Tests for Cointegration, Cobreaking and Cotrending in a System of Trending Variables

Transcription

1 Tests for Cointegration, Cobreaking and Cotrending in a System of Trending Variables Josep Lluís Carrion-i-Silvestre University of Barcelona Dukpa Kim y Korea University May 4, 28 Abstract We consider a set of variables with a deterministic trend and a stochastic trend. The deterministic trend is allowed to have changes in the intercept and slope. We develop three tests, a cointegration test, a joint test for cointegration and cobreaking, and a joint test for cointegration and cotrending. Our analysis in this paper is complementary to Carrion-i-Silvestre and Kim (27), which deals with deterministic trends with intecept shifts only. JEL Classi cation Number: C22 Keywords: cointegration, cobreaking, cotrending, multiple structural breaks AQR-IREA research group. Department of Econometrics, Statistics and Applied Economics, Faculty of Economics and Business, University of Barcelona, Av. Diagonal, 69, 834 Barcelona (carrion@ub.edu) y Corresponding author, Department of Economics, Korea University, 45 Anam-ro, Seongbuk-gu, Seoul 284 Korea (dukpakim@korea ac.kr)

2 Introduction The econometrics literature on unit root and cointegration tests is extremely mature. However, it is relatively recently that variables with both a stochastic and a deterministic trend with breaks draw attention. For the examples of unit root tests, see Carrion-i- Silvestre, Kim, and Perron (29) Harris, Harvey, Leybourne, and Taylor (29) Harvey, Leybourne, and Taylor (22, 23) and Kim and Perron (29). Choi (25) also provides a more extensive review of the unit root literature. For the examples of cointegration tests in the presence of breaks or structural break tests in cointegration models, see Johansen, Mosconi, and Nielsen (2) Saikkonen and Lütkepohl (2) Lütkepohl, Saikkonen, and Trenkler (24) Trenkler, Saikkonen, and Lütkepohl (27) Harris, Leybourne, and Taylor (26) Carrion-i-Silvestre and Sansó (26) Arai and Kurozumi (27) Qu (27) Kejriwal and Perron (2) and Carrion-i-Silvestre and Kim (27). This paper is closely related to Carrion-i-Silvestre and Kim (27). The main idea therein is to view the breaks in the cointegration equation as resulting from the breaks in each variable. This is di erent from a more traditional view that breaks are exogenously given to cointegration equations and the variables themselves do not have breaks. This kind of di erence in modelling the origin of breaks in the cointegration equation might matter less at least asymptotically if breaks are limited to intercepts. However, when slopes are changing, the origin of breaks can even a ect the limiting distributions of test statistics. The analysis in this paper takes the same view as Carrion-i-Silvestre and Kim (27). However, this paper deals with a model with both intercept shifts and slope changes while Carrion-i-Silvestre and Kim (27) deals with models with intercept shifts only. The existence of a slope change indeed makes it possible to consistently estimate break dates even in the presence of a stochastic trend, which in turn leads to a testing strategy di erent from Carrion-i-Silvestre and Kim (27). We develop three statistics, namely a robust cointegration test, a joint test for cointegration and cobreaking, and a joint test for cointegration and cotrending. We devise our test statistics as (quasi) log-likelihood ratio tests and derive their limiting distributions. The adequateness of our asymptotics in nite samples is shown via Monte Carlo simulation experiments. This paper is organized as follows. Section 2 introduces the models and tests and provides the asymptotic results. Section 3 o ers Monte Carlo experiment results. Section 4 concludes. The appendix collects some technical derivations.

