Research Journal of Recent Sciences _________________________________________________ ISSN 2277-2502 Vol. 2(5), 39-43, May (2013) Res.J.Recent Sci. International Science Congress Association 39 Testing Goodness-of-Fit in Autoregressive Fractionally Integrated Moving-Average Models with Conditional Hetroscedastic Errors of Unknown formAli Amjad , Salahuddin2 and AlamgirDepartment of Statistics, Islamia College University Peshawar, PAKISTAN Department of Statistics, University of Peshawar, Peshawar, PAKISTAN Available online at: www.isca.in Received 8th December 2012, revised 18th January 2013, accepted 11th February 2013Abstract This paper considers testing goodness-of-fit in Autoregressive fractionally integrated moving-average models with conditional hetroscedasticity. We extend the applicability of Hong’s and power transformed Hong’s test statistics as goodness-of-fit tests in ARFIMA-GARCH models, where the structural form of GARCH model is unknown. Simulation study is performed to assess the size and power performance of both tests. Key Words: Conditional hetroscedasticity, ARFIMA, GARCH, Goodness-of-Fit Tests. IntroductionIt is a nontrivial task to find an appropriate or a parsimonious model in regression and time series data analysis. Residuals analysis is commonly used as model diagnostics in time series model building. The adequacy of the fitted time series model is commonly tested by checking the assumption of white noise residuals. If the appropriate model has been chosen, there will be zero autocorrelation in the residuals series. Let be the series of the residuals from the fitted model, then in hypothesis testing settings we can state our null and alternative hypothesis as for all ¹ versus for some ¹ . In frequency domain approach the above hypothesis can be stated as 2/1, versus 2/1 for some p p - Î , where ik  2( is the normalized spectral density function of . Rejecting the above null hypothesis implies the inadequacy of the fitted model. Several tests have been developed to test the hypothesis of zero autocorrelation. Box and Pierce have developed a portmanteau test to test the adequacy of the fitted time series model. The test statistic is given as: (1) where is the autocorrelation of at lag and m is assumed to be fixed. They showed that for large n (sample size), the statistic has chi-square distribution with m degrees of freedom assuming that series is independently and identically distributed. If are the residuals from a fitted time series model, then is distributed as with p m - degrees of freedom, where p is the number of parameters in the model. Davis et al. showed that the distribution of can deviate from chi-squareand the true significance level is likely to be lower than the predicted significance level. A modified version of Box and pierce test statistic was proposed by Ljung and Box , which has the following form: /))2 (2) They preformed a comparative study of their test with the test of Box and Pierce and showed that their test has substantially improved approximation to chi-square distribution. For various choices of m , Ljung examined the properties of Box and Pierce test statistic. They suggested a modified version of Box and Pierce test statistic that allowed the use of various values of m . Their simulation studies showed that the modified test is more powerful under various innovations distributions. Hong introduced three classes of consistent one sided tests for testing serial correlation of the residuals of the linear dynamic model that include both lagged dependent and independent variables. Under the null hypothesis of zero autocorrelation, they showed that the standardized form of all these test statistics is asymptotically)1,0(. To improve asymptotic normality of Hong’s tests, Chen and Deo6 introduced power transformed Hong’s test. They examined the performance of Hong’s and power transformed Hong’s test statistics as goodness-of-fit tests Research Journal of Recent Sciences ______________________________________________________________ ISSN 2277-2502Vol. 2(5), 39-43, May (2013) Res. J. Recent Sci. International Science Congress Association 40 for different time series models with identically