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1-1-2013

Size and power of tests with non-normal GARCH error distributions

Chaiwat Kosapattarapim University of Wollongong, [email protected]

Yan-Xia Lin University of Wollongong, [email protected]

Michael McCrae University of Wollongong, [email protected]

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Recommended Citation Kosapattarapim, Chaiwat; Lin, Yan-Xia; and McCrae, Michael, "Size and power of cointegration tests with non-normal GARCH error distributions" (2013). Faculty of Engineering and Information Sciences - Papers: Part A. 2381. https://ro.uow.edu.au/eispapers/2381

Research Online is the open access institutional repository for the University of Wollongong. For further information contact the UOW Library: [email protected] Size and power of cointegration tests with non-normal GARCH error distributions

Abstract Several tests for cointegration among non-stationary financial have been developed including the Dicky Fuller (1979) tests, the Cointegration Regression Durbin-Watson test (1983), the Wild Bootstrap test (2003) and the Johansen likelihood ratio tests (1988). The Johensen's tests appeared to provide superior results when the tests were originally applied to situations where the cointegration errors were normally distributed. However, substantial empirical evidences show that financial time series tend to be nonnormal in their distribution which may, in turn, lead to non-normal GARCH type cointegration error distributions. The question addressed in this paper is whether the Johansen's tests are still more powerful than the alternative tests when the underlying cointegration errors are non-normally distributed. Lee and Tse (1996) examined the performance of Johansen's tests compared with DF tests and CRDW test when cointegration errors are fitted by GARCH(1,1) model with normal and student-t error distributions. This paper extends their work. More cointegration tests with a wide of error distributions are considered. Our simulation results indicate that the performance of power of the Johansen's tests in capturing cointegration between financial time series is still higher than alternative tests even when the cointegration errors are not normally distributed (a skewed student-t distribution gives the best results). However, the best size performance is given by the Dicky Fuller test using the skewed generalized error distribution.

Keywords tests, distributions, normal, error, garch, size, power, cointegration, non

Disciplines Engineering | Science and Technology Studies

Publication Details Kosapattarapim, C., Lin, Y. & McCrae, M. (2013). Size and power of cointegration tests with non-normal GARCH error distributions. International Journal of and Economics, 10 (1), 65-74.

This journal article is available at Research Online: https://ro.uow.edu.au/eispapers/2381 Size and power cointegration tests with non-normal GARCH error distributions

Chaiwat Kosapattarapim1 Yan-Xia Lin2 Michael McCrae3

Centre for Statistical and , School of Mathematics and Applied Statistics , University of Wollongong, Wollongong, NSW 2522, AUSTRALIA

Abstract

Several tests for cointegration among non-stationary financial time series have been developed including the Dicky Fuller (1979) unit root tests, the Cointegration Regression Durbin-Watson test (1983), the Wild Bootstrap test (2003) and the Johansen likelihood ratio tests (1988). The Johensen tests appeared to provide superior results when the tests were originally applied to situations where the cointegration errors were normally distributed. However, substantial empirical evidence shows that financial time series tend to be non-normal in their distribution which may, in turn, lead to non-normal GARCH type cointegration error distributions. The question addressed in this paper is whether the Johansen’s tests are still more powerful than the alternative tests when the underlying cointegration errors are non-normally distributed. Our simulation results indicate that the performance of power of the Johansen’s tests in capturing cointegration between financial time series is still higher than alternative tests even when the cointegration errors are not normally distributed (a skewed student-t distribution gives the best results). However, the best size performance is given by the Dicky Fuller test using the skewed generalized error distribution.

Key words: Cointegration test, GARCH Model, Heteroskedasticity

1 Introduction

Cointegrated methods are now widely employed in and financial time series to inves- tigate if among non-normally distributed, non-stationary series there may exist a combination of the series which is stationary. If there exists a linear combination of these time series that are integrated of order zero (T(0)), then such a system implies a long-run stationary equilibrium relationship between them despite short-run deviations from that equilibrium relationship.