3 2 Model and Tests The observed variables y t and x t are assumed to be generated by y t x t!! = y d x t + y t x t!, () where y t is a scalar random variable, x t is a p x random vector, d t is a p d vector of deterministic functions in time, y and x are coe cient matrices, and yt and x t are stochastic components. The deterministic component d t is speci ed as d t = [DU t (T ) : : : DU t (T m ) B t (T ) : : : B t (T m )], where DU t (T j ) = for t > T j and elsewhere, B t (T j ) = (t T j )DU t (T j ) and T =. DU t (T j ) stands for a shift in the intercept and B t (T j ) stands for a change in the slope of the linear trend. Models with intercept changes only are analyzed in Carrion-i-Silvestre and Kim (27). In fact, our strategy to devise test statistics and to derive their limiting distributions extremely resembles theirs. It is assumed that yt and x t are integrated of order one, which means that both y t and x t have a stochastic trend in addition to the deterministic trend represented by d t. We are interested in linear combinations of (y t x t) that cancel out the stochastic trends and/or deterministic trends existing in y t and x t. Let ( ) be such a linear combination. The equation in () can be written as y t = x t + d t + v t, for t = : : : T, (2) where = y x and v t = y t x t. For, we introduce more parameters = ( : : : m : : : m ) where j is the intercept change at T j and j is the slope change. a y and x are similarly de ned by ( y j y j ) and (x j x j ), respectively. denotes the rst di erence operator. Assume that v t = " t " t, with v = " =. Following Carrion-i-Silvestre and Kim (27), we will use the following de nitions throughout the paper:. (y t x t) is cointegrated (CI) if and only if =. 2. (y t x t) is cobreaking (CB) if and only if = = m = = = m =. 2

4 3. (y t x t) is cotrending (CT) if and only if it is CB and =. 2. Known Break Dates We rst consider testing for the null hypothesis of CI ( = ) against the alternative of no CI ( = < ). We devise our test statistic from the following simple case. Let u x t = x t, and assume (" t u x t ) IIN!! 2 ". (3) xx The normality assumption will be relaxed when the limiting distribution is derived for our test. On the other hand, we deal with only the case in which u x t and " t are independent. See Carrion-i-Silvestre and Kim (27) for the case of endogenous regressors. The model is expressed in matrix notation as y = D + X + =2 ", (4) where y = (y : : : y T ), D = (d : : : d T ), X = (x : : : x T ), " = (" : : : " T ), and =2 = C A. Since and are irrelevant for CI, they can be concentrated out from the likelihood function. Also, we set 2 " = for simplicity on the exposition, although the nal test statistic will be adjusted by a long-run variance estimate. The concentrated log likelihood function is given by L T ( ) = const: 2 y =2 M =2 y, (5) where = ( : : : m ) is a vector of break fractions where each i satis es T i = [ i T ], < < m, and M = I T Z (Z Z ) Z with Z = [ =2 D =2 X]. When we extend our analysis to the case of unknown break dates, the true break dates and generic break dates must be di erentiated. Thus, we will use and for the true dates and generic ones. 3

5 Our test statistic for CI is the likelihood ratio given by Q r 2 L T ( ) L T ( ). (6) As explained in detail by Carrion-i-Silvestre and Kim (27), this test is invariant to the CB and CT. In other words, this test provides a way to test for CI irrespective of CB and CT. Now, we make the following assumptions to derive the asymptotic distribution of our test statistics. Assumption T i = [ i T ], and i j a > for all j < i m + where = and m+ =. Assumption 2 " t = P i= c i t i with P i= i kc ik < C and t i:i:d:( ), u x t = P i= G i x t i with P i= i kg ik < C, G = P i= G i is of full column rank, and x t i:i:d:( I px ), and t and x t are independent. Assumption 3 x is of full column rank so that ( x x ) exists. These assumptions are identical to those in Carrion-i-Silvestre and Kim (27). The next theorem states the asymptotic distribution of the Q r statistic. Theorem Let = =T, = =T and!" = P i= c i. Then, under Assumptions 2, we have Q r ) ' r ( )! 2 " where = ( ) 2 ( m + m + p x ) + 2 ( m + m + p x ), ( ) = 2 2 ( p q n) = V Z V (s) dv (s) 2 Z V (s)2 ds, Z Z Z Q (s) dv (s) Q (s) Q (s) ds (s) = V (s) Z s V (s) = V (s) + Z s e (s V (r)dr, Q (s) = Q (sjp q n) Z s r) V (r)dr, e (s r) Q (rjp q n)dr, Q (sjp q n) = (du(s ) : : : du(s p ) b(s ) : : : b(s q ) W n (s) ), du(r j ) = (r > j ), b(r j ) = (r j )(r > j ), 4 Q (s) dv (s),