independent errors. In the current study, we consider model diagnostic checking of ARFIMA models when its innovations are conditionally hetroscedastic of unknown form. The ModelLong memory processes have been widely used in the analysis of time series data. Nile river data is an outstanding example which exhibits long memory behaviour. Other examples are the Ethernet traffic time series studied by Leland et.al. and foreign exchange rate returns studied by Goodhart and Hare. The common feature of these time series is that the decay of the autocorrelation function is like a power function rather than exponential as in the case of short memory time series. The spectral density of such processes behaves just like a power function and diverges as the frequency goes to zero. Autoregressive fractionally integrated moving average process ARFIMA(p, d, q)) is a well known class of long memory time series. These models take into account the hyperbolic decay of autocorrelation function. ARFIMA(p, d, q) were independently introduced by Granger and Joyeux10 and Hosking11. This model is a generalization of the ARIMA(p, d, q) model, where d is taken to be an integer. It is defined as )( (3) where and , , ( and B is the backward shift operator ), are the autoregressive and moving-average operators respectively; f and q have no common roots, 1( is fractionally differencing operator defined by the binomial expansion .......,.........2,1,0)11( + G (4) for 0.5, d 0, -1, -2, ......... and is a white noise sequence with finite variance. If &#x-3.3;夀 0, the series exhibit long memory. ARFIMA models have proven useful tools in the analysis of long range dependence processes. Autoregressive fractionally integrated moving average (ARFIMA) models with GARCHerrors have been widely used in time series data analysis. Baillie et al.12 used ARFIMAGARCH models to analyze the inflation of ten different countries. To model daily data on the Swiss 1- month Euromarket interest rate during the period 1986–1989, Hauser and Kunst13 used fractionally integrated models with ARCH errors. Other applications of fractionally integrated models with conditionally hetroscedastic errors can be found in Hauser and Kunst13, Lien and Tse14, Eleck and Markus15 and Koopman et al16. A two stage model building strategy is generally used to fit an ARFIMAGARCH model. In the first step an ARFIMA model is fitted to the given series and then a GARCH model to the residuals of the ARFIMA model. So, it is important to select a correct ARFIMA model in the first stage. The misspecification of ARFIMA model in the first stage will lead to misspecification of the GARCH model in the second stage17. The tests developed by Chen and Deo18, Delgado et al.19, Delgado and Velasco20 and Hidalgo and Kreiss21 all work for long memory time series models. However, they assumed Gaussian or linear processes with conditionally homoscedastic noise processes. Ling and Li22 and Li and Li23 have studied BP type tests for model diagnostics of ARFIMAGARCH models but assuming that the parametric form of GARCH model is known . In the present work, we considered model diagnosis of ARFIMA models with GARCH errors of unknown form. We investigate the performance of Hong;s statistic as a goodness of fit test for ARFIMAGARCH models through simulation study. We also examine the performance of power transformed Hong’s statistic of Chen and Deo in the above settings. The Test StatisticsIn a seminal paper Hong introduced several test statistics that are generalization of the Box and pierce test statistic. These tests are based on the distance between the kernel based spectral density estimator and the spectral density of the noise under the null hypothesis. The standardized form of the Hong’s test statistics with quadratic distance is given by /(2/1 (5) where 1(, /)1)(1(, (.) is the kernel function which is non-negative and symmetric and is the bandwidth that depends on the sample size. Under the assumption of i.i.d errors of the model, when and , Hong showed that the asymptotic null distribution of is standard normal. Hong and Lee24 extended the above result relaxing the assumption of i.i.d errors and established the results