# The Value-At-Risk Estimate of Stock and Currency-Stock Portfolios’ Returns

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. The Variance-Correlation Type of Bivariate GARCH Models

**A**,

**B**, and

**G**are the $2\times 2\text{}$parameter matrices; $\overline{Q}$ and $\overline{N}$ are the unconditional correlation matrices of ${u}_{t}$ and ${n}_{t}$; and ${n}_{t}=I\left[{u}_{t}<0\right]\circ {u}_{t}$, $I[\xb7]$ is a $2\times 1$ indicator function that takes on the value of 1 if the argument is true and 0 otherwise; ‘$\circ $’ denotes a Hadamard product; and $\overline{Q}=\mathrm{E}\left[{u}_{t}{u}_{t}^{\prime}\right]$, and $\overline{N}=\mathrm{E}\left[{n}_{t}{n}_{t}^{\prime}\right]$. Notably, the $\overline{Q}$ and $\overline{N}$ can be replaced with the sample analogues, ${\mathrm{T}}^{-1}{\sum}_{\mathrm{t}=1}^{\mathrm{T}}{u}_{t}\xb7{u}_{t}^{\prime}$ and ${\mathrm{T}}^{-1}{\sum}_{\mathrm{t}=1}^{\mathrm{T}}{n}_{t}\xb7{n}_{t}^{\prime}$, respectively. T denotes the sample size of the estimation period. If

**A, B**, and

**G**are respectively replaced by scalars $\overline{\mathrm{a}}$, $\overline{\mathrm{b}}$, and $\overline{\mathrm{g}}$, then the AGDCC model degenerates into the ADCC model. Hence, the ADCC model can be expressed as the following form.

_{3}and the values of parameters $\overline{\mathrm{a}},\text{}\overline{\mathrm{b}},\text{}\mathrm{and}\overline{\mathrm{g}}$ are obtained from the estimation of the correlation matrix equation: ${Q}_{t}=\left(\overline{Q}-{\mathrm{a}}^{\prime}\overline{Q}-{\mathrm{b}}^{\prime}\overline{Q}-{\mathrm{g}}^{\prime}\overline{N}\right)+{\mathrm{a}}^{\prime}{u}_{t-1}{u}_{t-1}^{\prime}+{\mathrm{g}}^{\prime}{n}_{t-1}{n}_{t-1}^{\prime}+{\mathrm{b}}^{\prime}{Q}_{t-1}$ where ${\mathrm{a}}^{\prime}={\overline{\mathrm{a}}}^{2}$, ${\mathrm{b}}^{\prime}={\overline{\mathrm{b}}}^{2}$, and ${\mathrm{g}}^{\prime}={\overline{\mathrm{g}}}^{2}$. Notably, the two-step parameters estimate approach can be employed when the dimension of the multivariate GARCH model is greater than two.

#### 2.2. The Variance-Covariance Type of Bivariate GARCH Models

**first**, the variance-covariance matrix ${H}_{t}$ is a symmetric matrix (i.e., ${\mathrm{h}}_{12,\mathrm{t}}={\mathrm{h}}_{21,\mathrm{t}}$).

**Second**, the model in Equation (10) includes eleven parameters (i.e., the elements in matrices ${C}_{0}$, ${A}_{\mathbf{11}}$, and ${G}_{\mathbf{11}}$).

**Third**, the elements of this variance-covariance matrix ${H}_{t}$ is a function of ${\mathrm{e}}_{1,\mathrm{t}-1}^{2}$, ${\mathrm{e}}_{2,\mathrm{t}-1}^{2}$, ${\mathrm{e}}_{1,\mathrm{t}-1}{\mathrm{e}}_{2,\mathrm{t}-1}$, ${\mathrm{h}}_{11,\mathrm{t}-1}$, ${\mathrm{h}}_{22,\mathrm{t}-1}$ and ${\mathrm{h}}_{12,\mathrm{t}-1}$ and is expressed as follows.

**first**obtain that the variance-covariance matrix ${H}_{t}$ is a symmetric matrix (i.e., ${\mathrm{h}}_{21,\mathrm{t}}={\mathrm{h}}_{12,\mathrm{t}}$).

**Second,**the number of parameters of the model in Equation (12) will decrease to nine (i.e., the elements in matrices ${C}_{\mathbf{0}}$, ${A}_{\mathbf{11}}$, ${A}_{\mathbf{12}}$, ${G}_{\mathbf{11}}$ and${G}_{\mathbf{12}}$).

**Third**, the elements of this variance-covariance matrix ${H}_{t}$ are expressed as the similar form of univariate GARCH(1,1), and are represented as follows.

## 3. Assessment Methods of Alternative VaR Models

#### 3.1. The Failure Rate and Unconditional Coverage Test

_{uc}) and gives a penalty of one to each exception of the VaR. Hence, the BLF of a long position is a Bernoulli random variable and it can be defined as follows.

_{uc}) which is a likelihood ratio test for testing the model accuracy. The null hypothesis of this test is that the probability of failure for each trial ($\widehat{\mathsf{\pi}}$) equals the specified model probability (p). The likelihood ratio test statistic is given by

_{uc}test can be employed to test whether the sample point estimate is statistically consistent with the VaR model’s prescribed confidence level.

#### 3.2. Conditional Coverage Test

_{uc}test can reject a model that either overestimates or underestimates the true but unobservable VaR, it cannot inspect whether the exceptions are randomly distributed. In a risk management framework, it is significant that the VaR exceptions must be uncorrelated over time, which hints that both the independence and unconditional coverage tests based on the evaluation of interval forecasts must be simultaneously considered when the competition of a group of VaR models is performing. Christoffersen (1998) thus proposed a conditional coverage test (LR

_{cc}) to jointly test the correct unconditional coverage and serial independence. The LR

_{cc}test is a joint test of these two properties and the corresponding test statistics are LR

_{cc}= LR

_{uc}+ LR

_{ind}when we condition on the first observation. The LR

_{ind}test denotes the likelihood ratio statistic that tests whether exceptions are independent, and the LR

_{uc}is defined in the previous subsection. Therefore, under the null hypothesis of the expected proportion of exceptions equals p and the failure process is independent, the appropriate likelihood ratio test statistic is expressed as follows:

#### 3.3. Dynamic Quantile Test

#### 3.4. Market Risk Capital and the Superior Predictive Ability Test

## 4. Data and Descriptive Statistics

## 5. Empirical Results and Analyses

#### 5.1. Estimation Results for Alternative Bivariate GARCH Models

_{1}(−4800.17) for the first univariate GARCH(1,1) model corresponding to the first component stock index are produced, and they are listed in the column ‘Ny-Sp’ in panel A of Table 2. Similarly, the coefficients${\text{}\mathsf{\mu}}_{2}\left(0.0491\right),$ ${\mathsf{\omega}}_{2}$(0.0168), ${\mathsf{\alpha}}_{2}$(0.0880), ${\mathsf{\beta}}_{2}$(0.9000) and one value of the log-likelihood LL

_{2}(−4803.92) for the second univariate GARCH(1,1) model corresponding to the second component stock index are produced and they are listed in the column ‘Ny-Sp’ in panel B of Table 2. In addition, the two variance series $(\mathrm{i}.\mathrm{e}.,\text{}{\mathrm{h}}_{11,\mathrm{t}}$, and ${\mathrm{h}}_{22,\mathrm{t}}$) and two residue series ($\mathrm{i}.\mathrm{e}.,\text{}{\mathrm{e}}_{1,\mathrm{t}}\text{}\mathrm{and}\text{}{\mathrm{e}}_{2,\mathrm{t}}$) are also obtained in this step. Second, the above two standardized residual return series (${\mathrm{u}}_{\mathrm{i},\mathrm{t}}\text{}\mathrm{for}\text{}\mathrm{i}=1,\text{}2$) and their corresponding negative component residual return series (${\mathrm{n}}_{\mathrm{i},\mathrm{t}}\text{}\mathrm{for}\text{}\mathrm{i}=1,2$) are used to estimate the intercept parameters of the conditional correlation matrix equation listed in Equation (4). During this step of estimation, the values of parameters $\overline{\mathrm{a}}$(0.2041), $\overline{\mathrm{b}}$(0.9738), and $\overline{\mathrm{g}}$(−2 × 10

^{−5}) and one log-likelihood value LL

_{3}(−4386.75) are obtained, and they are listed in panel C of Table 2. As shown in Table 2, the ${\mathsf{\omega}}_{\mathrm{i}},{\mathsf{\alpha}}_{\mathrm{i}}$, and ${\mathsf{\beta}}_{\mathrm{i}}$ coefficients where $\mathrm{i}=1,\text{}2$ are positive and significant at the 1% level for all six NYSE-based portfolios. Notably, the values of ${\mathsf{\omega}}_{1},{\mathsf{\alpha}}_{1}$, and ${\mathsf{\beta}}_{1}$ coefficients are all equal for all NYSE-based portfolios owing to having the same first component asset within the two-step estimate procedure. Moreover, the values of parameters $\overline{\mathrm{a}}$ and $\overline{\mathrm{b}}$ are all significantly positive, whereas the values of parameter $\overline{\mathrm{g}}$ are very small and are not significant for most cases, indicating that the asymmetric property of correlation seems not to exist in the stock-based portfolios. Notably, the values of ${\overline{\mathrm{a}}}^{2}+{\overline{\mathrm{b}}}^{2}+{\overline{\mathrm{g}}}^{2}$ for the six NYSE-based portfolios are less than 1, indicating that the correlation matrix ${Q}_{t}$ is positive definite. In addition, the mean conditional correlation for the overall period is between 0.4944 (Ny-Sm) and 0.9746 (Ny-Sp) for all NYSE-based portfolios. Finally, the values of ${\mathrm{Q}}_{1}^{2}\left(24\right)$ and ${\mathrm{Q}}_{2}^{2}\left(24\right)$ test statistics are significant for most of the six pairs of data. However, the values of the above statistics are significantly lower than those appearing in Table 1. These results indicate that the serial correlation has been significantly reduced in standard residuals, confirming that the NS-ADCC model addressed in this study is sufficient to correct the serial correlation that exists in the conditional variance equation of these six pairs of returns series.