6 V (r) and W (r) are independent standard Wiener processes of dimensions and p x respectively, and W n (r) collects the rst n elements of W (r). Note that ' r ( ) stands for the null distribution when = and stands for the alternative distribution when >. In practice, the value of has to be decided by the researcher. Following the custom in the econometrics literature, we recommend to set the value of so that the local asymptotic power curve is tangent to the theoretical power envelop with 5% power. Table reports the suggested values for (= =T ). With this choice of, selected percentiles of '( ) are reported in Table 2. 2 Now, we develop joint tests for CI and CB and for CI and CT. Now that is of interest, we work with the unconcentrated log likelihood L T ( j ). Let Q cb be the test statistic for the joint null of CI and CB and Q ct be the joint null of CI and CT. Then, we propose where B Q cb = max L T ( j ) s:t: R cb = B Q ct = max L T ( j ) s:t: R ct= max L T ( j C ) A, max L T ( j C ) A, R cb = ( : : : m : : : m ), R ct = ( : : : m : : : m ). The asymptotic null distributions for the Q cb and Q ct statistics are reported in the following theorem. Theorem 2 Let =, = =T and!" = P i= c i. Suppose Assumptions 3 hold. Let ( ) and 2 ( p q n) be the same as in Theorem. (i) If CB holds and p x m, we have Q cb ) ' cb ( )! 2 " ( ) 2 ( m + p x m) + 2 ( m + m + p x ). In Tables and 2, we report the results for the case of p x = and 2. The extended tables for p x = 3 : : : 5 are available upon request from the authors. 2 To generate these critical values, a Wiener process is approximated with 2, steps and the number of simulation repetitions is roughly 2,. 5

7 (ii) If CT holds and p x m +, we have Q ct ) ' ct ( )! 2 " ( ) 2 ( m + p x m ) + 2 ( m + m + p x ). It is worth mentioning that when there are more breaks than stochastic regressors (p x m for CB and p x m + for CT), the limiting distributions depend on nuisance parameters, and thus cannot be tabulated. Relevant percentiles of ' cb ( ) can be found in Tables 3 and Unknown Break Dates Now we consider using estimated break dates instead of the true ones. When the slopes are changing, the break dates can be estimated regardless of CI. More speci cally, the break dates can be estimated by minimizing the sum of squared residuals from a regression of q t = (y t x t) on d t where d t is de ned with a generic. Let q = [q 2 : : : q T ] and D = [d 2 : : : d T ]. Then the break date estimator is de ned to be e = (e : : : e m ) = arg min tr q (I D (D D ) D )q. The Q r, Q cb and Q ct tests can be constructed at the estimated break dates, e. Under Assumption 3, e is known to be consistent at rate T. Now, our test statistics are given by eq r = 2 max L T ( j e) max B eq cb = max L T ( j ) s:t: R cb = B eq ct = max L T ( j ) s:t: R ct= max max L T ( j e) L T ( j C e) A L T ( j C e) A. The next theorem states that these test statistics have the asymptotic distributions that have been derived for the known break date case. Theorem 3 Let =, = =T and!" = P i= c i. Suppose that Assumptions 3 hold and jje jj = O p (T ). (i)! 2 " eq r ) '( ), 6

