The Cusum Test for Parameter Change in GARCH(1,1) Models

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1 The Cusum Test for Parameter Change in GARCH(,) Models Sangyeol Lee, Yasuyoshi Tokutsu and Koichi Maekawa 3 Department of Statistics, Seoul National University, Seoul, 5-74, Korea Graduate School of Social Sciences, Hiroshima University, Japan 3 Faculty of Economics, Hiroshima University, Japan Abstract In this paper we consider the problem of testing for a parameter change in GARCH(,) models based on the residual cusum test. It is shown that the limiting distribution of the residual cusum test statistic is the sup of a Brownian bridge. Through a simulation study, it is demonstrated that the proposed test circumvents the drawbacks of Kim, Cho and Lee (000) s cusum test. For illustration, we apply the residual cusum test to the return of yen/dollar exchange rate data. Subtitle: Residual cusum test Keywords: Test for parameter change, GARCH(,) model, residual cusum test, Brownian bridge, weak convergence. JEL Classification Number: C, C4, C5, C Introduction Since Page (955), the problem of testing for a parameter change has been an important issue in statistics. It first started in the quality control context and quickly moved to other fields such as economics, engineering and medicine. So far, a large number of articles have been published in various journals. See, for instance, Brown, Durbin and Evans (975), Wichern, Miller and Hsu (976), Zacks (983), Krishnaiah and Miao (988) and Csörgő and Horváth (997). The change point problem has drawn much attention from many researchers in time series analysis since time series often suffer from structural changes owing to changes of policy and critical social events. It is well known that detecting a change point is a crucial task and ignoring it can lead to a false conclusion. A standard example can be found in Hamilton (994,

2 page 450). For relevant references, we refer to Wichern, Miller and Hsu (976), Picard (985), Inclán and Tiao (994), Mikosch and Stărică (999), Lee and Park (00), Lee et al. (003) and the papers cited in those articles. In this paper, we concentrate ourselves on Inclán and Tiao (994) s cusum test in GARCH(,) models. The GARCH model has long been popular in the financial time series analysis. For a general review, see Gouriéroux (997). Inclán and Tiao (994) s cusum test was originally designed for testing for variance changes and allocating their locations in iid samples. Later, it was demonstrated that the same idea can be extended to a large class of time series models (cf. Lee et all, 003). In fact, Kim, Cho and Lee (000) considered to apply the cusum test to GARCH(,) models taking account of the fact that the variance is a functional of GARCH parameters, and their change can be detected by examing the existence of the variance change. Although this reasoning was correct, it turned out that the cusum test suffers from severe size distortions and low powers. Hence, there was a demand to improve their cusum test. Here, in order to circumvent such drawbacks, we propose to use the cusum test based on the residuals, given as the squares of observations divided by estimated conditional variances. We intend to use residuals since the residual based test conventionally discard correlation effects and enhance the performance of the test. In fact, a significant improvement was observed in our simulation study. The organization of this paper is as follows. In Section, we introduce the residual cusum test, and show that its limiting distribution is the sup of a Brownian bridge. In Section 3, we perform a simulation study to compare our test with Kim, Cho and Lee s test. The result indicates that our method outperforms their cusum test. Then, for illustration, we apply our test to a real data set. Finally, in Section 4, we provide concluding remarks.

3 Residual cusum test Suppose that {y t } satisfies the following difference equation: where ξ t y t = c + ε t ε t = h t ξ t h t = ω + αε t + βh t, () are iid random variables with zero mean and unit variance, c is a real number, and ω, α, β are nonnegative real numbers with α + β <. We assume that E ε t 4+δ < and E ξ t 4+δ < for some δ > 0, and ξ t is independent y s, s < t. Suppose that the observations y,..., y n are given. The objective here is to test the following hypotheses: H 0 : θ = (ω, α, β) remains the same for the whole series. vs. H : not H 0. For a test, one can construct a cusum test based on {y t } as in Inclán and Tiao (994) and Kim, Cho and Lee (000). However, as observed in the simulation study in Section 3, the test is unstable and produces low powers. Thus one has to develop a better test which may not be much affected by the GARCH effects. As a candidate, can naturally consider the cusum test based on {ξt }, say, T n := ( ) n k max ξ nτ t k n n ξ t, () where τ = V ar (ξ t ), since T n is free from the GARCH parameters. In this case, however, one may speculate whether T n can detect any changes since T n itself has no information about the GARCH parameters. But since ξ t are not observable, one should replace ξ t s by the residuals ξ t = (y t ĉ) /ĥ t, which are obtained by estimating the unknown paramters ω, α, β, c. Those estimators play an important role to detect changes in the GARCH parameters in the presence of changes, while the iid property of the true errors remains when there are no changes. From this reasoning, one can anticipate that the residual cusum test should be more stable and produce better powers. 3