#### 5.2. The Performance Assessments of VaR Forecasts

#### 5.2.1. Preliminary Analysis of Average VaR Performance

_{1,95}underneath S1. In the same inference, regarding the 99% level, the total number of portfolios with the lower value of failure rate that the S-CCC model has is 2, that is, the summation of three numbers 1, 1, and 0. This number, 2, is listed in the entry corresponding to S-CCC model of the column S

_{1,99}underneath S1. Finally, we sum the above two numbers, 4 and 2, and record this number, 6 at the entry corresponding to the row S-CCC and column SS1 underneath S1. As shown in column SS1 underneath S1 in panel A of Table 5, we find that the NS-CCC model has a better VaR forecast performance for the CCC type of bivariate variance-covariance specification since this model has the larger number, 19 between the S-CCC (6) and NS-CCC (19) models, where the number in the brackets beside each model denotes the summation of the total number of portfolios having the lower value of failure rate for both 95% and 99% levels. In the same inference process, the S-DCC (respectively, S-BEKK) model has a better VaR forecast performance for the DCC (respectively, BEKK) type of bivariate variance-covariance specification. These results indicate that the standard approach has better VaR forecast performance for the DCC and BEKK types of bivariate variance-covariance specification whereas the non-standard or two-step approach has a better VaR forecast performance only for the CCC type of bivariate variance-covariance specification. As reported in column SS2 underneath S2 in panel A of Table 5, we find that the S-DCC model has the best VaR forecast performance since this model has the larger number, 34, among the S-CCC (1), S-DCC (34), and S-BEKK (11) models. On the contrary, the NS-CCC, NS-DCC, and NS-BEKK models seem to have the same VaR forecast performance since these three models have a nearly equal number among the NS-CCC (15), NS-DCC (16), and NS-BEKK (16) models. These results indicate that the DCC type of bivariate variance-covariance specification has the best VaR forecast performance for the standard approach, whereas the three types of bivariate variance-covariance specification seem to have the same VaR forecast performance for the non-standard approach. As listed in column SS3 underneath S3 in panel A of Table 5, we find that the S-DCC model has the best VaR forecast performance since this model has the largest number, 34 among all six bivariate GARCH models. These results indicate that the DCC type of bivariate variance-covariance specification with a standard approach has the best VaR forecast performance. Finally, as shown in column SS4 underneath S4 in panel A of Table 5, we find that the NS-DCC model has a better VaR forecast performance since this model has the larger number, 6 between the NS-DCC (6), and NS-ADCC (3) models. These results indicate that the asymmetric DCC model does not have a better forecast performance than its corresponding symmetric one.

#### 5.2.2. Summary Comparison Results Based on Alternative Accuracy Measures

_{uc}, LR

_{cc}, and DQ) are utilized to perform the back-testing of each of the seven bivariate GARCH models: the S-CCC, NS-CCC, S-DCC, NS-DCC, S-BEKK, NS-BEKK, and the NS-ADCC models, and then the produced results are used to assess the VaR forecast performance for the above seven models according to the issues explored in this study. In principle, the model with the greater number that passes the above three back-testing bears a better performance than the model with the smaller that number.

_{uc}, LR

_{cc}, and DQ test statistics of long position for each of the seven bivariate GARCH models under a 95% confidence level over the entire out-of-sample period18. Moreover, regarding a specified model, the total number of portfolios that pass the LR

_{uc}, LR

_{cc}, and DQ types of back-testing are counted and are respectively listed in column Sum of Panels A, B, and C in Table 6. For example, regarding the first subpanel of Panel A in Table 6, both the S-CCC and NS-CCC models pass the LR

_{uc}test only for the Ny-Sp and Ny-Sm portfolios. Hence, the numbers in column Sum corresponding to the rows S-CCC and NS-CCC models of the first subpanel of Panel A in Table 6 are 2 and 2, respectively. Concerning the other five models and the other two subpanels, the results are summarized in column Sum of Panel A in Table 6 corresponding to the specified model and the specified subpanel with the same inference process. Finally, the results in column Sum corresponding to three subpanels of Panel A in Table 6 are also respectively summarized at the three columns underneath LR

_{uc}that are also underneath 95% in the first panel of Table 7. In the same inference, the results in column Sum corresponding to three subpanels of Panel B (respectively, C) in Table 6 are also respectively summarized at the three columns underneath LR

_{cc}(respectively, DQ) that are also underneath 95% in the first panel of Table 7.

_{uc}, LR

_{cc}, and DQ) for four levels (i.e., 90%, 95%, 99%, and 99.5%) based on 21 equal weight bi-component stock portfolios as a sample. Subsequently, regarding all 21 equal weight bi-component stock portfolios, we sum the total number of portfolios that pass the three accuracy measures (i.e., LR

_{uc}, LR

_{cc}, and DQ) under the 90% (respectively, 95%) level for each of the seven bivariate GARCH models, and list them in column S

_{90}(respectively, S

_{95}) underneath the 90% (respectively, 95%) level at the first panel in Table 7. In other words, the numbers in column S

_{90}(respectively, S

_{95}) underneath the 90% (respectively, 95%) level at the first panel of Table 7 denote the summation of three corresponding numbers in columns S

_{90,uc}, S

_{90,cc}and S

_{90,dq}(respectively, S

_{95,uc}, S

_{95,cc}and S

_{95,dq}). Moreover, the numbers in column S

_{90,uc}(respectively, S

_{90,cc}) underneath the 90% denote the summation of three corresponding numbers in column LR

_{uc}(respectively, LR

_{cc}) underneath the 90% whereas those in column S

_{90,dq}underneath the 90% denote the summation of three corresponding numbers in column DQ underneath the 90%. In the same inference process, we sum the total number of portfolios that pass the three accuracy measures (i.e., LR

_{uc}, LR and DQ) under a 99% (respectively, 99.5%) level for each of the seven bivariate GARCH models and list them in column S

_{99}(respectively, S

_{995}) underneath a 99% (respectively, 99.5%) level at the second panel of Table 7.

_{90}underneath the 90% level, S

_{95}underneath the 95% level, S

_{99}underneath the 99% level, and S

_{995}underneath the 99.5% level in Table 7. The first category of the model performance competition is used to inspect which approach of parameters estimate (i.e., the standard or non-standard approach) has a better VaR forecast performance based on the same bivariate variance-covariance specification. We find that the NS-CCC model has a better VaR forecast performance than the S-CCC model for the CCC type of bivariate variance-covariance specification for most of the four levels since the NS-CCC model has the larger total number of portfolios that pass three accuracy measures except the 99.5% level. For example, regarding the 90% level, the NS-CCC model has the larger number, 38 between the S-CCC (33) and NS-CCC (38) models, where the number in the brackets beside each model denotes the summation of the total number of portfolios that pass three accuracy measures for the 90% level. In the same inference process, the S-DCC (respectively, S-BEKK) model has a better VaR forecast performance for the DCC (respectively, BEKK) type of bivariate variance-covariance specification for all levels. These results indicate that the standard approach has better VaR forecast performance for the DCC and BEKK types of bivariate variance-covariance specification whereas the non-standard approach has a better VaR forecast performance only for the CCC type of bivariate variance-covariance specification. These results are consistent with those found in the failure rate and mean VaR.

**5**(respectively, 5) and 0 (respectively,

**7**) for the NS-DCC and NS-ADCC models, respectively.

#### 5.3. Robust Check for the Performance Assessments of VaR Forecasts

#### 5.3.1. Can the Weight Combinations of Portfolios Affect the Performance of VaR Forecasts?

_{90,uc}(respectively, S

_{90,cc}) of Table 7 are recorded in column LR

_{uc}(respectively, LR

_{cc}) underneath the 90% level in panel A of Table 8. Similarly, the numbers in column S

_{90,dq}(respectively, S

_{90}) of Table 7 are recorded in column DQ (respectively, Sum) underneath the 90% level in panel A of Table 8. Subsequently, via following the procedure in the case of equal-weight bi-component portfolios, we summarize the results of three accuracy tests for the weight combinations: ${\mathrm{w}}_{1}=25\%,\text{}\mathrm{and}\text{}{\mathrm{w}}_{2}=75\%$ (respectively, ${\mathrm{w}}_{1}=75\%,\text{}\mathrm{and}\text{}{\mathrm{w}}_{2}=25\%$) in panel B (respectively, C) of Table 8. Table 8 lists all summary results of the out-of-sample VaR forecasts performance for the three weight combinations of stock-based portfolios. Then we execute four groups of the model performance comparisons as listed in Section 5.2.2. As reported in Table 8, regarding the three weight combinations, we find that, first, the NS-CCC model has a better VaR forecast performance than the S-CCC model for the CCC type of bivariate variance-covariance specification for most cases because the NS-CCC model has a larger total number of portfolios that pass the three accuracy measures except for the cases of the 99.5% level in panel A and both the 99% and 99.5% levels in panel C. Using the same inference process, the S-DCC model has a better VaR forecast performance than the NS-DCC model for all cases. Conversely, the S-BEKK model has a better VaR forecast performance than the NS-BEKK model except for the 90%, 95%, and 99% levels in panel B. Second, the S-DCC model has the best VaR forecast performance because this model has the largest total number of portfolios that pass the three accuracy measures among the S-CCC, S-DCC, and S-BEKK models for all cases. In contrast, the NS-CCC, NS-DCC, and NS-BEKK models seem to have the same VaR forecast performance because, for these three models, the corresponding total numbers of portfolios that pass the three accuracy measures are almost equal. Third, the S-DCC model has the best VaR forecast performance because this model has the largest total number of portfolios that pass the three accuracy measures among all seven bivariate GARCH models for all cases. Fourth, both the NS-ADCC and NS-DCC models have almost the same VaR forecast performance because, for these two models, the corresponding total numbers of portfolios that pass the three accuracy measures are almost equal.

#### 5.3.2. Can the Component Combinations of Portfolios Affect the Performance of VaR Forecasts?

_{uc}(respectively, LR

_{cc}) underneath the 90% level in panel A of Table 10. Conversely, the numbers in column Sum in panel C in Table 9 are recorded in column DQ underneath the 90% level in panel A of Table 10. Table 10 lists all the summary results of the out-of-sample VaR forecast performance for three weight combinations of currency-stock-based portfolios. Then we execute four groups of the model performance comparisons imitating the same process performed in Table 8. Before we perform this analysis, we find a specific phenomenon existing in Table 10 compared to Table 8. For example, the total numbers of portfolios that pass three accuracy measures are almost equal for most cases, such as all four levels of panel A; the 99% and 99.5% levels of panel B; the 90% and 95% levels of panel B except for the S-CCC model; and the 90%, 95%; and 99.5% levels of panel C except for the S-CCC model. Taking an example of the 90% level of panel A, the total numbers of portfolios that pass the three accuracy measures are 17, 18, 20, 20, 19, 20, and 21 for the S-CCC, NS-CCC, S-DCC, NS-DCC, S-BEKK, NS-BEKK, and NS-ADCC models, respectively. This phenomenon indicates that all seven models seem to have the same VaR forecast performance because the seven numbers are almost the same.