8 where '( ) is as de ned in Theorem. (ii) If CB holds and p x m,! 2 " eq cb ) ' cb ( ), where ' cb ( ) is as de ned in Theorem 2. (iii) If CT holds and p x m +,! 2 " eq ct ) ' ct ( ), where ' ct ( ) is as de ned in Theorem 2. 3 Simulation The data generation process (DGP) is identical to (). Much of the simulation design resembles the one employed in Carrion-i-Silvestre and Kim (27) to make the comparison easier. We let x be a matrix of 2, and be a p x vector of =p x s. y is determined in such a way that can have a desired value from the relationship = y x. = in all cases as this does not a ect the results. x t is an accumulation of independent standard normal errors. yt is a sum of x t and v t = P t j= (" j " j ) with " t being an independent standard normal variate and v = " =. The following 6 DGPs are simulated with p x = 2. The number of simulation repetitions is, in all cases. DGP (CI + CT) =, i = j =, =. DGP 2 (CI + CB) =, i = j =, = :3. DGP 3 (CI + No CB) =, i = j = :5 2:5 in steps of :5 and 3 2 in steps of 3, = :3. DGP 4 (No CI + CT) =, :25, :5 and :5 :25 in steps of :5, i = j =, =. DGP 5 (No CI + CB) =, :25, :5 and :5 :25 in steps of :5, i = j =, = :3. DGP 6 (No CI + No CB) = :5 :25 in steps of :5 with i = j = ( ), and = with i = j = 3 2 in steps of 3, = :3. For all tests, the long-run variance is estimated by a heteroskedasticity and autocorrelation consistent covariance estimator with the quadratic spectral kernel, for which the 7

9 bandwidth parameter is selected using Andrews s (99) data dependent method with an AR() approximation. Finite Sample Size Table 5 reports the nite sample sizes for nominal 5% signi cance. In the simulation, the break dates are estimated by minimizing the sum of squared residuals from a regression of rst di erences of (y t x t) on intercept shifts and impulse dummies. Overall, the actual sizes are quite close to the nominal level, which suggests that the asymptotic critical values o er a good approximation. Power Comparison Table 6 reports size adjusted powers. We focus on the tests using estimated break dates. For comparison, also simulated are the Gregory and Hansen (996) test (GH), the Shin (994) test, and the Kejriwal and Perron (2) test (KP). The nite sample critical values are obtained under DGP for all tests but the GH test. The critical value for the GH test is obtained under DGP4 because this test takes the null of no CI. As a result, the rejection probabilities are 5% exactly for all tests but the GH test. The GH test rejects the null with %, which is exactly as it is intended. With DGP 2, CI and CB still hold but CT does not. Hence, the Q e ct () test correctly detects the breakdown of CT, while all other tests show the same rejection probabilities as in DGP. With DGP 3, both Q e cb () and Q e ct () reject the joint null with large probabilities since CB does not hold. Q e ct () appears more powerful than Q e cb (). On the other hand, Q e r () still remains at 5% because CI still holds. This di erence in rejection probabilities shows that our tests are working exactly in the way they are designed. Note that our tests are more powerful than the Shin test and the KP test. Also noteworthy is the fact that the power of the Shin test is decreasing as gets larger, which is the well-documented non-monotonic power problem. With DGP 4, not only the Q e cb () and Q e ct () tests but also the Q e r () test rejects with large probabilities because CI does not hold. Again note that our tests are performing better than the Shin test and the KP test. The GH test is supposed to control the size, but it apparently fails. With DGP 5, the rejection rates for Q e ct () become very close to one due to the break down of CT. With DGP 6, the results are similar to the ones for DGP 5. 4 Conclusion We consider a system of trending variables and develop three statistics, a CI test, a joint test for CI and CB, and a joint test for CI and CT. Our analysis in this paper is complementary to Carrion-i-Silvestre and Kim (27), with the notable di erence that we allow for slope changes. When slopes are changing, the break dates are estimated 8