assuming the conditional heteroscedastic errors of unknown form. Simulation results of Chen and Deo found that for small samples the distribution of Hong’s test is right skewed, which results to the size distortion of the test. To deal with this problem, Chen and Deo introduced a power transformed version of Hong’s test statistics. The idea behind this transformation is to induce normality. Research Journal of Recent Sciences ______________________________________________________________ ISSN 2277-2502Vol. 2(5), 39-43, May (2013) Res. J. Recent Sci. International Science Congress Association 41 They showed that the appropriate power b to be used such that become approximately normal is given by . (6) Monte Carlo study of Chen and Deo18 showed that for the above choice of b , the distribution of could be well approximated by normal distribution. In our Monte Carlo simulations the above value of b is used.Monte Carlo EvidenceIn this section, we investigate, through simulations, the finite sample performance of the Hong’s and power transformed Hong’s test statistics as goodness-of-fit tests for ARFIMA(p, d, q) models with dependent errors. We use two sample sizes 100 = and300 = . The error distribution is taken to be standard normal. We use the following four kernels for both tests to examine the effect of different kernels. Daniel (DAN): /)sin( p = ),( ¥ -¥ Î Parzen(PAR): otherwise)61(2)6(6 QS: { } 3/5cos(3/5/)3/5sin(5/9( ¥ -¥ Î Bartlett(BAR): otherwise To investigate the effect of , we use three different rates: [ ] ln(, [ ] 2.0 and [ ] 3.0. For 100 = these rates deliver 5, 8, 12 and for 300 = these rates make 8, 10, 17. To examine the size performance of Hong’s and power transformed Hong’s test statistics we consider the following models. M1: 0.1),0.85) 5,GARCH((0.00,0.4,0)ARFIMAM2: 0.1),0.85) 5,GARCH((0.0,0.4,0)ARFIMA(0.5For power performance of both tests the following models are used. M3:0.1),0.85) 5,GARCH((0.00.4,0) 2,ARFIMA(0,. alternative fitting model as ,0) d ARFIMA(0,M4:0.1),0.85) 5,GARCH((0.0.4,0.2)ARFIMA(0,0 alternative fitting model as ,0) d ARFIMA(0,M5: 0.1),0.85) 5,GARCH((0.0,0.4,0.2)ARFIMA(0.5 alternative fitting model as 0) d, ARFIMA(1,. The results for M1 – M5 have been shown in table 1 to 5. These results report the percentage rejection rates at nominal levels of 5% and 10% based on 5000 replications. For small sample size n=100 size distortions occur for both tests but come close to the nominal size for n=300. The power transformed test is more undersized as compared to Hong’s test statistics. The size is better for M2 compared to M1. There is no significant effect of different kernels on the size of both tests. The size becomes better as we increase the bandwidth . This is true for both sample sizes, tests and different kernels. Both tests have good power performance for different sample sizes but the power increases as we increase the sample size from 100 to 300. Different kernels have no significant effect on the power of both tests. Table-1 Rejection rate under the ARFIMA(0.2,0.4,0)-GARCH((0.05, 0.1),0.85) alternative, fitting model ARFIMA(0,d,0) n=100 n=300 p=5 5% 10%p=8 5% 10%p=12 5% 10%p=8 5% 10%p=10 5% 10%p=17 5% 10% H BAR TUK QS DAN B BAR TUK QS DAN 33.12 40.56 33.76 41.26 31.08 38.66 32.04 39.58 26.56 40.36 27.38 41.48 25.08 38.08 26.28 38.68 29.78 37.56 29.26 36.58 27.16 34.38 27.96 35.18 24.26 36.60 23.68 35.88 22.30 33.20 22.86 34.16 27.64 35.04 26.40 33.60 25.62 32.10 25.80 32.62 22.94 34.06 22.02 32.28 21.32 30.88 21.80 31.34 57.12 64.22 56.34 63.62 53.70 61.14 54.62 61.94 51.10 63.18 50.58 62.86 48.28 59.94 49.32 60.66 55.36 62.72 54.26 61.42 51.84 59.42 52.66 60.10 49.74 61.70 48.62 60.46 46.32 58.04 47.56 58.90 51.74 59.34 49.72 57.54 47.54 55.32 48.22 55.90 46.92 58.12 44.60 55.98 42.76 53.92 43.70 54.54 Research Journal of Recent Sciences ______________________________________________________________ ISSN 2277-2502Vol. 2(5), 39-43, May (2013) Res. J. Recent Sci. International Science Congress Association 42 Table-2 Rejection rate under the ARFIMA(0,0.4,0)-GARCH((0.05, 0.1),0.85) n=100 n=300 p=5 5% 10%p=8 5% 10%p=12 5% 10%p=8 5% 10%p=10 5% 10%p=17 5% 10% H BAR TUK QS DAN B BAR TUK QS DAN 6.84 