4 Now we construct the residual cusum test. In analogy of h t = a + α β j ε t j, we define where a = ĥ t = â + α j=0 q β j ε t j, j=0 ω β, ε t = y t ĉ, and â, ĉ, α, β are the estimators for a, c, α, β with n (â a) = Op (), n (ĉ c) = OP (), n ( α α) = Op (), n ( β β ) = O P (), (3) and q = q n is a sequence of positive integers such that q and q/n 0 as n. Then we have the following result. Theorem. Let where τ = ( n ξ t4 n n T n := n τ max T n where 0 B is a Brownian bridge. k n ( ) n k ξ t n ξ t, (4) ) n ξ t. Then if nρ q 0 for all ρ 0, ), under H 0, d sup 0 u 0 B (u), n, Remark. The condition for q implies that q should be large enough to make a good approximation. A typical example of q is [(log n) ]. Proof. Split ξ t into J t + J t + J 3t, where J t = ξ t h t ĥ t, J t = ε t(c ĉ) ĥ t, J 3t = ε t (c ĉ). We first deal with J t. Write that ( ) J t = ξt h t + ξt = ξt + I t + II t, where I t = h t ĥ t h t ĥ t ξ t and II t = 4 ( ) h t ĥ t h t ĥ t ĥ t ξ t.

5 We first show that Note that n max k n I t h t ĥ t = (a â) + (α α) + ˆα + α := q j=0 q j=0 ( ) k I t = o P (). (5) n q β j ε t j j=0 β j (ε t j ˆε t j) (β j β j ) ε t j + α j=q+ β j ε t j 5 I it / ( ) ξt /h t. (6) i= In order to show (5) with I t replaced by I 4t, it suffices to verify that for any λ > 0, [ q ] l n := P β j β j Λnj > λ = o (), n, (7) j= where Λ nj = { ε t j ξ [ t ε t j ξt max E n k n h t h t ]}, which is O P () owing to the invariance principle for the strong mixing process: see Carrasco and Chen (00) and Theorem.7 of Peligrad (986). Observe that for any M > 0, ( M ) ( l n P β j β j Λnj > λ + P j= := l n + l n, j=m+ ) β j β j Λnj > λ and for any η > 0, ( n ) l n P β β > η ( n ε + P γ j t j ξt n h t j=m+ [ ε t j ξt E h t ] ) > λ, 5

6 where we have used (3), and γ is a number in (0, ). By choosing sufficiently large M and η, and utilizing (3) and Markov s inequality, we can make l n arbitrarily small. Since l n = o () for each M, we can see that (7) holds. This proves (5) with I t replaced by I 4t. In a similar fashion, one can readily show that (5) with I t replaced by 3 I it is o P (). Then (5) follows from the fact i= [ E n ] n I 5t = O ( nβ q) = o (). Now, we prove n max k n II t ( ) n k II t = o P (). (8) n In view of (6), we have that It is easy to see that II t C 5 I it/ĥ t for some C > 0. n i= n 4 I it/ĥ t = o P (), of which detailed proofs are omitted for brevity. Since n n i= I 5t = O P ( nβ q ), (8) follows. Combining (5) and (8), we have ( ) k max J t J t max ξ n k n n n t k n ( ) k n ξ t = o P (). (9) Meanwhile, in a similar fashion to the above, one can readily show that ( ) k max J n k n it J it = o P (), i =, 3, (0) n of which detailed proof are omitted for brevity. Also, we can easily verify that τ P τ. Hence, it follows from (9) and (0) that T n T n = o P (), and the theorem follows from Donsker s invariance principle. This completes the proof. 6