#### 5.3.3. Efficiency Evaluation Test via Market Risk Capital

_{uc}, LR

_{cc}, and DQ) that were executed previously. On the other hand, the second stage includes an efficiency test based on specific loss functions such as the MRC loss function. To repeat, the MRC is the amount of regulatory capital a bank must hold with respect to its market risk exposure. Regarding the above efficiency test, we use the superior predictive ability (SPA) test by Hansen (2005). The null hypothesis of this test is that none of the models is better than the benchmark. If the p-value of this test statistic is greater than the 10% level, then the null hypothesis is accepted, or the benchmark model shows better performance than the other competing models. In the SPA test, each competing model has to take turns being the benchmark model. Hence, we perform this test seven times for the seven bivariate GARCH models.

_{1}= 0.5, w

_{2}= 0.5” underneath Stock portfolios in Table 12. With the same inference process, the total numbers of portfolios that pass the SPA test for the S-DCC, NS-DCC, NS-BEKK, and NS-ADCC models respectively are 5, 0, 7, and 1 when taking seven currency-stock portfolios as a whole. These results are recorded in the column Sum of panel B in Table 11 and also summarized in column “w

_{1}= 0.5, w

_{2}= 0.5” underneath Currency-stock portfolios in Table 12. In Table 11, we also show that when a model passes the SPA test, the corresponding MRC is almost the smallest among all competing models’ MRCs.

_{1}= 0.5, w

_{2}= 0.5” underneath Stock portfolios in Table 12, the total number of portfolios that pass the SPA test is equal to 21 for the NS-BEKK model. This number, 21, is the largest among the 8 (the S-DCC), 2 (the NS-DCC), 21 (the NS-BEKK), and 2 (the NS-ADCC). These results indicate that irrespective of the weight combination of the portfolios, the NS-BEKK is the most suitable model to be used in the stock- and currency-stock-based portfolio by the bank risk manager. In addition, we are surprised that the NS-BEKK model is selected from the efficiency evaluation test for both the stock- and currency-stock-based portfolios, whereas the S-DCC model is chosen from the accuracy tests only for the stock-based portfolio. The reason we guess for this is that the three accuracy tests are based on the one-day VaR, whereas the efficiency evaluation test is based on the MRC that depends on the 10-day VaR at the 99% confidence level and the MRA’s multiplication factor. As to the other reasons, they are left for future investigations.

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Baba, Yoshi, Robert F. Engle, Dennis Kraft, and Kenneth F. Kroner. 1990. Multivariate Simultaneous Generalized ARCH. Oakland: University of California, unpublished manuscript. [Google Scholar]
- Baldi, Lucia, Massimo Peri, and Daniela Vandone. 2016. Stock markets’ bubbles burst and volatility spillovers in agricultural commodity markets. Research in International Business and Finance 38: 277–85. [Google Scholar] [CrossRef]
- Bams, Dennis, Gildas Blanchard, and Thorsten Lehnert. 2017. Volatility measures and Value-at-Risk. International Journal of Forecasting 33: 848–63. [Google Scholar] [CrossRef]
- Bauwens, Luc, Sebastien Laurent, and Jeroen V. K. Rombouts. 2006. Multivariate GARCH models: A survey. Journal of Applied Econometrics 21: 79–109. [Google Scholar] [CrossRef]
- Bayer, Sebastian. 2018. Combining Value-at-Risk forecasts using penalized quantile regressions. Econometrics and Statistics 8: 56–77. [Google Scholar] [CrossRef]
- Bollerslev, Tim. 1986. Generalized autoregressive conditional heteroscedasticity. Journal of Econometrics 31: 307–27. [Google Scholar] [CrossRef]
- Bollerslev, Tim. 1990. Modeling the coherence in short-run nominal exchange rates: A multivariate generalized ARCH model. Review of Economics and Statistics 72: 498–505. [Google Scholar] [CrossRef]
- Caporale, Guglielmo Maria, John Hunter, and Faek Menla Ali. 2014. On the linkages between stock prices and exchange rates: Evidence from the banking crisis of 2007–2010. International Review of Financial Analysis 33: 87–103. [Google Scholar] [CrossRef] [Green Version]
- Cappiello, Lorenzo, Robert F. Engle, and Kevin Sheppard. 2006. Asymmetric Dynamics in the Correlations of Global Equity and Bond Returns. Journal of Financial Econometrics 4: 537–72. [Google Scholar] [CrossRef]
- Chang, Chia-Lin, Hui-Kuang Hsu, and Michael McAleer. 2013. Is small beautiful? Size effects of volatility spillovers for firm performance and exchange rates in tourism. North American Journal of Economics and Finance 26: 519–34. [Google Scholar] [CrossRef]
- Christoffersen, Peter. 1998. Evaluating interval forecasts. International Economic Review 39: 841–62. [Google Scholar] [CrossRef]
- Engle, Robert. 2002. Dynamic conditional correlation: A simple class of multivariate generalized autoregressive conditional heteroskedasticity models. Journal of Business and Economic Statistics 20: 339–50. [Google Scholar] [CrossRef]
- Engle, Robert, and Kenneth F. Kroner. 1995. Multivariate simultaneous GARCH. Econometric Theory 11: 122–50. [Google Scholar] [CrossRef]
- Engle, Robert F., and Simone Manganelli. 2004. CAViaR: Conditional autoregressive value at risk by regression quantiles. Journal of Business and Economic Statistics 22: 367–81. [Google Scholar] [CrossRef]
- Hansen, Peter Reinhard. 2005. A test for superior predictive ability. Journal of Business & Economic Statistics 23: 365–80. [Google Scholar]
- Hansen, Peter R., and Asger Lunde. 2005. A forecast comparison of volatility models: Does anything beat a GARCH(1,1)? Journal of Applied Econometrics 20: 873–89. [Google Scholar] [CrossRef]
- Jarque, Carlos M., and Anil K. Bera. 1987. A test for normality of observations and regression residuals. International Statistics Review 55: 163–72. [Google Scholar] [CrossRef]
- Jorion, Philippe. 2000. Value at Risk: The New Benchmark for Managing Financial Risk. New York: McGraw-Hill. [Google Scholar]
- Kupiec, Paul H. 1995. Techniques for verifying the accuracy of risk measurement models. Journal of Derivatives 3: 73–84. [Google Scholar] [CrossRef]
- Laporta, Alessandro G., Luca Merlo, and Lea Petrella. 2018. Selection of Value at Risk models for energy commodities. Energy Economics 74: 628–43. [Google Scholar] [CrossRef]
- Lee, Cheng-Few, and Jung-Bin Su. 2012. Alternative statistical distributions for estimating Value-at-Risk: Theory and evidence. Review of Quantitative Finance and Accounting 39: 309–31. [Google Scholar] [CrossRef]
- Li, Hong. 2012. The impact of China’s stock market reforms on its international stock market linkages. Quarterly Review of Economics and Finance 52: 358–68. [Google Scholar] [CrossRef]
- Lin, Boqiang, and Jianglong Li. 2015. The spillover effects across natural gas and oil markets: Based on the VEC-MGARCH framework. Applied Energy 155: 229–41. [Google Scholar] [CrossRef]
- Liu, Xueyong, Haizhong An, Shupei Huang, and Shaobo Wen. 2017. The evolution of spillover effects between oil and stock markets across multi-scales using a wavelet-based GARCH–BEKK model. Physica A 465: 374–83. [Google Scholar] [CrossRef]
- Lopez, Jose A. 1999. Regulatory evaluation of Value-at-Risk Models. Journal of Risk 1: 37–64. [Google Scholar] [CrossRef]
- McAleer, Michael. 2018. Stationarity and invertibility of a dynamic correlation matrix. Kybernetika 54: 363–74. [Google Scholar] [CrossRef]
- Moore, Tomoe, and Ping Wang. 2014. Dynamic linkage between real exchange rates and stock prices: Evidence from developed and emerging Asian markets. International Review of Economics and Finance 29: 1–11. [Google Scholar] [CrossRef] [Green Version]
- Moschini, Gian Carlo, and Robert J. Myers. 2002. Testing for constant hedge ratios in commodity markets: A multivariate GARCH approach. Journal of Empirical Finance 9: 589–603. [Google Scholar] [CrossRef]
- Ozkan, Ibrahim, and Lutfi Erden. 2015. Time-varying nature and macroeconomic determinants of exchange rate pass-through. International Review of Economics & Finance 38: 56–66. [Google Scholar]
- Politis, Dimitris N., and Joseph P. Romano. 1994. The stationary bootstrap. Journal of the American Statistical Association 89: 1303–13. [Google Scholar] [CrossRef]
- Sarma, Mandira, Susan Thomas, and Ajay Shah. 2003. Selection of value-at-risk models. Journal of Forecasting 22: 337–58. [Google Scholar] [CrossRef]
- Silvennoinen, Annastiina, and Timo Teräsvirta. 2009. Multivariate GARCH Models. Handbook of Financial Time Series. New York: Springer, pp. 201–29. [Google Scholar]
- Su, Jung-Bin. 2014a. How to mitigate the impact of inappropriate distributional settings when the parametric value-at-risk approach is used? Quantitative Finance 14: 305–25. [Google Scholar] [CrossRef]
- Su, Jung-Bin. 2014b. Empirical analysis of long memory, leverage, and distribution effects for stock market risk estimates. North American Journal of Economics and Finance 30: 1–39. [Google Scholar] [CrossRef]
- Su, Jung-Bin. 2014c. The interrelation of stock markets in China, Taiwan and Hong Kong and their constructional portfolio’s value-at-risk estimate. Journal of Risk Model Validation 8: 69–127. [Google Scholar] [CrossRef]
- Su, Jung-Bin. 2015. Value-at-risk estimates of the stock indices in developed and emerging markets including the spillover effects of currency market. Economic Modelling 46: 204–24. [Google Scholar] [CrossRef]
- Su, Jung-Bin, and Jui-Cheng Hung. 2011. Empirical analysis of jump dynamics, heavy-tails and skewness on value-at-risk estimation. Economic Modelling 28: 1117–30. [Google Scholar] [CrossRef]
- Su, Jung-Bin, Ming-Chih Lee, and Chien-Liang Chiu. 2014. Why does skewness and the fat-tail effect influence value-at-risk estimates? Evidence from alternative capital markets. International Review of Economics & Finance 31: 59–85. [Google Scholar]
- Tamakoshi, Go, and Shigeyuki Hamori. 2014. Co-movements among major European exchange rates: A multivariate time-varying asymmetric approach. International Review of Economics and Finance 31: 105–13. [Google Scholar] [CrossRef]
- Turhan, M. Ibrahim, Ahmet Sensoy, Kevser Ozturk, and Erk Hacihasanoglu. 2014. A view to the long-run dynamic relationship between crude oil and the major asset classes. International Review of Economics & Finance 33: 286–99. [Google Scholar]
- Wang, Ping, and Peijie Wang. 2010. Price and volatility spillovers between the Greater China Markets and the developed markets of US and Japan. Global Finance Journal 21: 304–17. [Google Scholar] [CrossRef]
- Wang, Kai-Li, Jau-Rong Li, and Meng-Jou Hsiao. 2010. A further investigation of dynamic information transmission between DRs and their underlying stocks with exchange rate threshold effects. Journal of Financial Studies 18: 93–130. [Google Scholar]
- Weber, Enzo. 2013. Simultaneous stochastic volatility transmission across American equity markets. Quarterly Review of Economics and Finance 53: 53–60. [Google Scholar] [CrossRef] [Green Version]
- Yaya, OlaOluwa S., Mohammed M. Tumala, and Christopher G. Udomboso. 2016. Volatility persistence and returns spillovers between oil and gold prices: Analysis before and after the global financial crisis. Resources Policy 49: 273–81. [Google Scholar] [CrossRef]
- Yu, Wenhua, Kun Yang, Yu Wei, and Likun Lei. 2018. Measuring Value-at-Risk and Expected Shortfall of crude oil portfolio using extreme value theory and vine copula. Physica A 490: 1423–33. [Google Scholar] [CrossRef]

