Monti (1994). Twenty-four different ARMA41 5 odels are considered. Table 3 shows the power of the three tests when, erroneously, an AR(1) or a MA(1) odel is tted to the data. In each case 1,000 series of 100 observations were generated, and the power was coputed for D 10 and D 0. The power of the three tests decreases as increases, as expected, but the loss of power in D is relatively sall. The test based on D is always the ost powerful, and the increase in power with respect to the best of the other two statistics, Q MT and Q LB 1 can be as high as 50% in soe cases (see odels 1, 11, and 3). In order to analyze the perforance of the test for sall saple size, Table 4 presents the sae power study with n D 30. The values of are 5 and 10. As before, the power of the three tests decreases as increases, and the test based on D is alost always the ost powerful (with the exception of odel ). Again the increase in power can be as high as 75% (see odels 5 and 11). As expected, the power of the three tests is generally low, although in soe cases (see odels 7, 15, and 19) the power of the proposed test can be higher than 70%. Table 3. Power Levels of the Tests Based on D, Q LB, and Q MT When the Data Are Generated Fro ARMA(, ) Models and AR(1) and MA(1) Models Are Fitted, n D 100 D 10 D 0 M 1 ˆ1 ˆ D Q LB Q MT D Q LB Q MT (a) Fitted by AR(1) odel 1 ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ (b) Fitted by MA(1) odel ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ00 ƒ ƒ ƒ

5 Peña and Rodríguez: A Powerful Portanteau Test of Lack of Fit for Tie Series 605 Table 4. Powers of the Tests Based on D, Q LB, and Q MT When the Data Are Generated Using ARMA(, ) Models and AR(1) and MA(1) Models Are Fitted, n D 30 D 5 D 10 M 1 ˆ1 ˆ D Q LB Q MT D Q LB Q MT (a) Fitted by AR(1) odel 1 ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ (b) Fitted by MA(1) odel ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ00 ƒ ƒ ƒ CHECKING THE LINEARITY ASSUMPTION The analysis of tie series using nonlinear odels has gained uch attention in recent years because of the liitations of linear odels in capturing soe observed real data structures and the advances in coputational power. A large nuber of tests of the linearity assuption have been proposed, see Peña, Tiao, and Tsay (001, chap. 10). These tests can be classi ed into two groups. The rst group is based on the Volterra expansion (Wiener, 1958, lecture 10) of a stationary tie series Y t, as Y t D Œ C ˆX a i tƒi C idƒˆ ˆX i1 jdƒˆ C a ij tƒi tƒj ˆX i1 j1 kdƒˆ a ijk tƒi tƒj tƒk C 1 (11) where Œ is the ean level of Y t and 8 t1 ƒˆ < t < ˆ9 is a strictly stationary process of independent and identically distributed rando variables. Obviously, Y t is nonlinear if any of the higher order coef cients, 8a ij 9, 8a ijk 91 : : :, is nonzero. Based on this expansion and on Tukey s one degree of freedo for the nonadditivity test, Keenan (1985) and Tsay (1986), aong others, have proposed speci c tests for nonlinearity. A second group of tests is based on the idea, proposed by Granger and Anderson (1978) and Maravall (1983), that looking at the autocorrelation function of the squared values of the tie series could be useful for identifying nonlinear tie series. If the residuals Õ t are independent then the Õ t will also be independent, but if the odel is nonlinear and the residuals Õ t are not independent, this feature can appear in the autocorrelation function of Õ t. McLeod and Li (1983) proposed detecting nonlinearity in tie series data, using the statistic Q LB 4 Õ t 5 D n4n C 5 X 4n ƒ k5 ƒ1 r k 4 Õ t 51 (1) where r k 4 Õ t 5 are the autocorrelation coef cients of the squared residuals coputed by P n r k 4 Õ t 5 D 4 tdkc1 Õ ƒ O t 54 Õ ƒ O tƒk 5 P n 4 td1 Õ ƒ O 4k D 11 1 : : : 1 51 t 5 where O D P Õ t =n. In a siilar way, it is possible to build a test siilar to that of Monti, using the partial autocorrelation of the squared residuals, Q MT 4 Õ t 5. We can extend the D statistic (10) for testing for nonlinearity to kd1 D 4 Õ t 5 D n61 ƒ er 4 Õ t 5 1= 71 where e R 4 Õ t 5 is the autocorrelation atrix (5), which is now built using the standardized autocorrelation coef cients Qr k 4 Õ t 5. In the following theore, we give the asyptotic distribution of the statistic D 4 Õ t 5 under the null hypothesis that the series follows an ARMA odel that is correctly identi ed. Theore. If the series follows an ARMA odel, the statistic D 4 Õ t 5 coputed fro the squared residuals of the correctly tted odel is asyptotically distributed as P w i11 i, where 11 i 4i D 11 : : : 1 5 are independent 1 rando variables and w i D 4 ƒ i C 15= 4i D 11 : : : 1 5.