10 consistently regardless of CI. This feature enables us to implement a test procedure that is di erent from Carrion-i-Silvestre and Kim (27). Acknowledgement Josep Lluís Carrion-i-Silvestre acknowledges nancial support from the Spanish Ministerio de Ciencia e Innovación under grant ECO C3--P. Dukpa Kim acknowledges nancial support from the National Research Foundation of Korea Grant funded by the Korean Government (NRF-24SA5A88632). Dukpa Kim also acknowledges that this research is supported by the Korea University Grant (K6344). <Appendix> Proof of Theorem : We only need to nd so that Z [T r] converges properly. Let =! T =2 G =2 x T =2 G =2, where = diagfi m+ T I m+ g. It follows that Z [T r] = DU [T r] (T ). DU [T r] (T m ) T =2 G =2 x [T r] T =2 G =2 x d [T r] C A = DU [T r] (T ). DU [T r] (T m ) T =2 G =2 x [T r] C A ) Q (r). Then, the rest of the proof is the same as the proof of Theorem in Carrion-i-Silvestre and Kim (27). Proof of Theorem 2: (27). See the proof for Theorem 2 in Carrion-i-Silvestre and Kim Proof of Theorem 3: The result follows from a similar argument to the proof of Theorem 4(i) in Carrion-i-Silvestre and Kim (27). Because we have the B t (T j ) terms in addition to the DU t (T j ) terms, let D e = [ d e : : : d e T ]. dt e is the same as d t except that DU t (Tj ) is replaced by DU t ( T e j ) and B t (Tj ) by B t ( T e j )+( T e j Tj )DU t (T j ) for j = : : : m, where ( T e : : : T e m ) = T e = T (e : : : e m ). Then, the columns of D D e are DU(Tj ) DU( T e j ) and B t (Tj ) B( T e j ) ( T e j Tj )DU( T e j ) for j = : : : m, and each of these columns has non-zero elements only between min(tj T e j ) + and max(tj T e j ), which are bounded. The rest of the proof is analogous to the proof of Theorem 4(i) in Carrion-i-Silvestre and Kim (27) with the simpli cation that the same break date estimates are used for both the null and alternative regressions. 9

11 References [] Andrews, D.W.K., 99. Heteroskedasticity and autocorrelation consistent covariance matrix estimation. Econometrica 59, [2] Arai, Y. and E. Kurozumi (27) Testing for the null hypothesis of cointegration with a structural break, Econometric Reviews 26, [3] Carrion-i-Silvestre, J. L., D. Kim and P. Perron (29) GLS-based unit root tests with multiple structural breaks both under the null and the alternative hypotheses, Econometric Theory 25, [4] Carrion-i-Silvestre, J. L. and D. Kim (27) Quasi-Likelihood Ratio Tests for Cointegration, Cobreaking and Cotrending, Econometric Reviews, forthcoming. [5] Carrion-i-Silvestre, J. L. and A. Sansó (26) Testing the null of cointegration with structural breaks, Oxford Bulletin of Economics and Statistics 68, [6] Choi, I. (25) Almost All About Unit Roots, Themes in Modern Econometrics, Cambridge University Press. [7] Gregory, A. W. and B. E. Hansen (996) Residual-based tests for cointegration in models with regime shifts, Journal of Econometrics 7, [8] Harris, D., D. Harvey, S. J. Leybourne, and A. M. Robert Taylor, (29) Testing for a unit root in the presence of a possible break in trend, Econometric Theory 25, [9] Harris, D., S. J. Leybourne and A. M. Robert Taylor (26) Tests of the cointegration rank in VAR models in the presence of a possible break in trend at an unknown point, Journal of Econometrics 92, [] Harvey, D., S. J. Leybourne and A. M. Robert Taylor (22) Unit root testing under a local break in trend, Journal of Econometrics 67, [] Harvey, D., S. J. Leybourne and A. M. Robert Taylor (23) Testing for unit root in the possible presence of multiple trend breaks using minimum Dickey-Fuller statistics, Journal of Econometrics 77, [2] Johansen, S., R. Mosconi, and B. Nielsen (2) Cointegration analysis in the presence of structural breaks in the deterministic trend, Econometrics Journal 3, [3] Kejriwal, M. and P. Perron (2) Testing for multiple structural changes in cointegrated regression models, Journal of Business and Economic Statistics 28, [4] Kim, D. and P. Perron (29) Unit root tests allowing for a break in the trend function at an unknown time under both the null and alternative hypotheses, Journal of Econometrics 48, -3. [5] Lütkepohl, H., P. Saikkonen, and C. Trenkler (24) Testing for the cointegrating rank of a VAR process with level shift at unknown time, Econometrica 72,