1 Email address: [email protected]. 2 Email address: [email protected] 3 Email address: [email protected]

1 Financial return time series frequently exhibit non-normality due to , leptokurtosis and heavy tail. Engle (1982) and Bollerslev (1986) developed a generalized autoregressive conditional het- eroskedasticity (GARCH) model (Engle, 1982: Bollerslev, 1986) to capture the conditional in the underlying return time series. The GARCH model was firstly developed by assuming that the random errors in model have normal conditional distribution and subsequently extended to has non-normal conditional distributions. The initial cointegration assumed that the cointegration errors could be fitted by a GARCH(1,1) model with normal random error distributions. A number of tests developed and used to examine the cointegration relationships among time series, including the Dickey Fuller (DF: Dickey and Fuller, 1979) unit root tests, Cointegration Regression Durbin-Watson test (CRDW: Sargan and Bhargava,

1983), the Johansen’s likelihood ratio tests (λtrace(Johansen trace test) and λmax (Johansen maximum eigenvalue test): Johansen, 1988) and the Wild Bootstrap test (WB: Gerolimetto and Procidano, 2003). The performance of size and power of cointegration tests have been examined with various GARCH error assumptions. Kim and Schmidt (1993) investigated the size of cointegration when the cointegration errors followed a GARCH model with normal random error distributions. They found that the DF tests tend to overreject the null hypothesis of no cointegration in the presence of GARCH errors. Lee and Tse (1996) examined the performance of Johansen’s likelihood ratio tests for cointegration when cointegration errors are fitted by GARCH(1,1) model with normal and student- t random error distributions. They compared the size and power of the Johansen’s tests with other cointegration tests, DF tests and CRDW test. They reported that although the Johansen cointegration tests tend to overreject the null hypothesis of no cointegration, the power of the Johansen’s tests remained higher than the other two tests when the cointegration errors were fitted by GARCH(1,1) model with normal error distributions. Moreover, Gerolimetto and Procidano (2003) examined the WB test compared with DF tests, CRDW test and Johansen’s tests when the cointegration errors followed by a GARCH(1,1) model with normal random error distributions. Their conclusions were the same as Lee and Tse ’s works when comparing the Johanses’s tests with the other three cointegration tests. As we know, the empirical distributions of financial time series are significantly non-normal which exhibit skewness and heavy tail. Numerous heavy tail distributions have been identified from financial time series and applied in modeling and estimating GARCH models. Cheung and Lai (1993) and Gonzalo (1994) examined the effect of non-Gaussian error distribution on the performance of the Johansen’s cointegration tests. They found that the Johansen tests are robust to both skewness and excess in cointegration errors. Our study is an extension ideas of their previous works whether the performance of Johansen’s tests remain higher compare with the other cointegration tests when the cointegration errors followed a GARCH(1,1) model with normal and non-normal randon error distributions. In this paper, we extend the work by Lee and Tse (1996) as well as the work of

2 Gerolimetto and Procidano (2003). Kosapattarapim et al. (2011) studied the volatility forecasting when the GARCH error terms were fitted by the six different types of error distributions, normal(N), skewed normal(SN), student- t(STD), skewed student-t(SSTD), generalized error distribution(GED) and skewed generalized error distribution(SGED). We will continue to use these six types of error distributions in this paper to investigate whether these six types of error distributions impact on the performance of the size and power of cointegration tests. We take into account the skewness, excess kurtosis of financial and generate the data from (ECM) when cointegration errors fitted by GARCH(1,1) with above six types of error distributions. We investigate the size and power of the Johansen trace test (trace,hereafter) and the Johansen maximum eigenvalue test (maxeigen). For comparison purpose, the DF (tρˆ) , T (ˆρ − 1), the CRDW and the WB test are also considered in this study. The aim of this paper is to investigate which cointegration test is the most powerful in detecting cointegration relationship when the cointegration errors follow GARCH model with non-normal error distribution. This paper is organized as follows. The next section, we use closing price indices from Thailand (SET) and Malaysia (KLCI) to demonstrates an evidence that two nonstationary financial time series are cointegrated and their cointegration errors follow GARCH(1,1) with skewed student-t error distribution. In order to investigate the performance of size and power of the cointegration tests, the Monte Carlo simulations are used. The simulation are described in Section 3. The simulation results are presented in Section 4. Finally, the last section gives our conclusions.