6 606 Journal of the Aerican Statistical Association, June 00 The proof of this theore is given in the Appendix. The asyptotic distribution of Theore can be approxiated as before by a Satterthwaite-type approach using a distribution of the for ab. Thus, if the series follows an ARMA odel, the asyptotic distribution of D 4 Õ t 5 can be approxiated by a gaa distribution, 4 D b=1 D 1=a5, where D 34 C 15=44 C 15 and D 3=4 C 15. The rst colun of Table 1 shows the percentile corresponding to D 005 for the ost usual values of for this approxiated distribution. 5.1 Power Study In this section we copare the powers of the statistics D 4 Õ t 5, Q LB 4 Õ t 5, and Q MT 4 Õ t 5 for testing for linearity. We will calculate the test powers for the four nonlinear odels used by Keenan (1985), Model 11 Y t D e t ƒ 04e tƒ1 C 03e tƒ C 05e t e tƒ 3 Model 1 Y t D e t ƒ 03e tƒ1 C 0e tƒ C 04e tƒ1 e tƒ ƒ 05e tƒ 3 Model 31 Y t D 04Y tƒ1 ƒ 03Y tƒ C 05Y tƒ1 e tƒ1 C e t 3 Model 41 Y t D 04Y tƒ1 ƒ 03Y tƒ C 05Y tƒ1 e tƒ1 C 08e tƒ1 C e t 3 where the e t s are independent N Table 5 suarizes the power results. For each odel 1,000 replications of saple size n D 04 were generated. An AR4p5 odel is tted to the data, where p is selected by the Akaike inforation criterion (A IC) (Akaike, 1974) with p The power of the proposed test, D 4 Õ t 5, is broadly between 6% (Model 4, D 7) and 40% (Model, D 4) higher than the powers of the tests based on Q LB 4 Õ t 5 and Q MT 4 Õ t 50 None of these tests is powerful in handling Model 1, which contains a concurrent nonlinear ter e t e tƒ. The dif culties in observing the nonlinearity of Model 1 are seen in the study by Tsay (1986), which copares soe statistics for checking nonlinearity over the sae four odels. Fro the coparison of Table 5 and the results obtained by Tsay (1986), we conclude that the proposed test D 4 Õ t 5 is better than that proposed by Keenan for all lags, for D 7, 1, and 4, and for all odels except Model. However, the proposed portanteau test is slightly worse than the test proposed by Tsay (1986). We are interested in checking the behavior of the proposed statistics in the detection of nonlinearity in Threshold Autoregressive (TAR) odels, which are aong the ost popular Table 6. Powers of the Tests Based on D, Q LB, and Q MT Give Data Generated Fro a TAR() Model When AR() Is Fitted, D.05 D 1 D 4 D 36 n D Q LB Q MT D Q LB Q MT D Q LB Q MT nonlinear tie series odels in applied research. The siplest, two-regie TAR(1) odel is given by y t D C y tƒ1 C 1t 1 if y tƒ1 µ c1 y t D 45 0 C 45 1 y tƒ1 C t 1 if y tƒ1 > c0 With this ai, 1,000 replications were generated with saple sizes 100, 00, and 500 fro the TAR() odel with four regies proposed by Tiao and Tsay (1994). This odel is an alternative to the AR() odel for capturing the structure of the U.S. real GNP series fro the rst quarter of 1947 to the rst quarter of 1991 with a total of 177 observations. The regies and their econoic interpretation are described in Tiao and Tsay (1994). Table 6 shows the power of the three statistics D, Q LB, and Q MT applied to the residual autocorrelations after tting an AR() odel. The values shown in the table are close to the noinal signi cance level used, (.05), con ring the lack of power of these statistics to detect threshold nonlinear structure. Table 7 displays the power of the statistics based on the autocorrelations of the squared residuals of the AR() t, D 4 Õ t 5, Q LB 4 Õ t 5, and Q MT 4 Õ t 5. The power of the three statistics is now uch larger, especially in large saples, and again the statistic D 4 Õ t 5 is the ost powerful for all saple sizes and lags. The increase in power of D 4 Õ t 5 with respect to the best of Q LB and Q MT for D 1 goes fro.9% to 40.8%. Finally we analyze an iportant class of nonlinear tie series odels with changing conditional variance. Engle (198) proposes Autoregressive Conditional Heteroscedasticity (ARCH) odels, which have been generalized to the Autoregressive Stochastic Volatility (ARSV(1)) odels, proposed by Taylor (1986), and the Generalized Autoregressive Conditional Heteroscedasticity (GARCH(1,1)) odels, proposed by Bollerslev (1986). These odels have been used for analyzing nancial tie series (see Carnero, Peña, and Ruiz (001), for a review of their applications in nance) and for Table 5. Powers of the Tests Based on D, Q LB, and Q MT When the Data Are Generated by Four Nonlinear Models, and the Fitted Model Is an AR( p), D.05 D 7 D 1 D 4 Model D Q LB Q MT D Q LB Q MT D Q LB Q MT Table 7. Powers of the Tests Based on D ( Õ t ), Q LB ( Õ t ), and Q MT ( Õ t ) Give Data Generated Fro a TAR() Model When AR() Is Fitted, D.05 D 1 D 4 D 36 n D Q LB Q MT D Q LB Q MT D Q LB Q MT

7 Peña and Rodríguez: A Powerful Portanteau Test of Lack of Fit for Tie Series 607 Table 8. Powers of the D ( Õ ), Q t LB( Õ ), and Q t MT( Õ t ) Tests for Two ARCH(p) Models D 7 D 1 D 4 p n D Q LB Q MT D Q LB Q MT D Q LB Q MT odeling environental variables (see Tol, 1996, for a eteorological application). The test ost often used for checking this type of nonlinearity is Q LB 4 Õ t 5 as given by (1). To analyze the power of the new proposed test for these odels, we generate rst 1,000 series of sizes n D 100, 50, 500, and 1,000 for the covariance stationary process ARCH4p5, y t D t t t D 0 C 1 y tƒ1 C C py tƒp 1 where 0 > 0 and P p i < 1. We consider the cases p D 1 and p D 3. The paraeters i have been sapled fro a unifor U and are rescaled by an auxiliary variable, s, fro a unifor distribution U so that P p i D s. Table 8 shows the power of the tests based on D 4 Õ 5, t Q LB 4 Õ t 5, and Q MT 4 Õ t 5 for the ARCH4p5 odels, with D 7, 1, and 4. The test based on D 4 Õ t 5 is again the ost powerful for all saple sizes and lags, but the advantage of this test over the other two is lower here than with other nonlinear odels. For the ARCH(3) odel the differences in power of the three tests are sall, and the increase in power of D 4 Õ t 5 goes fro 1.0% to 1.7%. Tables 9 and 10 show the power of these tests for four covariance stationary GARCH(1,1) odels, y t D t t 1 t D 1 C y tƒ1 C tƒ1 1 where 0, 0, and C < 1. Again 1,000 replicates have been ade. The paraeters in the rst two odels (see Table 9), are taken fro real nancial tie series (see Carnero et al., 001), whereas those in Table 10 are taken fro environental data (see Tol, 1996). An interesting result is that the power of the proposed test depends very uch on the paraeter values: it is slightly less than the one for Q LB 4 Õ t 5 (although greater than the one for Q MT 4 Õ t 55 in nancial data with long persistence, whereas it is the ost powerful of the three in Table 10 with paraeter values obtained fro the environental data. A possible reason for this different behavior is given in the following section. Table 11 shows the power for 1,000 replicates fro an ARSV(1) odel, log t y t D ü t t 1 D log tƒ1 C t1 where ü > 0, < 1, and t and t are assued to be utually independent and norally distributed white noise processes with zero ean and variances 1 and, respectively. The siulation experient follows the design of Sandann and Koopan (1998). The autoregressive paraeters are D , and the paraeter is selected so that the coef cient of variation of log t takes the values 10, 1, and 001. High values of the ratio of volatility variance to its squared ean indicate pronounced relative strength of a stochastic volatility process. We can observe that the proposed test is ore powerful than the other tests for D 4 and D 36. However, for D 1 and 09 the power of D 4 Õ t 5 is siilar to or slightly slower than the test based on Q LB. These results are consistent with the behavior observed in Table 9, since both odels, GARCH(1,1) and ARSV(1), are used to t nancial tie series with long persistence. 