1 | Owing to the wide application of the DCC model, McAleer (2018) derived the stationarity and invertibility conditions of the DCC model in order to provide a solid statistical foundation for the estimates of the DCC parameters. |

2 | For more details about these two types of models, please see Bauwens et al. (2006) and Silvennoinen and Teräsvirta (2009). |

3 | The parameters of the standard CCC and DCC models are estimated by the GARCH instruction provided by the Rats 6.0 program. The parameters of these models are estimated only by one step compared with the two steps’ CCC and DCC models. |

4 | In a real case, if an institution wants to evaluate the operation performance of several fund managers that respectively have different values of assets measured with different currencies, indicating that it is hard to evaluate their operation performance when ‘the VaR expressed in actual monetary value’ is used. However, it is easy to evaluate their performance as ‘the VaR expressed in return’ is utilized since the return is dimensionless. Notably, we can convert the above expression via the following equation. ‘The VaR expressed in actual monetary value’ = ‘the VaR expressed in return’ * the value of asset’s position. Taking an example to illustrate it, if the value of an asset is USD 1000, and its VaR expressed in return is 1.4091%, then ‘the VaR expressed in actual monetary value’ is equal to USD 14.091 (=1.4091%* USD1000). |

5 | The BEKK model is named after Baba et al. (1990). |

6 | The parameters of the standard BEKK model are estimated by the GARCH instruction provided by the Rats 6.0 program. The parameters of these models are estimated only by one step. This approach is the same as the standard CCC and DCC models mentioned above. |

7 | According to Bauwens et al. (2006), there are three non-mutually exclusive approaches to construct multivariate GARCH models: (i) direct generalizations of the univariate GARCH model of Bollerslev (1986); (ii) linear combinations of univariate GARCH models; and (iii) nonlinear combinations of univariate GARCH models. Notably, both the VEC and BEKK models belong to the above first approach. In the general VEC model, each element of the conditional variance matrix (${H}_{\mathrm{t}}$) is a linear function of the lagged squared errors and cross-products of errors and lagged values of the elements of ${H}_{\mathrm{t}}$. The BEKK model is a special case of the VEC model. Hence, the number of parameters in the BEKK model is less than that in the VEC model. For example, the numbers of parameters in the VEC(1,1) and BEKK(1,1,1) models are $\mathrm{n}\left(\mathrm{n}+1\right)\left[\mathrm{n}\left(\mathrm{n}+1\right)+1\right]/2$ and $\mathrm{n}\left(5\mathrm{n}+1\right)/2$, respectively. The BEKK(1,1,1) model is expressed as Equation (10) in this study. |

8 | Please see the Proposition 2.1 of Engle and Kroner (1995) for more details. |

9 | Regarding the seven stock indices, the total number of bi-component portfolios can be calculated by ${C}_{2}^{7}=21$. |

10 | For each pair of data, they are retained for the same trade date and are deleted otherwise. Taking the Ny-Da pair of data as an example, both NYSE and DAX are traded on 31 January 2002, thus the close prices of both data are retained on this date. Conversely, if only NYSE is traded on 25 May 2003, whereas DAX is not traded on this date, then the close price of NYSE on this date must be deleted, and vice versa. |

11 | When we conduct a hypothesis test there are two kinds of errors: type I and type II errors. Briefly, type I errors happen when we reject a true null hypothesis whereas type II errors happen when we fail to reject a false null hypothesis. Although the errors cannot be completely eliminated, we can minimize one type of error. However, when we try to decrease the probability of one type of error, the probability for the other type increases. The only way to decrease these two types of errors is to increase the sample size. Thus, in this study, we set the sample size of the estimation (respectively, forecast) period as 3300 (respectively, 500). They are large enough in order to decrease type I and type II errors as much as we can. |

12 | The out-of-sample VaR forecast is executed via a rolling window approach. That is, the seven bivariate GARCH models are estimated for each of 28 pair-wise data series, with a sample of 3300 daily returns, and then a one-day-ahead VaR forecast of the bi-component portfolio for the next period is obtained. Subsequently, the estimation period is then rolled forward by adding one new day and dropping the most distant day. Via repeating this procedure, the out-of-sample VaR forecasts are computed for the next 500 days. |

13 | Due to the limited space, the empirical results of the other 22 bi-component portfolios for the NS-ADCC model, and the empirical results for the other six bivariate GARCH models (i.e. the S-CCC, NS-CSS, S-DCC, NS-DCC, S-BEKK, and NS-BEKK models) are all omitted here and are available upon request. |

14 | The mean VaR is the average of all the VaR values over the out-of-sample period, and can be calculated by the following equation: $\mathrm{Mean}\text{}\mathrm{VaR}=\left({\sum}_{\mathrm{t}=1}^{500}{\mathrm{LVaR}}_{\mathrm{p},\mathrm{t}}\right)/500$, where ${\mathrm{LVaR}}_{\mathrm{p},\mathrm{t}}$ denotes the value of the portfolio’s VaR at time t, and can be calculated by Equation (8) or Equation (16). The sample size of the out-of-sample period is equal to 500 in this study. |

15 | Actually, it is very hard to compete against the models’ forecasting performance via the failure rate since it cannot provide the significance level for the obtained conclusion. Owing to the above reason, the forecasting performance comparison of alternative models based on the failure rate is listed in the section of ‘Preliminary analysis of average VaR performance’. |

16 | Notably, the failure rate and mean VaR is regarded as the preliminary analysis of the average VaR performance. They cannot provide precise results. Moreover, due to the limited space, the detailed results of the VaR forecasting performance at the 99% level based on failure rate are omitted here and are available upon request. However, the summary results of this level are also listed in Table 5. |

17 | Due to the limited space, the detailed results of the VaR forecasting performance at the 99% level based on mean VaR are omitted here and are available upon request. However, the summary results of this level are also listed in Table 5. |

18 | Due to the limited space, the detailed results of the VaR forecasting performance at the other three levels (90%, 99%, and 99.5%) based on the LR _{uc}, LR_{cc}, DQ tests are omitted here and are available upon request. However, the summary results of these three levels are also listed in Table 7. |

19 | Even if the total number of passing three accuracy tests is zero for the NS-ADCC model at panel A of Table 8 ($\mathrm{i}.\mathrm{e}.,{\text{}\mathrm{w}}_{1}=50\%,{\text{}\mathrm{and}\text{}\mathrm{w}}_{2}=50\%$ weight combination of stock portfolio), we considered this model since the NS-DCC model is the special case of NS-ADCC model. |

20 | Due to the limited space, the results of the efficiency evaluation test based on MRC for the other two weight combinations stock and currency-stock portfolios are omitted here and are available upon request. However, the summary results of the above results are also listed in Table 12. |

Mean | Std. Dev. | Max. | Min. | Skewness | Kurtosis | J-B | Q^{2} (24) | |
---|---|---|---|---|---|---|---|---|

NYSE | 0.0089 | 1.2783 | 11.5257 | −10.232 | −0.2954 ^{c} | 9.169 ^{c} | 13,370.5 ^{c} | 7455.9 ^{c} |

S&P500 | 0.0074 | 1.2701 | 10.9571 | −9.4695 | −0.1789 ^{c} | 8.1254 ^{c} | 10,476.6 ^{c} | 6366.9 ^{c} |

Nasdaq | 0.0042 | 1.5947 | 11.1594 | −9.5876 | 0.0327 | 4.6001 ^{c} | 3352.0 ^{c} | 5291.3 ^{c} |

CAC40 | −0.0100 | 1.5396 | 10.5945 | −9.4715 | 0.0105 | 4.6143 ^{c} | 3372.2 ^{c} | 3455.4 ^{c} |

DAX | 0.0079 | 1.5817 | 10.7974 | −9.5756 | −0.0399 | 4.4070 ^{c} | 3076.9 ^{c} | 3645.6 ^{c} |

FTSE | −0.0015 | 1.2463 | 9.3842 | −9.2645 | −0.0890 ^{b} | 5.9661 ^{c} | 5642.3 ^{c} | 4986.8 ^{c} |

SMI | −0.0011 | 1.2668 | 10.7876 | −10.518 | −0.1255 ^{c} | 8.2662 ^{c} | 10,831.7 ^{c} | 2591.4 ^{c} |