2 An empirical example of cointegration errors fitted by GARCH model with non-normal error distribution

The purpose of this section is to identify if there exists financial times series which are cointegrated and the cointegration errors can be fitted by a GARCH models with one of the six types of error distributions mentioned in Section 1. We show an empirical example where returns are cointegrated and the cointegration errors fitted by a GARCH model with skewed student-t error distribution. The Johansen cointegration tests are used in this study. The Johansen cointegration tests were developed by Johansen (1988) and used for testing the number of cointegration relation (Johansen, 1991). It is based on the (VAR) approach and the maximum likelihood procedure. The first step of the Johansen tests is to use the Akaike Information Criterion (AIC) to identify a VAR model with the appropriated lag length for underlying series. The second step is to determine the number of cointegrating relations in underlying series based on the VAR model. Either the trace (λtrace) or maximum eigenvalue (λmax) statistics are used to test the cointegration relations. When the trace statistics is used, the null hypothesis of

3 the test is that there are at most r cointegrating relations against the alternative of n cointegrating relation, where n is the number of entry of underlying time series vector r = 0, 1, 2, . . . , n−1 (Johansen and Juselius; 1990). While the null hypothesis of maximum eigenvalue statistics test is that there are r cointegrating relations against the alternative of r + 1 cointegrating relations. If the tested time series are cointegrated, the VAR model will be transformed into a Vector Error Correction Model (VECM) which can be used to analyze the short-run and long-run relationship among the tested time series. In this Section, we investigate two closing price indices from Thailand (SET) and Malaysia (KLCI). Each stock data employed in this study comprise 1,133 daily closing price from 1/07/1998 to 31/12/2002. We show that the price indexes SET and KLCI stock are cointegrated and their cointegration errors satisfy a GARCH model with SSTD error distribution. In preparation for the Johansen cointegration analysis, two stock market indices need to be tested whether all data series are nonstationary with the same integrated order. The Augmented Dicky-Fuller test (ADF) is applied to each log of closing price index and return series. After the first differences, we also have to examine whether both two stock indices are integrated of order one I(1). The results are reported in Table 1.

Table 1: Unit root test results for two stock indices

Log Price Return Stock market t-prob t-ADF t-prob t-ADF SET 0.0416 0.2552 0.0366 -9.9906∗∗ KLCI 0.0051 0.4673 0.0062 -14.9220∗∗

Notes: Return series are defined by rt = ln[pt/pt−1] and pt denotes the closing price of index.

From Table 1, the ADF statistics of Log Price are clearly not significant but the ADF statistics of Return are significant. This implies that Log Price series of two stocks are nonstationary while returns of two stock indices are stationary. Therefore, both two stock indices are I(1). The second step of the Johansen test is to specify an appropriate lag length of VAR system. We examine the cointegrated analysis between SET and KLCI. The hypothesis for tests have been used. If the null hypothesis cannot be rejected, it will that statistically, there is no error autocorrelation in the model. A VAR(8) model is chosen for our data. To determine whether SET and KLCI are cointegrated and the number of cointegrating vector, the Johansen’s tests are employed. Two tests, trace test (λtrace) and the maximum eigenvalue test

(λmax), are considered. The values of tests statistics are presented in Table 2. From Table 2, we should not expect two Jonhansen’s tests will provide the same conclusion. The trace and maximum eigenvalue statistics might give different cointegration test results. The value of

λtrace is 6.82 which is less than the 5% critical value (8.18). We cannot reject the null hypothesis at 5% significant level. It indicates that there are not more than one cointegrating relation at 5% level.

4 Table 2: Results and Critical Values for the λtrace and λmax test

H0 H1 λtrace CV(trace,5%) λmax CV(max,5%) r = 0 r > 0 18.54∗∗ 17.95 11.72 14.90 r ≤ 1 r > 1 6.82 8.18 6.82 8.18

Notes: ** denotes rejection of null hypothesis at 5% level.

But the value of λmax is not significant at 5% level. By using λtrace, we conclude that there is only one cointegrating relation for SET and KLCI. The cointegrating coefficient vector is given as, β0 = (1.0000,-0.7532) and normalized cointegration equation with respect to SET is shown in the equation below :

SET = 0.7532KLCI + et (2.1)

where et is the cointegration errors. The eight lag lengths of the series in VECM model (VECM(8)) are determined according to the autocorrelation test. We use PcGiven program to obtain the residuals of the VECM(8) and fit the residuals by GARCH(p,q) models. According to the results of sensitivity analysis (Kosapattarapim et al, 2011), the order of the best fitted GARCH model is not sensitive to the type of error distribution in the underlying GARCH model. In this study, the order of the best fitted model is determined base on normal error distribution. It was found that a GARCH(1,1) is the best fitted model for the residuals. Considering the particular six types of error distributions, N, SN, STD, SSTD, GED and SGED, we investigate which type of error distribution in the GARCH(1,1) model provides the smallest AIC values. The results of AIC values are reported in Tables 3.