5. An Exaple with Real Data In this section we apply the D 4 Õ t 5 statistic to nd nonlinearities in the well-known sunspot tie series. This tie series has been studied previously by various authors who have applied both linear and nonlinear odels. A detailed description of soe of the different odels proposed can be found in Priestley (1989). Following Priestley (1989, p. 88), we have tted an AR(9) odel to the saple of 46 observations corresponding to the rst 46 observations fro 1,700. The order of the linear odel was deterined by the A IC. The odel is estiated by axiu likelihood, and no structure is found in the residuals using the statistics D, Q LB, Table 9. Powers of the D ( Õ ), Q t LB( Õ ), and Q t MT( Õ t ) Tests for Two GARCH(1,1) Models of Financial Tie Series D 1 D 4 D 3 n D Q LB Q MT D Q LB Q MT D Q LB Q MT

8 608 Journal of the Aerican Statistical Association, June 00 Table 10. Powers of the D ( Õ ), Q t LB( Õ ), and Q t MT( Õ t ) Tests for Two GARCH(1,1) Models of Meteorological Tie Series D 7 D 1 D 4 n D Q LB Q MT D Q LB Q MT D Q LB Q MT and Q MT, with lags D 1 and 4 and D 005. However, when the statistic D 4 Õ t 5 is used, a clear indication of nonlinear structure appears. Table 1 presents the values of the three portanteau statistics applied to the squared residuals. To facilitate the coparison, in addition to the value of each statistic the ratio between the statistic and the percentile corresponding to D 005 for each distribution is also given (see coluns 3, 5, and 7). The statistic D 4 Õ t 5 clearly suggests a nonlinear structure in the residuals of the sunspot series, whereas the results fro the Ljung Box and Monti statistics on the squared residuals are not decisive. 6. ADVANTAGES AND LIMITATIONS OF THE PROPOSED TEST The proposed test has an interesting property that ay explain why it works so well in soe cases copared with other portanteau goodness-of- t tests. Note that, for large n, both the Ljung Box and Monti tests are syetric on the autocorrelation coef cients; that is, given a sequence of autocorrelation coef cients 4r 1 1 : : : 1 r 5, a perutation of this sequence does not affect the test. However, the proposed test is highly asyetric with respect to the autocorrelation coef- cients. In fact, br includes 4 ƒ i C 15 ties the ith autocorrelation coef cient, which iplies that the rst autocorrelation coef cient appears ties in the atrix, whereas the th autocorrelation coef cient appears only twice. One would expect that this asyetry will iply a larger sensitivity of br to changes in the low-order coef cients, and this sees to be the case. We have checked that for D , we have d b R dr j r45 D 1 0 D ƒ 4C1ƒj541ƒ 5 ƒ1 1 j D11: : : 11 where the derivative of the deterinant with respect to the jth autocorrelation coef cient is evaluated at r 45 D : : : We can see that, in these particular cases, the effect of a change in the jth autocorrelation coef cient decreases with the lag. Thus, we can conclude that the proposed test is ore sensitive to the low-order autocorrelation coef cients than to the large order ones, in contrast to Q LB and Q MT, which have the sae sensitivity for all order autocorrelation coef cients. This sae property applies to the partial autocorrelation coef cients. In the proposed statistic they also have a weight that decreases with the lags. Fro (9) it is easy to see that d b R d j D ƒ4 C 1 ƒ j5 O j 1 ƒ O j b R 1 j D 11 : : : 1 1 and evaluating this function at 45 D : : : 1 5 0, d OR d j 45 D 1 0 D ƒ4 C 1 ƒ j5 41 ƒ 5 4C15=ƒ1 1 we conclude that the effect on the deterinant of a change in the jth partial autocorrelation coef cient decreases linearly with the lag. When the inforation about the lack of t is ainly included in the low lag autocorrelation coef cients (siple or partial), D sees to be ore powerful because it gives ore weight to these coef cients than do both Q LB and Q MT. On the other hand, when this is not the case and the inforation is spread over a long nuber of lags, this advantage ay disappear and D has a power siilar to those of Q LB and Q MT. For instance, for heteroscedastic data with long persistence (see Tables 9 and 11) we have seen that the perforances of Table 11. Power of the D ( Õ ), Q t LB( Õ ), and Q t MT( Õ t ) Tests for ARSV(1) Models (n D 50) D 1 D 4 D 36 n ü CV D Q LB Q MT D Q LB Q MT D Q LB Q MT