UDI | −0.0028 | 0.4755 | 2.1552 | −4.1066 | −0.2346 ^{c} | 3.3553 ^{c} | 1817.9 ^{c} | 1062.7 ^{c} |

Ny-Sp | Ny-Na | Ny-Ca | Ny-Da | Ny-Ft | Ny-Sm | |
---|---|---|---|---|---|---|

Panel A. The univariate GARCH(1,1) model for the first component stock index | ||||||

${\mathsf{\mu}}_{1}$ | 0.0505 (0.015) ^{c} | 0.0505 (0.015) ^{c} | 0.0505 (0.015) ^{c} | 0.0505 (0.015) ^{c} | 0.0505 (0.015) ^{c} | 0.0505 (0.015) ^{c} |

${\mathsf{\omega}}_{1}$ | 0.0178 (0.003) ^{c} | 0.0178 (0.003) ^{c} | 0.0178 (0.003) ^{c} | 0.0178 (0.003) ^{c} | 0.0178 (0.003) ^{c} | 0.0178 (0.003) ^{c} |

${\mathsf{\alpha}}_{1}$ | 0.0886 (0.003) ^{c} | 0.0886 (0.003) ^{c} | 0.0886 (0.003) ^{c} | 0.0886 (0.003) ^{c} | 0.0886 (0.003) ^{c} | 0.0886 (0.003) ^{c} |

${\mathsf{\beta}}_{1}$ | 0.8986 (0.006) ^{c} | 0.8986 (0.006) ^{c} | 0.8986 (0.006) ^{c} | 0.8986 (0.006) ^{c} | 0.8986 (0.006) ^{c} | 0.8986 (0.006) ^{c} |

${\mathrm{Q}}_{1}^{2}\left(24\right)$ | 39.228 ^{b} | 39.228 ^{b} | 39.228 ^{b} | 39.228 ^{b} | 39.228 ^{b} | 39.228 ^{b} |

LL_{1} | −4800.17 | −4800.17 | −4800.17 | −4800.17 | −4800.17 | −4800.17 |

Panel B. The univariate GARCH(1,1) model for the second component stock index | ||||||

${\mathsf{\mu}}_{2}$ | 0.0491 (0.015) ^{c} | 0.0656 (0.018) ^{c} | 0.0503 (0.018) ^{c} | 0.0811 (0.018) ^{c} | 0.0407 (0.014) ^{c} | 0.0512 (0.016) ^{c} |

${\mathsf{\omega}}_{2}$ | 0.0168 (0.001) ^{c} | 0.0174 (0.001) ^{c} | 0.0220 (0.001) ^{c} | 0.0239 (0.002) ^{c} | 0.0132 (0.001) ^{c} | 0.0387 (0.001) ^{c} |

${\mathsf{\alpha}}_{2}$ | 0.0880 (0.002) ^{c} | 0.0758 (0.001) ^{c} | 0.0872 (0.002) ^{c} | 0.0926 (0.002) ^{c} | 0.0945 (0.002) ^{c} | 0.1242 (0.002) ^{c} |

${\mathsf{\beta}}_{2}$ | 0.9000 (0.001) ^{c} | 0.9163 (0.001) ^{c} | 0.9040 (0.001) ^{c} | 0.8982 (0.001) ^{c} | 0.8972 (0.002) ^{c} | 0.8511 (0.001) ^{c} |

${\mathrm{Q}}_{2}^{2}\left(24\right)$ | 38.818 ^{b} | 44.580 ^{c} | 30.213 | 26.734 | 30.074 | 8.831 |

LL_{2} | −4803.92 | −5598.51 | −5557.72 | −5588.51 | −4751.05 | −4833.02 |

Panel C. The conditional correlation matrix equation | ||||||

$\overline{\mathrm{a}}$ | 0.2041 (0.009) ^{c} | 0.1925 (0.019) ^{c} | 0.1000 (0.011) ^{c} | 0.0715 (4 × 10 ^{−8}) ^{c} | 0.1451 (0.000) ^{c} | −0.101 (5 × 10 ^{−10}) ^{c} |

$\overline{\mathrm{b}}$ | 0.9738 (0.002) ^{c} | 0.9782 (0.003) ^{c} | 0.9939 (0.001) ^{c} | 0.9934 (3 × 10 ^{−8}) ^{c} | 0.9619 (1 × 10 ^{−10}) ^{c} | 0.6633 (1 × 10 ^{−9}) ^{c} |

$\overline{\mathrm{g}}$ | −2 × 10^{−5}(3 × 10 ^{−5}) | 0.0701 (0.064) | −9 × 10^{−6}(0.035) | 0.1044 (2 × 10 ^{−11}) ^{c} | −1 × 10^{−5}(0.000) ^{c} | −1.3 × 10^{−4}(0.0) ^{c} |

${\mathsf{\rho}}_{12}$ | 0.9746 (0.014) | 0.8748 (0.059) | 0.6219 (0.093) | 0.6326 (0.090) | 0.5769 (0.051) | 0.4944 (0.011) |

LL_{3} | −4386.75 | −7768.51 | −9449.89 | −9428.01 | −8758.57 | −9078.06 |

_{1}and LL

_{2}respectively indicate the log-likelihood value for two independent univariate GARCH equations whereas LL

_{3}denotes the log-likelihood value for the bivariate ADCC equation. (5) ${\mathrm{Q}}_{1}^{2}\left(24\right)$ and ${\mathrm{Q}}_{2}^{2}\left(24\right)$ respectively denote the Ljung-Box Q test for the 24th order serial correlation of the squared returns for the first and second component stock indices of an equal weight bi-component portfolio. (6) ${\mathsf{\rho}}_{12}$ is the mean correlation between two component stock indices of an equal weight bi-component portfolio during the first estimate period.