Table 3: AIC values given by GARCH(1,1) with different type of error distributions

Normal Skewd normal Student-t Skewed Student-t GED Skewed GED -5.2791 -5.2842 -5.3132 -5.3166 -5.3123 -5.3163

Table 3 shows that, based on AIC criterion, GARCH(1,1) with Skewed student-t (SSTD) is the best fitted GARCH model for the residuals in VECM. The estimations of parameters in the GARCH(1,1) model with SSTD error are reported in Table 4. All parameters in Table 4 are significant at 5% level. The results confirm that the residuals in VECM can be well fitted by a GARCH model with SSTD error distribution.

5 Table 4: Estimated parameters and diagnostic of GARCH(1,1)model with SSTD

GARCH(1,1) parameter Coefficient P-value ω 1.168∗10−5 0.06687∗ −1 α1 1.213∗10 0.00022∗ ∗ ∗ −1 −16∗∗∗ β1 8.500∗10 < 2∗10 ξ 1.103 < 2∗10−16∗∗∗ ν 7.437 < 2.01∗10−6∗∗∗ LM Test 6.396 0.9723 Q(15) 5.685 0.9311

Notes: The GARCH(1,1) model is defined by (3.4)-(3.5), the ξ and ν are skewness and of SSTD respectively.

3 Simulation design

This Section deal with the comparable performance of size and power of the cointegration tests (Johansen, DF, CRDW and WB tests) when the cointegration errors follow GARCH(1,1) model with six different types of error distributions respectively. The comparison study is carried out by simulation approach. The size of the tests are evaluated by the of the null hypothesis stating that truly non- cointegrated is not cointegrated is rejected in 10,000 trials. For the power of the test, it is evaluated by the frequency that the null hypothesis stating that truly cointegrated is not cointegrated is rejected in 10,000 trials.

To examine the size of the tests, we simulated samples from a non-cointegrated system Xt =

(x1t, x2t) with GARCH(1,1) error. The system is given by.

∆x1t = e1t (3.2)

∆x2t = e2t (3.3)

The errors e1t and e2t follow GARCH(1,1) model.

p eit = ηit hit, (3.4)

2 hit = ωi + αieit−1 + βihit−1, (3.5)

where hit is the conditional on information available up to time t − 1 and ηit are i.i.d. with

E(ηit) = 0, V ar(ηit) = 1 for i=1,2 and t=1,2,. . . ,T+d. The first d = 500 observations are discarded. We independently simulated 10,000 samples with size T = 100 and 1,000 respectively from the system and apply the four cointegration tests (Johansen, DF, CRDW and WB tests) to the simulated data. The comparisons of the frequency of rejection non cointegration crossing the four tests are carried out.

6 In terms of size of the tests, we expect the minimum of proportion of rejection the null hypothesis. To examine the power of a cointegration tests, we generate data from a bivariate cointegrated system as follows

∆x1t = −0.2(x1,t−1 − x2,t−1) + e1t (3.6)

∆x2t = e2t (3.7)

where the errors e1t and e2t follow GARCH(1,1) model from (3.4) and (3.5) respectively. The frequency of rejections non cointegration are counted. If a test is robust and powerful then we expect to obtain the higher proportion of rejection the null hypothesis. We independently simulated 10,000 samples with size T=100 and 1,000 observations respectively from (3.6) and (3.7). Then the four cointegration tests are applied to the simulated data.

In our simulation studies, two sets of GARCH(1,1) parameter (ωi, αi, βi), i = 1, 2, are considered.

To simplify the notation, we use the same notation of ht and (ω, α, β) in each simulation, omitting the index i in reporting results. The two sets of GARCH(1,1) parameters in this simulation study are

(0.1, 0.3, 0.6) and (0.1, 0.65, 0.05). The probability distributions of the errors in GARCH models (e1t and e2t) are Normal, Skewed normal, Student-t, Skewed Student, GED and Skewed GED respectively.