9 Peña and Rodríguez: A Powerful Portanteau Test of Lack of Fit for Tie Series 609 Table 1. Nonlinearity Tests on the Residuals of the Sunspot Data When an AR(9) Model Is Fitted D ( Õ t ) D ( Õ t )=d C Q LB ( Õ t ) Q LB ( Õ t )=d C Q MT ( Õ t ) Q MT ( Õ t )=d C the three tests are siilar. We have checked that for seasonal odels the perforance of D is still better in general (and soeties equal) to the perforances of Q LB and Q MT. However, it is possible that if the relevant inforation for odel checking is ainly given by the high-order autocorrelation coef cients, D ight be less powerful than Q LB and Q MT. APPENDIX: PROOFS OF THEOREMS AND APPROXIMATIONS TO THE DISTRIBUTIONS Proof of Theore 1 Suppose that under the null hypothesis, b D is asyptotically distributed as the rando variable X. Then, applying the -ethod (e.g., in Arnold, 1990) to g4x5 D log41 ƒ x5, it follows that ƒn log R 1= is also asyptotically distributed as X. Fro 495 we obtain the equivalent expression, ƒn log b R 1= ƒ i C 1 D ƒn log 1 ƒ O i 0 (A.1) To nd the distribution of (A.1), suppose that 4n O 1 1 n O 1 : : : 1 no 5 is asyptotically distributed as Y. Then, applying the ultivariate -ethod (e.g., in Arnold, 1990) to g4o 1 1 : : : 1 O 5 D ƒ P 44ƒ i C 15=5log41 ƒ O i 5, it follows that ƒ i C 1 ƒn log41ƒ O i 5! 11 ƒ 1 1 : : : 1 1 Y 1 (A.) where! stands for convergence in distribution. Fro the Craer Wold theore (e.g., in Arnold, 1990), it follows that 11 ƒ 1 1 : : : 1 1 4nO 1 1 n O 1 : : : 1 n O 50! 11 ƒ 1 1 : : : 1 1 Y 0 (A.3) Using the fact that n 1= O 45 is asyptotically distributed as N 401 I ƒ Q 5 and fro the theore on quadratic fors given by Box (1954), it follows that 11 ƒ 1 1 : : : 1 1 4nO 1 1 n O 1 : : : 1 n O 50 D no 0 45 W O 45! i 11 i 0 (A.4) Finally, fro (A.3) and (A.4), 11 ƒ 1 1 : : : 1 1 Y! and fro (A.), b D! P i 11 i. i 11 i 1 Approxiation to the Distribution of D Box and Pierce (1970) and McLeod (1978) approxiate the atrix Q D X V ƒ1 X 0 by the projection atrix Q D X 4X 0 X 5 ƒ1 X 0 when is oderately high. This approxiation is useful for coputing an expression for a and b that does not depend on the ARMA paraeters and ˆ. We have i D tr44 I ƒ Q 5W 5 D tr4w 5ƒ tr4q 5 C 41=5tr4Q C 51 (A.5) where C is a diagonal atrix with eleents c i D i, i D 01 : : : 1 4ƒ 15, and i D tr44 I ƒ Q 5W 4 I ƒ Q 5W 5 D tr4w 5 ƒ tr4q 5 C 4=5tr4Q C 5 ƒ 4= 5tr4Q C 5C 41= 5tr4Q C Q C 50 (A.6) An alternative expression for P i and P i can be obtained using the Cholesky decoposition of the atrix 4 I ƒ Q 5 (see Velilla (1994)). As Q is an idepotent atrix with rank p C q, (A.5) and (A.6) can be written as a function of p, q,, q ii, and q ij, where the q ij are the eleents of Q, i D C 1 ƒ 4p C q5 C 1 4i ƒ 15q ii 1 id i D 1 4 C 154 C 15ƒ 4p C q5 6 C C 1 4i ƒ 15q ii ƒ id X id jd X id 4i ƒ 154j ƒ 15q ij 0 4i ƒ 15 q ii (A.7) (A.8) Now we will show that the ters in (A.8) which depend on q ij tend to zero when increases. Consider the sequences a i D i and b i D 4i ƒ 15q ii. Then P 4i ƒ 15q ii=i µ P q ii D p C q < ˆ as! ˆ, and by Kronecker s lea (Davidson, 1997, pp ) we obtain that 4=5 P td1 4i ƒ 15q ii! 0. A siilar arguent and the property of idepotent atrices, q ii D q ii C P q ij, is used to show that 4= 5 P 4i ƒ 15 q ii! 0 and 41= 5 P P id jd4i ƒ 154j ƒ 15q ij! 0. Thus, for large we can approxiate Equations (A.7) and (A.8) by Proof of Theore i D C 1 ƒ 4p C q51 i D 1 4C 154 C 15 ƒ 4p C q50 6 (A.10) (A.9) This is based on the result by McLeod and Li (1983) for the asyptotic distribution of n 1= r 45 4 Õ t 5, which is N 401 I 5. Applying this result to the one obtained by Monti (1994), the asyptotic distribution of n 1= O 45 4 Õ t 5 is N 401 I 5. Following the sae reasoning as in Theore 1, the asyptotic distribution for D 4 Õ t 5 is obtained. [Received Deceber 000. Revised August 001.]

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