Ny-Sp | Ny-Na | Ny-Ca | Ny-Da | Ny-Ft | Ny-Sm | Sp-Na | S1 | S2 | S3 | S4 | |

S-CCC | 0.056 | 0.072 | 0.078 | 0.086 | 0.082 | 0.068 | 0.072 | 2 | 0 | 0 | - |

NS-CCC | 0.064 | 0.072 | 0.078 | 0.084 | 0.080 | 0.068 | 0.074 | 2 | 3 | 0 | - |

S-DCC | 0.050 | 0.058 | 0.066 | 0.060 | 0.060 | 0.050 | 0.058 | 7 | 7 | 7 | - |

NS-DCC | 0.064 | 0.072 | 0.074 | 0.084 | 0.074 | 0.072 | 0.074 | 0 | 4 | 0 | 2 |

S-BEKK | 0.064 | 0.072 | 0.080 | 0.074 | 0.082 | 0.072 | 0.078 | 2 | 0 | 0 | - |

NS-BEKK | 0.066 | 0.072 | 0.078 | 0.086 | 0.078 | 0.072 | 0.072 | 3 | 1 | 0 | - |

NS-ADCC | 0.064 | 0.072 | 0.076 | 0.084 | 0.076 | 0.068 | 0.074 | - | - | - | 1 |

Sp-Ca | Sp-Da | Sp-Ft | Sp-Sm | Na-Ca | Na-Da | Na-Ft | |||||

S-CCC | 0.092 | 0.078 | 0.090 | 0.074 | 0.080 | 0.074 | 0.088 | 0 | 0 | 0 | - |

NS-CCC | 0.080 | 0.076 | 0.084 | 0.070 | 0.080 | 0.072 | 0.088 | 5 | 2 | 0 | - |

S-DCC | 0.060 | 0.058 | 0.074 | 0.058 | 0.058 | 0.070 | 0.064 | 7 | 6 | 6 | - |

NS-DCC | 0.078 | 0.074 | 0.082 | 0.070 | 0.076 | 0.072 | 0.086 | 0 | 5 | 0 | 1 |

S-BEKK | 0.076 | 0.068 | 0.090 | 0.074 | 0.078 | 0.068 | 0.082 | 3 | 1 | 1 | - |

NS-BEKK | 0.080 | 0.078 | 0.080 | 0.072 | 0.076 | 0.076 | 0.080 | 4 | 3 | 0 | - |

NS-ADCC | 0.080 | 0.074 | 0.082 | 0.070 | 0.076 | 0.070 | 0.082 | - | - | - | 2 |

Na-Sm | Ca-Da | Ca-Ft | Ca-Sm | Da-Ft | Da-Sm | Ft-Sm | |||||

S-CCC | 0.072 | 0.074 | 0.068 | 0.060 | 0.072 | 0.066 | 0.072 | 2 | 1 | 1 | - |

NS-CCC | 0.072 | 0.066 | 0.062 | 0.064 | 0.070 | 0.068 | 0.066 | 4 | 2 | 0 | - |

S-DCC | 0.042 * | 0.060 | 0.062 | 0.046 * | 0.058 | 0.068 | 0.052 | 5 | 5 | 5 | - |

NS-DCC | 0.072 | 0.064 | 0.062 | 0.060 | 0.072 | 0.068 | 0.068 | 0 | 2 | 0 | 1 |

S-BEKK | 0.074 | 0.070 | 0.058 | 0.062 | 0.064 | 0.066 | 0.064 | 4 | 2 | 2 | - |

NS-BEKK | 0.068 | 0.064 | 0.062 | 0.060 | 0.074 | 0.068 | 0.066 | 3 | 4 | 0 | - |

NS-ADCC | 0.074 | 0.064 | 0.062 | 0.060 | 0.072 | 0.068 | 0.068 | - | - | - | 0 |

Ny-Sp | Ny-Na | Ny-Ca | Ny-Da | Ny-Ft | Ny-Sm | Sp-Na | S1 | S2 | S3 | S4 | |

S-CCC | −1.4091 | −1.4483 | −1.5124 | −1.5304 | −1.2833 | −1.3269 | −1.4743 | 0 | 0 | 0 | - |

NS-CCC | −1.4098 | −1.4657 | −1.5443 | −1.5693 | −1.3090 | −1.3493 | −1.4983 | 7 | 1 | 0 | - |

S-DCC | −1.5538 | −1.5421 | −1.6369 | −1.7554 | −1.3867 | −1.4296 | −1.5800 | 7 | 7 | 7 | - |

NS-DCC | −1.4102 | −1.4910 | −1.5578 | −1.5693 | −1.3243 | −1.3507 | −1.5198 | 0 | 7 | 0 | 4 |

S-BEKK | −1.3908 | −1.4551 | −1.5733 | −1.5852 | −1.3034 | −1.3448 | −1.4913 | 4 | 0 | 0 | - |

NS-BEKK | −1.3820 | −1.4664 | −1.5387 | −1.5411 | −1.3126 | −1.3253 | −1.4967 | 3 | 0 | 0 | - |

NS-ADCC | −1.4102 | −1.4909 | −1.5551 | −1.5693 | −1.3234 | −1.3479 | −1.5198 | - | - | - | 0 |

Sp-Ca | Sp-Da | Sp-Ft | Sp-Sm | Na-Ca | Na-Da | Na-Ft | |||||

S-CCC | −1.4932 | −1.5176 | −1.2668 | −1.3117 | −1.5738 | −1.6023 | −1.3522 | 0 | 0 | 0 | - |

NS-CCC | −1.5298 | −1.5596 | −1.2943 | −1.3374 | −1.6024 | −1.6381 | −1.3720 | 7 | 1 | 0 | - |

S-DCC | −1.5915 | −1.6840 | −1.3337 | −1.4311 | −1.7643 | −1.7069 | −1.4406 | 7 | 7 | 7 | - |

NS-DCC | −1.5432 | −1.5586 | −1.3081 | −1.3427 | −1.6171 | −1.6402 | −1.3913 | 0 | 6 | 0 | 4 |

S-BEKK | −1.5550 | −1.5769 | −1.2827 | −1.3176 | −1.6229 | −1.6598 | −1.3950 | 6 | 0 | 0 | - |

NS-BEKK | −1.5219 | −1.5321 | −1.2919 | −1.3174 | −1.6058 | −1.6177 | −1.3758 | 1 | 0 | 0 | - |

NS-ADCC | −1.5415 | −1.5611 | −1.3061 | −1.3380 | −1.6178 | −1.6425 | −1.3865 | - | - | - | 3 |

Na-Sm | Ca-Da | Ca-Ft | Ca-Sm | Da-Ft | Da-Sm | Ft-Sm | |||||

S-CCC | −1.3946 | −1.9465 | −1.6814 | −1.7286 | −1.6709 | −1.7457 | −1.4840 | 0 | 0 | 0 | - |

NS-CCC | −1.4078 | −2.0114 | −1.7267 | −1.7713 | −1.7241 | −1.7910 | −1.5147 | 7 | 0 | 0 | - |

S-DCC | −1.6290 | −2.0847 | −1.7812 | −2.0290 | −1.9297 | −1.7918 | −1.6887 | 6 | 7 | 6 | - |

NS-DCC | −1.4211 | −2.0574 | −1.7398 | −1.7773 | −1.7435 | −1.7951 | −1.5154 | 1 | 7 | 1 | 2 |

S-BEKK | −1.3816 | −2.0203 | −1.7504 | −1.7624 | −1.7478 | −1.7912 | −1.5184 | 6 | 0 | 0 | - |

NS-BEKK | −1.4024 | −2.0156 | −1.7250 | −1.7125 | −1.7332 | −1.7133 | −1.4849 | 1 | 0 | 0 | - |

NS-ADCC | −1.4111 | −2.0557 | −1.7398 | −1.7773 | −1.7435 | −1.7951 | −1.5154 | - | - | - | 0 |

**Table 5.**The summary results for the out-of-sample VaR forecasts performance of equal-weight stock portfolios based on the mean VaR and failure rate.

Panel A. Failure Rate | ||||||||||||||||||||||||||||||||||||

S1 | S2 | S3 | S4 | |||||||||||||||||||||||||||||||||

95% Level | S_{1,95} | 99% Level | S_{1,99} | SS1 | 95% Level | S_{2,95} | 99% Level | S_{2,99} | SS2 | 95% Level | S_{3,95} | 99% Level | S_{3,99} | SS3 | 95% Level | S_{4,95} | 99% Level | S_{4,99} | SS4 | |||||||||||||||||

S-CCC | 2 | 0 | 2 | 4 | 1 | 1 | 0 | 2 | 6 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | - | - | - | - | - | - | - | - | - |

NS-CCC | 2 | 5 | 4 | 11 | 1 | 2 | 5 | 8 | 19 | 3 | 2 | 2 | 7 | 2 | 2 | 4 | 8 | 15 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | - | - | - | - | - | - | - | - | - |

S-DCC | 7 | 7 | 5 | 19 | 5 | 7 | 5 | 17 | 36 | 7 | 6 | 5 | 18 | 5 | 6 | 5 | 16 | 34 | 7 | 6 | 5 | 18 | 5 | 6 | 5 | 16 | 34 | - | - | - | - | - | - | - | - | - |

NS-DCC | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 4 | 5 | 2 | 11 | 2 | 1 | 2 | 5 | 16 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 2 | 1 | 1 | 4 | 1 | 0 | 1 | 2 | 6 |

S-BEKK | 2 | 3 | 4 | 9 | 2 | 4 | 5 | 11 | 20 | 0 | 1 | 2 | 3 | 4 | 1 | 3 | 8 | 11 | 0 | 1 | 2 | 3 | 4 | 1 | 3 | 8 | 11 | - | - | - | - | - | - | - | - | - |

NS-BEKK | 3 | 4 | 3 | 10 | 1 | 2 | 2 | 5 | 15 | 1 | 3 | 4 | 8 | 3 | 3 | 2 | 8 | 16 | 0 | 0 | 0 | 0 | 3 | 2 | 0 | 5 | 5 | - | - | - | - | - | - | - | - | - |

NS-ADCC | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | 1 | 2 | 0 | 3 | 0 | 0 | 0 | 0 | 3 |

Panel B. Mean VaR | ||||||||||||||||||||||||||||||||||||

S1 | S2 | S3 | S4 | |||||||||||||||||||||||||||||||||

95% Level | S_{1,95} | 99% Level | S_{1,99} | SS1 | 95% Level | S_{2,95} | 99% Level | S_{2,99} | SS2 | 95% Level | S_{3,95} | 99% Level | S_{3,99} | SS3 | 95% Level | S_{4,95} | 99% Level | S_{4,99} | SS4 | |||||||||||||||||

S-CCC | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | - | - | - | - | - | - | - | - | - |

NS-CCC | 7 | 7 | 7 | 21 | 6 | 7 | 7 | 20 | 41 | 1 | 1 | 0 | 2 | 1 | 1 | 0 | 2 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | - | - | - | - | - | - | - | - | - |

S-DCC | 7 | 7 | 6 | 20 | 7 | 7 | 6 | 20 | 40 | 7 | 7 | 7 | 21 | 7 | 7 | 6 | 20 | 41 | 7 | 7 | 6 | 20 | 7 | 7 | 6 | 20 | 40 | - | - | - | - | - | - | - | - | - |

NS-DCC | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 2 | 7 | 6 | 7 | 20 | 7 | 6 | 7 | 20 | 40 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 2 | 4 | 4 | 2 | 10 | 4 | 4 | 2 | 10 | 20 |

S-BEKK | 4 | 6 | 6 | 16 | 4 | 5 | 6 | 15 | 31 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | - | - | - | - | - | - | - | - | - |

NS-BEKK | 3 | 1 | 1 | 5 | 3 | 2 | 1 | 6 | 11 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | - | - | - | - | - | - | - | - | - |

NS-ADCC | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | 0 | 3 | 0 | 3 | 2 | 3 | 0 | 5 | 8 |

_{1,95}and S

_{1,99}underneath S1 at panel A (respectively, panel B), respectively, denote the total sum of three corresponding numbers in the columns 95% and 99% underneath S1 at panel A (respectively, panel B) and the numbers in the column SS1 underneath S1 at panel A (respectively, panel B) denote the total sum of two corresponding numbers in the columns S

_{1,95}and S

_{1,99}underneath S1 at panel A (respectively, panel B). (5) In the same inference process, the numbers in the column SS2 underneath S2 at panel A (respectively, panel B) denote the total sum of two corresponding numbers in the columns S

_{2,95}and S

_{2,99}underneath S2 at panel A (respectively, panel B) whereas the numbers in the column SS3 underneath S3 at panel A (respectively, panel B) denote the total sum of two corresponding numbers in the columns S

_{3,95}and S

_{3,99}underneath S3 at panel A (respectively, panel B). In addition, the numbers in the column SS4 underneath S4 at panel A (respectively, panel B) denote the total sum of two corresponding numbers in the columns S

_{4,95}and S

_{4,99}underneath S4 at panel A (respectively, panel B). (6) The bold font in all columns under S1 denotes the greater number when two numbers corresponding to two models are compared with each other and these two models have the same bivariate variance-covariance specification but with different approaches of parameter estimates (i.e., the S-CCC vs. NS-CCC; the S-DCC vs. NS-DCC; and the S-BEKK vs. NS-BEKK). (7) The bold font in all columns under S2 denotes the greatest number when three numbers corresponding to three models are compared with each other and these three models have the same approach of parameter estimates but with different bivariate variance-covariance specification (i.e., the S-CCC, S-DCC, and S-BEKK; the NS-CCC, NS-DCC, and NS-BEKK). (8) The bold font in all columns under S3 denotes the greatest number when six numbers corresponding to the six models are compared with each other, and these six models are the S-CCC, NS-CCC, S-DCC, NS-DCC, S-BEKK, and NS-BEKK. (9) The bold font in all columns under S4 denotes the greater number when two numbers corresponding to two models are compared with each other. The two models are the NS-DCC and NS_ADCC.