9.42 6.78 9.44 6.84 10.08 6.76 9.92 4.68 9.34 4.58 9.50 4.92 9.72 4.94 9.62 7.10 10.12 7.12 10.10 7.28 10.48 7.16 10.14 5.26 9.68 5.28 9.78 5.34 9.92 5.34 9.80 7.16 10.76 7.26 10.84 7.40 11.10 7.44 11.16 5.26 10.12 5.28 10.18 5.56 10.50 5.56 10.58 3.84 9.02 3.96 9.04 4.10 9.12 4.44 9.20 3.30 6.70 3.38 6.64 3.86 7.82 3.94 7.40 4.54 9.90 4.76 9.60 5.24 9.70 5.34 9.74 3.66 7.46 3.76 7.66 4.60 8.56 4.40 8.14 5.66 10.10 5.94 9.96 6.10 10.10 6.52 10.20 4.80 9.22 5.16 9.40 5.96 10.24 5.80 10.20 Table-3 Rejection rate under the ARFIMA (0,0.4,0.2)-GARCH((0.05, 0.1), 0.85) alternative fitting model ARFIMA(0,d,0) n=100 n=300 p=5 5% 10%p=8 5% 10%p=12 5% 10%p=8 5% 10%p=10 5% 10%p=17 5% 10% H BAR TUK QS DAN B BAR TUK QS DAN 33.88 41.58 34.40 42.10 32.00 39.50 32.54 21.00 27.12 40.34 27.76 42.38 25.74 38.84 26.60 39.52 30.84 37.94 30.18 37.38 28.12 34.94 29.02 21.48 24.56 35.90 24.22 36.48 22.22 34.02 23.28 34.72 28.12 35.20 26.86 33.96 24.80 32.30 25.44 20.66 22.94 32.64 21.72 32.80 21.18 30.80 21.36 31.28 63.10 70.12 62.48 69.34 59.52 66.84 60.28 68.00 56.48 69.08 56.02 68.70 53.42 65.88 55.10 66.90 61.22 68.68 60.00 67.46 57.16 65.02 58.04 65.86 55.14 67.64 54.00 66.58 51.76 63.72 53.16 64.66 56.80 65.06 54.78 62.52 52.26 60.70 52.62 61.38 51.58 63.74 49.12 61.24 46.72 59.44 47.90 60.18 Table-4 Rejection rate under the ARFIMA (0.5,0.4,0)-GARCH((0.05, 0.1),0.85) alternative fitting model ARFIMA(1,d,0) n=100 n=300 p=5 5% 10% p=8 5% 10% p=12 5% 10% p=8 5% 10% p=10 5% 10% p=17 5% 10% H BAR TUK QS DAN B BAR TUK QS DAN 2.36 3.82 2.06 3.48 2.78 4.48 2.62 4.32 1.54 5.20 1.28 3.48 1.88 4.36 1.76 4.18 3.38 5.38 3.22 5.32 4.30 6.18 3.96 6.98 2.40 5.94 2.28 5.08 3.12 5.98 2.98 5.68 4.84 6.62 4.70 6.76 5.84 7.64 5.52 8.78 3.58 7.36 3.56 6.32 4.32 7.30 4.16 6.96 4.74 7.20 4.70 6.98 4.10 8.50 4.76 8.26 3.36 6.84 3.22 6.62 3.52 7.12 3.22 7.82 4.98 8.34 4.88 8.24 4.94 10.08 5.52 9.54 4.16 7.78 4.18 7.80 4.28 8.58 4.32 8.96 5.20 10.42 5.10 10.64 5.20 10.94 5.84 10.62 5.18 9.68 5.30 10.22 5.32 10.26 5.04 10.40 Conclusion We applied the Hong’s and power transformed Hong test of Chen and Deo for goodness-of-fit of autoregressive fractionally integrated moving average models with conditionally hetroscedastic errors of unknown form. Our simulation study reveals that for large sample size (n = 300) both the tests have good size and power performance, when applied to different long memory models with conditionally hetroscedastic errors but for small sample (n = 100) both tests are undersized. The power transformed test is more undersized compared to Hong’s test. This size distortion occurs due to the fact that the mean and variance of these test statistics are based on the asymptotic theory and could be misleading in small samples as reported by Chen and Deo. The above results show that some size correction devices are needed in the above test statistics for ARFIMA models with dependent errors of unknown form. Research Journal of Recent Sciences ______________________________________________________________ ISSN 2277-2502Vol. 2(5), 39-43, May (2013) Res. J. Recent Sci. International Science Congress Association 43 Table-5 Rejection rate under the ARFIMA(0.5,0.4,0.2)-GARCH((0.05, 0.1),0.85) alternative fitting model ARFIMA(1,d,0) n=100 n=300 p=5 5% 10%p=8 5% 10%p=12 5% 10%p=8 5% 10%p=10 5% 10%p=17 5% 10% H BAR TUK QS DAN B BAR TUK QS DAN 15.52 20.72 14.72 20.04 16.24 21.14 16.08 20.78 10.72 20.68 10.16 20.22 11.64 20.92 11.76 20.40 16.58 22.00 16.50 21.88 16.84 22.42 16.82 22.24 12.44 21.32 12.34 21.36 13.08 21.50 13.06 21.34 16.68 22.28 16.74 22.32 16.46 21.78 16.60 22.06 13.32 21.42 13.24 21.42 13.84 20.90 13.80 21.20 46.00 53.10 46.22 53.20 45.42 52.88 45.88 53.14 39.94 52.28 39.94 52.42 39.44 51.82 40.30 51.88 45.66 52.82 45.56 52.94 43.96 51.84 44.96 52.08 39.24 51.98 39.76 51.80 38.44 50.58 39.30 50.92 42.98 50.90 42.06 50.26 40.82 48.56 41.32 48.58 38.14 49.58 37.52 48.90 35.86 47.20 36.56 47.58 References1.Box G.E.P and Pierce D.A., Distribution of the residuals autocorrelations in autoregressive-integrated moving average time series models, JASA, 65(332), 1509-25 (1970)2.Davies N., Triggs C.M. and Newbold P. 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