12 [6] Qu, Z. (27) Searching for cointegration in a dynamic system, Econometrics Journal,, [7] Saikkonen, P. and H. Lütkepohl (2) Testing for the cointegrating rank of a VAR process with structural shifts, Journal of Business and Economic Statistics 8, [8] Shin, Y. (994) A residual-based test of the null of cointegration against the alternative of no cointegration, Econometric Theory, 9-5. [9] Trenkler, C., P. Saikkonen, and H. Lütkepohl (27) Testing for the cointegrating rank of a VAR process with level shift and trend break, Journal of Time Series Analysis 29, Table. Suggested Values of, Local to Unity Parameter m 2 3 p x = 3:4 8: 22:6 27:7 p x = 2 4:9 9:4 23:7 28:9

13 Table 2. Upper Percentiles for ' r ( ) (i) One Break, m = p x = % 5% % :2 9:8 7:92 5:35 :3 9:4 8:36 6: :4 9:49 8:49 6:3 :5 9:52 8:48 6:38 :6 9:52 8:49 6:2 :7 9:38 8:3 6:3 :8 9: 7:9 5:56 p x = 2 % 5% % :2 :24 9: 6:78 :3 :38 9:3 6:76 :4 :48 9:46 7:4 :5 :5 9:45 7:3 :6 :46 9:46 6:99 :7 :43 9:29 6:95 :8 :6 9:3 6:67 (ii) Two Breaks, m = 2 p x = % 5% % (:2 :4) 2:54 :49 9: (:2 :5) 2:8 :73 9:44 (:2 :6) 2:87 :78 9:55 (:2 :7) 2:77 :67 9:26 (:2 :8) 2:59 :42 8:94 (:3 :5) 2:75 :66 9:28 (:3 :6) 2:96 :98 9:79 (:3 :7) 3: 2: 9:87 (:3 :8) 2:83 :72 9:29 (:4 :6) 2:8 :7 9:46 (:4 :7) 3: 2:3 9:88 (:4 :8) 2:89 :79 9:45 (:5 :7) 2:79 :7 9:46 (:5 :8) 2:74 :66 9:26 (:6 :8) 2:58 :44 9:3 p x = 2 % 5% % (:2 :4) 3:54 2:49 :5 (:2 :5) 3:68 2:6 :23 (:2 :6) 3:65 2:63 :24 (:2 :7) 3:69 2:62 :8 (:2 :8) 3:43 2:36 :3 (:3 :5) 3:66 2:55 :3 (:3 :6) 3:77 2:77 :48 (:3 :7) 3:8 2:72 :57 (:3 :8) 3:63 2:52 :24 (:4 :6) 3:7 2:69 :48 (:4 :7) 3:77 2:8 :64 (:4 :8) 3:62 2:58 :37 (:5 :7) 3:68 2:59 :22 (:5 :8) 3:62 2:56 :3 (:6 :8) 3:5 2:39 :2 (iii) Three Breaks, m = 3 p x = % 5% % (:2 :4 :6) 6:43 5:33 2:95 (:2 :4 :7) 6:54 5:47 3:22 (:2 :4 :8) 6:44 5:35 3:6 (:2 :5 :7) 6:5 5:44 3: (:2 :5 :8) 6:47 5:4 3:7 (:2 :6 :8) 6:39 5:28 2:95 (:3 :5 :7) 6:57 5:55 3:34 (:3 :5 :8) 6:55 5:48 3:26 (:3 :6 :8) 6:59 5:53 3:9 (:4 :6 :8) 6:43 5:34 2:89 p x = 2 % 5% % (:2 :4 :6) 7:37 6:27 3:92 (:2 :4 :7) 7:4 6:33 4:2 (:2 :4 :8) 7:28 6:23 3:63 (:2 :5 :7) 7:37 6:3 4: (:2 :5 :8) 7:38 6:33 3:98 (:2 :6 :8) 7:29 6:7 3:83 (:3 :5 :7) 7:37 6:25 4:7 (:3 :5 :8) 7:4 6:3 3:95 (:3 :6 :8) 7:4 6:36 4: (:4 :6 :8) 7:3 6:22 3:8 2