4 Results

Table 5 reports the results on the size of the tests when the errors e1t and e2t follow six types of errors in GARCH(1,1) model. The values of skewness of the SN, SSTD and SGED distribution are chosen as 0.1, 0.5 and 3 respectively. The degree of freedoms (υ) for Student-t distribution are 5 and 8 respectively. These values of υ are also considered by Lee and Tse (1996) in their work. The kurtosis of the Student-t density is given by 3(υ − 2)/(υ − 4) for υ > 4; it is 9 for υ = 5 and 4.5 for υ = 8. The shape parameter of the SN, GED and SGED is equal to 3. The size distortion decreases when the sample size (T ) increases. When (α + β) is larger, the size distortion is larger too. In terms of skewed distribution, when the scale of skewness is increased, the size distortion is not different significantly. For the different cointegration tests, the DF test with (tρˆ) is smaller than the others. The size distortion of DF tests becomes small when using SGED error distribution. We also investigate the power of the tests for all cointegration tests under the presence of GARCH(1,1) model. Results are reported in Table 6. The power of the Johansen tests for T = 100 is higher than all other cointegration tests. Particularly, the performance of λmax is slightly more powerful than the

λtrace, but not different significantly. According to Table 5, the size distortion of DF tests is outper- form the other cointegration tests. It turns out that DF tests are the worst for the power of the test

7 Table 5: The size of the test at 5% level

Parameter 100 observations (α = 0.3, β = 0.6) N SN STD SSTD GED SGED Shape 3 4.5 9 4.5 9 3 3 Skew 0.1 0.5 3 0.1 0.5 3 0.1 0.5 3 0.1 0.5 3 Trace 0.0930 0.0909 0.0894 0.0918 0.1070 0.0989 0.1029 0.1006 0.1057 0.0902 0.0966 0.0928 0.0862 0.0863 0.0868 0.0891 Maxeigen 0.0905 0.0940 0.0891 0.0899 0.1065 0.1027 0.1043 0.1035 0.1066 0.0950 0.0973 0.0927 0.0884 0.0899 0.0851 0.0860 CRDW 0.0730 0.0731 0.0719 0.0722 0.0822 0.0778 0.0903 0.0865 0.0877 0.0791 0.0776 0.0743 0.0695 0.0679 0.0657 0.0654 DF (tρˆ) 0.0762 0.0800 0.0717 0.0743 0.0854 0.0808 0.0908 0.0879 0.0893 0.0807 0.0768 0.0775 0.0758 0.0730 0.0671 0.0678 DF T (ˆρ − 1) 0.0665 0.0673 0.0649 0.0640 0.0761 0.0710 0.0824 0.0799 0.0798 0.0711 0.0685 0.0679 0.0618 0.0627 0.0572 0.0579 WB 0.0777 0.0797 0.0719 0.0744 0.0849 0.0790 0.0927 