Panel A. The LR_{uc} Test | ||||||||

Ny-Sp | Ny-Na | Ny-Ca | Ny-Da | Ny-Ft | Ny-Sm | Sp-Na | Sum | |

S-CCC | 0.3653 | 4.5110 | 7.1022 | 11.3307 | 9.1101 | 3.0805 | 4.5110 | 2 |

[0.5455] | [0.0336] | [0.0076] | [0.0007] | [0.0025] | [0.0792] | [0.0336] | ||

NS-CCC | 1.9027 | 4.5110 | 7.1022 | 10.1944 | 8.0790 | 3.0805 | 5.3168 | 2 |

[0.1677] | [0.0336] | [0.0076] | [0.0014] | [0.0044] | [0.0792] | [0.0211] | ||

S-DCC | 0.0000 | 0.6421 | 2.4591 | 0.9921 | 0.9921 | 0.0000 | 0.6421 | 7 |

[1.0000] | [0.4229] | [0.1168] | [0.3192] | [0.3192] | [1.0000] | [0.4229] | ||

NS-DCC | 1.9027 | 4.5110 | 5.3168 | 10.1944 | 5.3168 | 4.5110 | 5.3168 | 1 |

[0.1677] | [0.0336] | [0.0211] | [0.0014] | [0.0211] | [0.0336] | [0.0211] | ||

S-BEKK | 1.9027 | 4.5110 | 8.0790 | 5.3168 | 9.1101 | 4.5110 | 7.1022 | 1 |

[0.1677] | [0.0336] | [0.0044] | [0.0211] | [0.0025] | [0.0336] | [0.0076] | ||

NS-BEKK | 2.4591 | 4.5110 | 7.1022 | 11.3307 | 7.1022 | 4.5110 | 4.5110 | 1 |

[0.1168] | [0.0336] | [0.0076] | [0.0007] | [0.0076] | [0.0336] | [0.0336] | ||

NS-ADCC | 1.9027 | 4.5110 | 6.1810 | 10.1944 | 6.1810 | 3.0805 | 5.3168 | 2 |

[0.1677] | [0.0336] | [0.0129] | [0.0014] | [0.0129] | [0.0792] | [0.0211] | ||

Sp-Ca | Sp-Da | Sp-Ft | Sp-Sm | Na-Ca | Na-Da | Na-Ft | ||

S-CCC | 15.0408 | 7.1022 | 13.7549 | 5.3168 | 8.0790 | 5.3168 | 12.5179 | 0 |

[0.0001] | [0.0076] | [0.0002] | [0.0211] | [0.0044] | [0.0211] | [0.0004] | ||

NS-CCC | 8.0790 | 6.1810 | 10.1944 | 3.7650 | 8.0790 | 4.5110 | 12.5179 | 1 |

[0.0044] | [0.0129] | [0.0014] | [0.0523] | [0.0044] | [0.0336] | [0.0004] | ||

S-DCC | 0.9921 | 0.6421 | 5.3168 | 0.6421 | 0.6421 | 3.7650 | 1.9027 | 6 |

[0.3192] | [0.4229] | [0.0211] | [0.4229] | [0.4229] | [0.0523] | [0.1677] | ||

NS-DCC | 7.1022 | 5.3168 | 9.1101 | 3.7650 | 6.1810 | 4.5110 | 11.3307 | 1 |

[0.0076] | [0.0211] | [0.0025] | [0.0523] | [0.0129] | [0.0336] | [0.0007] | ||

S-BEKK | 6.1810 | 3.0805 | 13.7549 | 5.3168 | 7.1022 | 3.0805 | 9.1101 | 2 |

[0.0129] | [0.0792] | [0.0002] | [0.0211] | [0.0076] | [0.0792] | [0.0025] | ||

NS-BEKK | 8.0790 | 7.1022 | 8.0790 | 4.5110 | 6.1810 | 6.1810 | 8.0790 | 0 |

[0.0044] | [0.0076] | [0.0044] | [0.0336] | [0.0129] | [0.0129] | [0.0044] | ||

NS-ADCC | 8.0790 | 5.3168 | 9.1101 | 3.7650 | 6.1810 | 3.7650 | 9.1101 | 2 |

[0.0044] | [0.0211] | [0.0025] | [0.0523] | [0.0129] | [0.0523] | [0.0025] | ||

Na-Sm | Ca-Da | Ca-Ft | Ca-Sm | Da-Ft | Da-Sm | Ft-Sm | ||

S-CCC | 4.5110 | 5.3168 | 3.0805 | 0.9921 | 4.5110 | 2.4591 | 4.5110 | 3 |

[0.0336] | [0.0211] | [0.0792] | [0.3192] | [0.0336] | [0.1168] | [0.0336] | ||

NS-CCC | 4.5110 | 2.4591 | 1.4130 | 1.9027 | 3.7650 | 3.0805 | 2.4591 | 6 |

[0.0336] | [0.1168] | [0.2345] | [0.1677] | [0.0523] | [0.0792] | [0.1168] | ||

S-DCC | 0.7107 | 0.9921 | 1.4130 | 0.1728 | 0.6421 | 3.0805 | 0.0415 | 7 |

[0.3991] | [0.3192] | [0.2345] | [0.6775] | [0.4229] | [0.0792] | [0.8384] | ||

NS-DCC | 4.5110 | 1.9027 | 1.4130 | 0.9921 | 4.5110 | 3.0805 | 3.0805 | 5 |

[0.0336] | [0.1677] | [0.2345] | [0.3192] | [0.0336] | [0.0792] | [0.0792] | ||

S-BEKK | 5.3168 | 3.7650 | 0.6421 | 1.4130 | 1.9027 | 2.4591 | 1.9027 | 6 |

[0.0211] | [0.0523] | [0.4229] | [0.2345] | [0.1677] | [0.1168] | [0.1677] | ||

NS-BEKK | 3.0805 | 1.9027 | 1.4130 | 0.9921 | 5.3168 | 3.0805 | 2.4591 | 6 |

[0.0792] | [0.1677] | [0.2345] | [0.3192] | [0.0211] | [0.0792] | [0.1168] | ||

NS-ADCC | 5.3168 | 1.9027 | 1.4130 | 0.9921 | 4.5110 | 3.0805 | 3.0805 | 5 |

[0.0211] | [0.1677] | [0.2345] | [0.3192] | [0.0336] | [0.0792] | [0.0792] | ||

Panel B. The LR_{cc} Test | ||||||||

Ny-Sp | Ny-Na | Ny-Ca | Ny-Da | Ny-Ft | Ny-Sm | Sp-Na | Sum | |

S-CCC | 0.4874 | 4.5806 | 8.3725 | 14.2417 | 11.1676 | 5.8915 | 4.5806 | 4 |

[0.7836] | [0.1012] | [0.0152] | [0.0008] | [0.0037] | [0.0525] | [0.1012] | ||

NS-CCC | 2.3481 | 4.5806 | 8.3725 | 13.5015 | 10.4649 | 5.8915 | 5.3441 | 4 |

[0.3091] | [0.1012] | [0.0152] | [0.0011] | [0.0053] | [0.0525] | [0.0691] | ||

S-DCC | 0.4258 | 0.7050 | 3.8973 | 8.3789 | 3.3337 | 4.4727 | 1.6051 | 6 |

[0.8082] | [0.7029] | [0.1424] | [0.0151] | [0.1888] | [0.1068] | [0.4481] | ||

NS-DCC | 2.3481 | 4.5806 | 7.1218 | 13.5015 | 8.8586 | 6.6231 | 5.9173 | 3 |

[0.3091] | [0.1012] | [0.0284] | [0.0011] | [0.0119] | [0.0364] | [0.0518] | ||

S-BEKK | 2.3481 | 4.5806 | 10.4649 | 8.8586 | 11.1676 | 8.4983 | 7.4256 | 2 |

[0.3091] | [0.1012] | [0.0053] | [0.0119] | [0.0037] | [0.0142] | [0.0244] | ||

NS-BEKK | 2.7784 | 4.5806 | 8.3725 | 14.2417 | 9.8444 | 6.6231 | 4.5806 | 3 |

[0.2492] | [0.1012] | [0.0152] | [0.0008] | [0.0072] | [0.0364] | [0.1012] | ||

NS-ADCC | 2.3481 | 4.5806 | 7.7056 | 13.5015 | 9.3082 | 5.8915 | 5.9173 | 4 |

[0.3091] | [0.1012] | [0.0212] | [0.0011] | [0.0095] | [0.0525] | [0.0518] | ||

Sp-Ca | Sp-Da | Sp-Ft | Sp-Sm | Na-Ca | Na-Da | Na-Ft | ||

S-CCC | 16.9305 | 8.3725 | 17.5529 | 7.1218 | 10.4649 | 11.0641 | 15.0612 | 0 |

[0.0002] | [0.0152] | [0.0001] | [0.0284] | [0.0053] | [0.0039] | [0.0005] | ||

NS-CCC | 10.4649 | 9.3082 | 15.4585 | 4.7351 | 10.4649 | 8.4983 | 15.0612 | 1 |

[0.0053] | [0.0095] | [0.0004] | [0.0937] | [0.0053] | [0.0142] | [0.0005] | ||

S-DCC | 1.7585 | 3.3432 | 5.9173 | 3.3432 | 1.6051 | 8.2299 | 5.5328 | 6 |

[0.4150] | [0.1879] | [0.0518] | [0.1879] | [0.4481] | [0.0163] | [0.0628] | ||

NS-DCC | 8.3725 | 8.8586 | 12.8426 | 4.7351 | 7.7056 | 8.4983 | 14.2417 | 1 |

[0.0152] | [0.0119] | [0.0016] | [0.0937] | [0.0212] | [0.0142] | [0.0008] | ||

S-BEKK | 9.3082 | 5.891 | 17.5529 | 11.0641 | 8.3725 | 5.8915 | 11.1676 | 2 |

[0.0095] | 5[0.0525] | [0.0001] | [0.0039] | [0.0152] | [0.0525] | [0.0037] | ||

NS-BEKK | 9.1201 | 8.3725 | 12.2670 | 6.6231 | 7.7056 | 11.3752 | 12.2670 | 0 |

[0.0104] | [0.0152] | [0.0021] | [0.0364] | [0.0212] | [0.0033] | [0.0021] | ||

NS-ADCC | 10.4649 | 8.8586 | 12.8426 | 4.7351 | 7.7056 | 8.2299 | 12.8426 | 1 |

[0.0053] | [0.0119] | [0.0016] | [0.0937] | [0.0212] | [0.0163] | [0.0016] | ||

Na-Sm | Ca-Da | Ca-Ft | Ca-Sm | Da-Ft | Da-Sm | Ft-Sm | ||

S-CCC | 8.4983 | 11.0641 | 5.8915 | 5.5726 | 8.4983 | 7.9810 | 6.6231 | 2 |

[0.0142] | [0.0039] | [0.0525] | [0.0616] | [0.0142] | [0.0184] | [0.0364] | ||