7 3 Empirical study 3. Simulation study In this section, we evaluate the performance of the test statistic ˆT n through a simulation study. In particular, ˆTn is compared with Kim, Cho and Lee (000) s test statistic B T (Ĉ). In this simulation we perform a test at nominal level The empirical sizes and power are calculated as the rejection number of the null hypothesis out of 000 repetitions. In order to see the performance of ˆT n, we consider the model y t = h t ξ t h t = ω + αyt + βh t, where y 0 is assumed to be 0 and {ξ t } are iid standard nomal random variables. Now we consider the problem of test the following hypotheses: H 0 : θ = (ω, α, β) are constant during the time t =,, n. vs. H : θ changes to θ = (ω, α, β ) at n/. Here we evaluate ˆT n with sample sizes n = 500, 800, 000 and use q = [ (log n) ]. The empirical sizes and powers are summarized in Tables -3. The figures in the parentheses denote the sizes and powers of Kim, Cho and Lee s test. As we see in the tables, we can see that our test has no size distortions. In particular, the test is still stable even for the case that α + β is close to (see Tables and 3). As mentioned earlier, this is because ξ t behaves asymptoticall like iid ξt, unaffected by the GARCH parameters. Meanwhile, we can see that the powers are more than 0.9 at the sample size 000. Generally, the cusum test in GARCH models needs a much larger sample size to make accurate inference compared to iid samples. It seems that the GARCH data with volatility makes it harder to identify small changes. Compared to ours, Kim, Cho and Lee s test shows severe size distortions and much lower powers. Overall, our simulation study strongly supports the validity of the residual cusum test. ** Tables -3 here ** 7

8 3. Real data analysis In this section, we intend to demonstrate the validity of our method in actual practice. For this task, we analyze the return of yen/dollar exchange rate data from Jan 5, 998 to Jan 7, 003. Recall that the D k plot, defined in Inclán and Tiao (994), is a useful tool to detect multiple changes. In our case, we only detected one change point on Sep 8, 999 (see the vertical line in Figure ). The data in the first period from Jan 5, 998 to Sep 8, 999 appears to follow the model: y t = ε t ε t = h t ξ t h t = ε t h t, and the data in the second period follows the model y t = ε t ε t = h t ξ t h t = ε t h t. Theis result indicates that the parameters experience significant changes. Meanwhile, we ignored the change on purpose and fitted the GARCH(,) model to the whole observations. Consequently, we obtained an IGARCH(,) model as follows: y t = ε t ε t = h t ξ t h t = ε t h t. The result vividly shows that ignoring changes can lead to a false conclusion in statistical inference. This misspecification result coincides with the one reported by Maekawa et al. (003). ** Figure here ** 8

9 4 Concluding remarks In this paper, we proposed a residual based cusum test based and derived that the test statistic is asymptotically distributed as the sup of a Brownian bridge under regularity conditions. In the proof, we used the invariance principle result for beta (strong) mixing processes, which was possible owing to the results of Carrasco and Chen (00) and Pelrigrad (986). The proof was of an independent interest since the mixingale approach adopted by Kim, Cho and Lee (000) is not easy to apply, and the proof would be much lengthier without the beta mixing condition. In fact, the present paper was motivated to circumvant the drawbacks of the cusum test proposed by Kim, Cho and Lee. The idea in developing our test is explained in Section. As seen in Subsection 3., the simulation result appeared to be remarkably favorable to our test: the sizes and powers are greatly improved compared to the original cusum test. This indicates that the residual cusum test is highly trustful. In Subsection 3., the test was applied to the yen/dollar exchange rate data and detected one change point. It was also seen that ignoring the change leads to a wrong conclusion. Overall, we believe that our test consititutes a functional tool for testing for a paramter change in GARCH(,) models. We anticipate that the residual cusum test can be extended to other type of GARCH models. We leave the task of extension as our future study. Acknowledgements. The second author wishes to thank Professor Y. Suzuki for helpful discussion. The first author acknowledges that this work was supported by KOSEF in part through the Statistical Reserach Center for Complex System at Seoul National University. Also, the third author acknowledges the support from Grant-in-Aid for Scientific Research by the Ministry of Education, Science and Technology, Japan. 9