0.0879 0.0882 0.0778 0.0784 0.0780 0.0749 0.0711 0.0669 0.0662 (α = 0.65, β = 0.05) N SN STD SSTD GED SGED Shape 3 4.5 9 4.5 9 3 3 Skew 0.1 0.5 3 0.1 0.5 3 0.1 0.5 3 0.1 0.5 3 Trace 0.0887 0.0866 0.0867 0.0866 0.0926 0.0933 0.0936 0.0911 0.0947 0.0817 0.0886 0.0864 0.0902 0.0813 0.0880 0.0848 Maxeigen 0.0911 0.0891 0.0879 0.0872 0.0932 0.0984 0.0965 0.0978 0.1002 0.0852 0.0916 0.0909 0.0932 0.0840 0.0909 0.0828 CRDW 0.0849 0.0888 0.0831 0.0891 0.0860 0.0854 0.1103 0.0994 0.1058 0.0930 0.0896 0.0857 0.0813 0.0784 0.0768 0.0767 DF (tρˆ) 0.0834 0.0912 0.0821 0.0865 0.0910 0.0888 0.1081 0.0983 0.1023 0.0902 0.0881 0.0903 0.0848 0.0815 0.0764 0.0761 8 DF T (ˆρ − 1) 0.0749 0.0816 0.0754 0.0817 0.0818 0.0802 0.1013 0.0926 0.0975 0.0851 0.0799 0.0818 0.0736 0.0732 0.0703 0.0702 WB 0.0751 0.0882 0.0823 0.0810 0.0738 0.0749 0.0989 0.0994 0.0997 0.0890 0.0886 0.0827 0.0839 0.0809 0.0754 0.0755 Parameter 1,000 observations (α = 0.3, β = 0.6) N SN STD SSTD GED SGED Shape 3 4.5 9 4.5 9 3 3 Skew 0.1 0.5 3 0.1 0.5 3 0.1 0.5 3 0.1 0.5 3 Trace 0.0719 0.0753 0.0736 0.0746 0.0839 0.0764 0.0955 0.0998 0.0969 0.0858 0.0830 0.0870 0.0646 0.0774 0.0703 0.0764 Maxeigen 0.0759 0.0761 0.0748 0.0735 0.0855 0.0761 0.0988 0.0998 0.1005 0.0889 0.0834 0.0878 0.0696 0.0747 0.0692 0.0741 CRDW 0.0676 0.0689 0.0702 0.0753 0.0822 0.0721 0.1050 0.1010 0.1008 0.0894 0.0776 0.0842 0.0648 0.0642 0.0655 0.0717 DF (tρˆ) 0.0660 0.0651 0.0708 0.0738 0.0785 0.0701 0.1024 0.0978 0.0967 0.0855 0.0734 0.0794 0.0614 0.0676 0.0675 0.0644 DF T (ˆρ − 1) 0.0583 0.0596 0.0630 0.0653 0.0724 0.0640 0.0936 0.0909 0.0915 0.0800 0.0664 0.0747 0.0564 0.0566 0.0578 0.0585 (α = 0.65, β = 0.05) N SN STD SSTD GED SGED Shape 3 4.5 9 4.5 9 3 3 Skew 0.1 0.5 3 0.1 0.5 3 0.1 0.5 3 0.1 0.5 3 Trace 0.0656 0.0651 0.0709 0.0680 0.0697 0.0673 0.0763 0.0755 0.0770 0.0729 0.0742 0.0747 0.0660 0.0733 0.0664 0.0699 Maxeigen 0.0671 0.0670 0.0706 0.0668 0.0665 0.0685 0.0806 0.0760 0.0764 0.0744 0.0730 0.0758 0.0647 0.0718 0.0667 0.0691 CRDW 0.0694 0.0755 0.0705 0.0733 0.0743 0.0704 0.0974 0.0916 0.0974 0.0795 0.0729 0.0797 0.0626 0.0665 0.0619 0.0677 DF (tρˆ) 0.0653 0.0706 0.0672 0.0723 0.0718 0.0724 0.0930 0.0887 0.0906 0.0756 0.0726 0.0735 0.0626 0.0667 0.0615 0.0646 DF T (ˆρ − 1) 0.0598 0.0655 0.0605 0.0645 0.0638 0.0670 0.0870 0.0814 0.0838 0.0688 0.0638 0.0688 0.0552 0.0582 0.0527 0.0565