14 Table 3. Upper Percentiles for ' cb ( ) p x = % 5% % :2 6:4 4:47 :85 :3 6:36 4:83 :52 :4 6:34 4:85 :39 :5 6:34 4:85 :3 :6 6:46 4:95 :45 :7 6:3 4:72 :39 :8 6:4 4:5 :3 p x = 2 % 5% % :2 7:6 5:44 :83 :3 7:22 5:67 2:4 :4 7:4 5:85 2:38 :5 7:39 5:89 2:38 :6 7:43 6:2 2:67 :7 7:3 5:89 2:29 :8 7:3 5:5 :89 Table 4. Upper Percentiles for ' ct ( ) p x = 2 % 5% % :2 5:66 3:94 :24 :3 5:9 4:25 :44 :4 5:83 4:7 :47 :5 5:88 4: :35 :6 5:94 4:29 :55 :7 5:84 4:4 :47 :8 5:77 3:95 :39 Table 5. Finite Sample Sizes, DGP (CI+CT) T e Qr (m) e Qcb (m) e Qct (m) Q r (m) Q cb (m) Q ct (m) (:5) 2 :49 :58 :65 :48 :59 :67 36 :55 :58 :49 :56 :48 :5 (:3, :7) 2 :43 :6 :43 :53 36 :57 :55 :58 :55 (:2, :5, :8) 2 :35 :32 36 :42 :44 3

15 Table 6. Power Comparison, Null Rejection Probabilities e Qr () e Qcb () e Qct () Shin KP GH DGP : : : :5 :5 :5 :5 :5 : DGP 2 : : :3 :5 :5 : :5 :5 : DGP 3 : :5 :3 :5 :3 : :2 : : : :5 :4 : :35 :26 : :5 :5 :73 : :57 :55 : 2: :5 :92 : :73 :8 : 2:5 :5 :99 : :83 :94 : 3: :5 : : :88 :99 : 6: :5 : : :85 : : 9: :5 : : :64 : : 2: :5 : : :4 : : DGP 4 : : : : : : :42 :78 :5 :25 : : : :2 :72 :48 :5 : : : :4 :78 :99 :5 :8 :9 :3 :9 :7 : : :7 :2 :39 :9 :6 : :5 :32 :38 :66 :35 :29 : :2 :48 :56 :8 :48 :43 : :25 :63 :7 :88 :56 :56 : DGP 5 : : :3 : : : :42 :78 :5 :25 : : : :2 :72 :48 :5 : : : :4 :78 :99 :5 :8 :9 : :9 :7 : : :7 :2 : :9 :6 : :5 :32 :38 : :35 :29 : :2 :48 :56 : :48 :43 : :25 :63 :7 : :56 :56 : DGP 6 :5 :5 :3 :8 :7 : :6 :2 : : : :7 :5 : :4 :36 : :5 :5 :32 :77 : :58 :59 : :2 2: :49 :92 : :68 :72 : :25 2:5 :63 :97 : :72 :8 : : 3: : : : :44 :78 :4 6: : : : :43 :8 :2 9: : : : :45 :86 :32 2: : : : :48 :92 :59 5: : : : :5 :97 :8 4