10 References [] Carrasco, M and Chen, X. (00). Mixing and moment properties of various GARCH and stochastic volatility models. Econometric Theory, 8, [] Csörgő, M. and Horváth, L. (997) Limit Theorems in Change-Point Analysis. Jhon Wiley & Sons Ltd, West Sussex, England. [3] Inclán, C. and Tiao, G. C. (994). Use of cumulative sums of squares for retrospective detection of changes of variances. J. Amer. Statist. Assoc. 89, [4] Kim, S., Cho, S. and Lee, S. (000). On the cusum test for parameter changes in GARCH(,) models. Commun. in Statist. Theory & Meth. 9, [5] Krishnaiah, P. R. and Miao, B. Q. (988). Review about estimation of change points. In Handbook of Statistics Vol. 7. P. R. Krishnaiah and C. P. Rao eds Elsevier, New York. [6] Gouriéroux, C. (997). ARCH Model and Financial Application. Springer, New York. [7] Hamilton, J. D. (994). Time Series Analysis. Princeton University Press, New Jersey. [8] Lee, S., Ha, J., Na, O. and Na, S. (003). The Cusum Test for Parameter Change in Time Series Models. Scand. J. Statist. 30, [9] Lee, S. and Park, S. (00). The cusum of squares test for scale changes in infinite order moving average processes. Scand. J. Statist. 8, [0] Maekawa, K., Lee, S. and Tokutsu, Y. (003). Structural change and spurious volatility persistence. Submitted for publication. [] Mikosch, T. and Stărică, C. (999). Change of structure in financial time series, long range dependence and the GARCH model. Technical report. University of Groningen. 0

11 [] Page, E. S. (955). A test for change in a parameter occurring at an unknown point. Biometrika 4, [3] Picard, D. (985). Testing and estimating change-points in time series. Adv. Appl. Prob. 7, [4] Peligrad, M. (986). Recent advances in the central limit theorem and its weak invariance principle for mixing sequences of random varaibles. In Dependence in Probability and Statistics. Eberlein, E. and Taqqu, M. S. eds. 93-3, Birkhäuser, Boston. [5] Wichern, D. W., Miller, R. B. and Hsu, D. A. (976). Changes of variance in first-order autoregressive time series models - with an application. Appl. Statist. 5, [6] Zacks, S. (983). Survey of classical and Bayesian approaches to the change-point problem : fixed sample and sequential procedures of testing and estimation. In Recent Advances in Statistics. M. H. Rivzi et al. eds , Academic Press, New York.

12 Table. θ = (0.5, 0., 0.) θ = (ω, α, β ) n = 500 n = 800 n = 000 n = 500 size 0.06 (0.0) (0.05) (0.035) (0.039) (3.0, 0., 0.) (0.077) (0.03) 0.99 (0.009) (0.5, 0.6, 0.) (0.44) (0.349) 0.90 (0.43) (0.5, 0., 0.6) (0.) (0.69) 0.90 (0.38) Table. θ = (0., 0.4, 0.4) θ = (ω, α, β ) n = 500 n = 800 n = 000 n = 500 size (0.009) (0.004) (0.005) 0.04 (0.00) (0.4, 0.4, 0.4) (0.98) (0.387) (0.449) (0., 0., 0.4) 0.56 (0.57) (0.493) 0.98 (0.646) Table 3. θ = (0., 0., 0.7) θ = (ω, α, β ) n = 500 n = 800 n = 000 n = 500 size 0.0 (0.00) 0.03 (0.003) 0.03 (0.008) 0.04 (0.0) (0.4, 0., 0.7) 0.9 (0.73) 0.7 (0.8) 0.99 (0.7) (0., 0., 0.) 0.66 (0.07) 0.97 (0.94) (0.33) Figure : Plot of Foreign Exchange rate data //5 998/5/5 998/9/5 999//5 999/5/5 999/9/5 000//5 000/5/5 000/9/5 00//5 00/5/5 00/9/5 00//5 00/5/5 00/9/5 003//5

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