Notes: For T = 1,000 results of the Wild Bootstrap have not been done because of the time limitation for running simulated programme. (see Table 6 with T = 100). The power of Johanses’s test remain higher compared with the other cointegration tests. Par- ticularly, the power of the Johansen tests slightly increase when the cointegrating errors follow GARCH(1,1) with skewed student-t error distribution.

5 Conclusion

We examined the performance of four cointegration tests under the presence of GARCH model with six types of error distributions by simulations. These tests tend to overreject the null hypothesis of no cointegration. The size distortion can be improved by the Dickey-Fuller tests but the power performance of DF tests is significantly lower than the other cointegration tests. The size distortion of Dickey-Fuller tests tend to be smaller for the errors of GARCH model with skewed generalized error distribution. The power of Johansen’s tests provide the best performance compare with other cointegration tests and slightly increase when the errors of GARCH model given by the skewed student- t error distribution.

9 Table 6: The power of the test at 5% level

Parameter 100 observations (α = 0.3, β = 0.6) N SN STD SSTD GED SGED Shape 3 4.5 9 4.5 9 3 3 Skew 0.1 0.5 3 0.1 0.5 3 0.1 0.5 3 0.1 0.5 3 Trace 0.8027 0.8030 0.8015 0.8033 0.8000 0.8010 0.8135 0.8113 0.8098 0.8145 0.8063 0.8109 0.7978 0.8014 0.8029 0.8002 Maxeigen 0.8254 0.8278 0.8266 0.8270 0.8243 0.8227 0.8362 0.8344 0.8351 0.8284 0.8278 0.8301 0.8225 0.8206 0.8267 0.8225 CRDW 0.6731 0.7044 0.6951 0.6988 0.6731 0.6738 0.7359 0.7163 0.7264 0.7160 0.6994 0.7134 0.6856 0.6934 0.6868 0.6913 DF (tρˆ) 0.5614 0.5902 0.5781 0.5834 0.5672 0.5628 0.6234 0.6177 0.6142 0.6027 0.5883 0.6044 0.5681 0.5777 0.5796 0.5786 DF T (ˆρ − 1) 0.6166 0.6464 0.6413 0.6422 0.6170 0.6181 0.6811 0.6656 0.6741 0.6625 0.6435 0.6624 0.6269 0.6339 0.6363 0.6325 WB 0.5750 0.6023 0.5987 0.5994 0.5825 0.5834 0.6224 0.6201 0.6224 0.6199 0.6020 0.6209 0.5843 0.5952 0.5933 0.5940 (α = 0.65, β = 0.05) N SN STD SSTD GED SGED Shape 3 4.5 9 4.5 9 3 3 Skew 0.1 0.5 3 0.1 0.5 3 0.1 0.5 3 0.1 0.5 3 Trace 0.8078 0.8090 0.8123 0.8071 0.8056 0.8026 0.8155 0.8178 0.8150 0.8190 0.8141 0.8186 0.8061 0.8045 0.8076 0.8039 Maxeigen 0.8336 0.8338 0.8372 0.8325 0.8287 0.8296 0.8380 0.8381 0.8411 0.8363 0.8337 0.8420 0.8292 0.8305 0.8296 0.8271 CRDW 0.6726 0.7094 0.6996 0.7031 0.6640 0.6745 0.7390 0.7252 0.7335 0.7246 0.7088 0.7236 0.6863 0.6945 0.6936 0.6936 DF (tρˆ) 0.5668 0.6051 0.5967 0.6061 0.6700 0.5691 0.6431 0.6310 0.6365 0.6227 0.6072 0.6263 0.5783 0.5921 0.5873 0.5807 10 DF T (ˆρ − 1) 0.6167 0.6571 0.6498 0.6549 0.6148 0.6201 0.6902 0.6806 0.6860 0.6724 0.6595 0.6743 0.6346 0.6432 0.6425 0.6376 WB 0.5792 0.5955 0.6097 0.5545 0.5772 0.5542 0.5912 0.5742 0.5906 0.5886 0.5823 0.5880 0.5926 0.6082 0.5992 0.5965 Parameter 1,000 observations (α = 0.3, β = 0.6) N SN STD SSTD GED SGED Shape 3 4.5 9 4.5 9 3 3 Skew 0.1 0.5 3 0.1 0.5 3 0.1 0.5 3 0.1 0.5 3 Trace 1.0000 0.9999 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9999 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 Maxeigen 1.0000 0.9999 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 1.0000 1.0000 0.9999 1.0000 1.0000 1.0000 1.0000 CRDW 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 DF (tρˆ) 1.0000 0.9999 1.0000 0.9999 1.0000 1.0000 1.0000 1.0000 0.9999 0.9998 1.0000 0.9999 1.0000 1.0000 1.0000 1.0000 DF T (ˆρ − 1) 1.0000 0.9999 1.0000 0.9999 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 1.0000 1.0000 1.0000 1.0000 1.0000 (α = 0.65, β = 0.05) N SN STD SSTD GED SGED Shape 3 4.5 9 4.5 9 3 3 Skew 0.1 0.5 3 0.1 0.5 3 0.1 0.5 3 0.1 0.5 3 Trace 0.9999 0.9999 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 Maxeigen 0.9999 0.9999 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 1.0000 0.9999 1.0000 1.0000 1.0000 1.0000 CRDW 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 DF (tρˆ) 0.9999 0.9999 1.0000 0.9999 1.0000 1.0000 0.9999 1.0000 0.9998 0.9998 1.0000 0.9998 0.9999 1.0000 1.0000 1.0000 DF T (ˆρ − 1) 1.0000 1.0000 1.0000 0.9999 1.0000 1.0000 1.0000 1.0000 1.0000 0.9998 1.0000 0.9999 1.0000 1.0000 1.0000 1.0000

Notes: For T = 1,000 results of the Wild Bootstrap have not been done because of the time limitation for running simulated programme . References

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