NS-CCC | 8.4983 | 10.7855 | 3.4251 | 5.5328 | 6.2122 | 8.0564 | 2.7784 | 3 |

[0.0142] | [0.0045] | [0.1804] | [0.0628] | [0.0447] | [0.0178] | [0.2492] | ||

S-DCC | 7.5978 | 1.7585 | 2.0071 | 5.7610 | 1.6051 | 5.8915 | 4.0175 | 6 |

[0.0223] | [0.4150] | [0.3665] | [0.0561] | [0.4481] | [0.0525] | [0.1341] | ||

NS-DCC | 8.4983 | 10.9746 | 3.4251 | 3.3337 | 8.4983 | 8.0564 | 3.2955 | 3 |

[0.0142] | [0.0041] | [0.1804] | [0.1888] | [0.0142] | [0.0178] | [0.1924] | ||

S-BEKK | 11.0641 | 10.7251 | 3.3432 | 3.4251 | 3.6140 | 7.9810 | 2.3481 | 4 |

[0.0039] | [0.0046] | [0.1879] | [0.1804] | [0.1641] | [0.0184] | [0.3091] | ||

NS-BEKK | 5.8915 | 10.9746 | 3.4251 | 5.5726 | 8.8586 | 8.0564 | 2.7784 | 4 |

[0.0525] | [0.0041] | [0.1804] | [0.0616] | [0.0119] | [0.0178] | [0.2492] | ||

NS-ADCC | 11.0641 | 10.9746 | 3.4251 | 3.3337 | 8.4983 | 8.0564 | 3.2955 | 3 |

[0.0039] | [0.0041] | [0.1804] | [0.1888] | [0.0142] | [0.0178] | [0.1924] | ||

Panel C. The DQ Test | ||||||||

Ny-Sp | Ny-Na | Ny-Ca | Ny-Da | Ny-Ft | Ny-Sm | Sp-Na | Sum | |

S-CCC | 10.3218 | 12.2181 | 12.0662 | 33.4966 | 26.8956 | 12.3289 | 10.0792 | 5 |

[0.1710] | [0.0936] | [0.0983] | [0.0000] | [0.0003] | [0.0902] | [0.1841] | ||

NS-CCC | 10.2944 | 12.0725 | 11.9839 | 28.8697 | 27.2485 | 12.5786 | 10.8587 | 5 |

[0.1724] | [0.0981] | [0.1010] | [0.0001] | [0.0003] | [0.0830] | [0.1448] | ||

S-DCC | 3.7374 | 8.5843 | 7.3272 | 19.6552 | 18.7722 | 10.3007 | 13.4539 | 5 |

[0.8094] | [0.2838] | [0.3956] | [0.0063] | [0.0089] | [0.1721] | [0.0617] | ||

NS-DCC | 10.3132 | 12.1512 | 11.2779 | 37.2880 | 22.0679 | 13.0092 | 12.2997 | 5 |

[0.1715] | [0.0956] | [0.1269] | [0.0000] | [0.0024] | [0.0718] | [0.0911] | ||

S-BEKK | 10.0575 | 12.2142 | 16.0366 | 19.1733 | 23.0047 | 20.5890 | 14.0009 | 3 |

[0.1853] | [0.0937] | [0.0247] | [0.0076] | [0.0017] | [0.0044] | [0.0511] | ||

NS-BEKK | 10.1730 | 12.5699 | 11.8918 | 34.6125 | 24.0851 | 11.2893 | 10.2065 | 5 |

[0.1789] | [0.0833] | [0.1041] | [0.0000] | [0.0011] | [0.1264] | [0.1771] | ||

NS-ADCC | 10.3132 | 12.1526 | 11.5718 | 33.5806 | 23.4834 | 12.5590 | 12.3032 | 5 |

[0.1715] | [0.0956] | [0.1155] | [0.0000] | [0.0014] | [0.0836] | [0.0910] | ||

Sp-Ca | Sp-Da | Sp-Ft | Sp-Sm | Na-Ca | Na-Da | Na-Ft | ||

S-CCC | 30.3576 | 18.2263 | 37.9757 | 16.0300 | 15.2448 | 24.5045 | 31.0751 | 0 |

[0.0000] | [0.0109] | [0.0000] | [0.0248] | [0.0329] | [0.0009] | [0.0000] | ||

NS-CCC | 20.2060 | 18.0954 | 37.3363 | 12.2757 | 15.3185 | 18.6607 | 31.1347 | 1 |

[0.0051] | [0.0115] | [0.0000] | [0.0918] | [0.0321] | [0.0093] | [0.0000] | ||

S-DCC | 4.3366 | 6.6243 | 13.7254 | 11.9702 | 3.6055 | 15.4307 | 12.6713 | 6 |

[0.7402] | [0.4690] | [0.0562] | [0.1015] | [0.8239] | [0.0308] | [0.0805] | ||

NS-DCC | 14.5434 | 18.7120 | 30.9949 | 12.2764 | 14.4964 | 18.6412 | 30.5141 | 1 |

[0.0423] | [0.0091] | [0.0000] | [0.0918] | [0.0430] | [0.0093] | [0.0000] | ||

S-BEKK | 16.1927 | 13.1625 | 35.5377 | 20.6767 | 15.3437 | 13.9051 | 20.7264 | 2 |

[0.0234] | [0.0682] | [0.0000] | [0.0042] | [0.0318] | [0.0528] | [0.0041] | ||

NS-BEKK | 12.8649 | 18.0528 | 39.2391 | 13.9991 | 13.9110 | 25.1249 | 29.5862 | 3 |

[0.0754] | [0.0117] | [0.0000] | [0.0511] | [0.0527] | [0.0007] | [0.0001] | ||

NS-ADCC | 20.5613 | 18.7242 | 31.0229 | 12.2829 | 14.4518 | 18.1772 | 34.3521 | 1 |

[0.0044] | [0.0090] | [0.0000] | [0.0916] | [0.0437] | [0.0111] | [0.0000] | ||

Na-Sm | Ca-Da | Ca-Ft | Ca-Sm | Da-Ft | Da-Sm | Ft-Sm | ||

S-CCC | 25.8101 | 22.3909 | 10.9822 | 12.0453 | 17.3897 | 17.2360 | 26.4417 | 2 |

[0.0005] | [0.0021] | [0.1393] | [0.0990] | [0.0150] | [0.0159] | [0.0004] | ||

NS-CCC | 25.8747 | 21.4803 | 7.6053 | 12.5239 | 13.7256 | 17.2310 | 7.1334 | 4 |

[0.0005] | [0.0031] | [0.3686] | [0.0845] | [0.0562] | [0.0159] | [0.4151] | ||

S-DCC | 13.3830 | 3.3975 | 4.8187 | 10.4288 | 13.2134 | 17.0012 | 10.4312 | 6 |

[0.0633] | [0.8459] | [0.6820] | [0.1655] | [0.0670] | [0.0173] | [0.1654] | ||

NS-DCC | 25.8073 | 21.9385 | 7.7121 | 7.7693 | 19.4782 | 16.9916 | 9.4860 | 3 |

[0.0005] | [0.0026] | [0.3586] | [0.3533] | [0.0068] | [0.0174] | [0.2196] | ||

S-BEKK | 23.7673 | 21.6152 | 7.5296 | 9.2160 | 10.0008 | 17.0505 | 6.3742 | 4 |

[0.0012] | [0.0029] | [0.3758] | [0.2375] | [0.1885] | [0.0170] | [0.4967] | ||

NS-BEKK | 12.0903 | 21.9022 | 7.6893 | 11.8549 | 17.9357 | 16.7977 | 17.1000 | 3 |

[0.0976] | [0.0026] | [0.3607] | [0.1054] | [0.0122] | [0.0187] | [0.0167] | ||

NS-ADCC | 31.0924 | 21.9193 | 7.7121 | 7.7693 | 19.4782 | 16.9965 | 9.4886 | 3 |

[0.0000] | [0.0026] | [0.3586] | [0.3533] | [0.0068] | [0.0174] | [0.2194] |

_{uc}(respectively, LR

_{cc}) test statistic is asymptotically χ

^{2}(1) (respectively, χ

^{2}(2)) distributed and its corresponding critical values at the 5% significance level is 3.841 (respectively, 5.991). On the contrary, the DQ test statistic is asymptotically χ

^{2}(7) distributed and its corresponding critical values at the 5% significance level is 14.067. (5) The numbers in column Sum denote the total number of portfolios passing the specific test at the 5% significance level.

**Table 7.**The summary results for the out-of-sample VaR forecasts performance of equal-weight stock portfolios based on alternative accuracy tests.

90% Level | 95% Level | |||||||||||||||||||||||||

LR_{uc} | S_{90,uc} | LR_{cc} | S_{90,cc} | DQ | S_{90,dq} | S_{90} | LR_{uc} | S_{95,uc} | LR_{cc} | S_{95,cc} | DQ | S_{95,dq} | S_{95} | |||||||||||||

S-CCC | 7 | 7 | 4 | 18 | 5 | 2 | 1 | 8 | 3 | 2 | 2 | 7 | 33 | 2 | 0 | 3 | 5 | 4 | 0 | 2 | 6 | 5 | 0 | 2 | 7 | 18 |

NS-CCC | 7 | 7 | 5 | 19 | 6 | 3 | 2 | 11 | 3 | 3 | 2 | 8 | 38 | 2 | 1 | 6 | 9 | 4 | 1 | 3 | 8 | 5 | 1 | 4 | 10 | 27 |

S-DCC | 7 | 7 | 7 | 21 | 7 | 7 | 4 | 18 | 6 | 4 | 3 | 13 | 52 | 7 | 6 | 7 | 20 | 6 | 6 | 6 | 18 | 5 | 6 | 6 | 17 | 55 |

NS-DCC | 7 | 7 | 6 | 20 | 6 | 5 | 3 | 14 | 3 | 3 | 2 | 8 | 42 | 1 | 1 | 5 | 7 | 3 | 1 | 3 | 7 | 5 | 1 | 3 | 9 | 23 |

S-BEKK | 7 | 7 | 7 | 21 | 5 | 5 | 4 | 14 | 3 | 2 | 3 | 8 | 43 | 1 | 2 | 6 | 9 | 2 | 2 | 4 | 8 | 3 | 2 | 4 | 9 | 26 |

NS-BEKK | 7 | 7 | 4 | 18 | 7 | 4 | 2 | 13 | 3 | 2 | 2 | 7 | 38 | 1 | 0 | 6 | 7 | 3 | 0 | 4 | 7 | 5 | 3 | 3 | 11 | 25 |

NS-ADCC | 7 | 7 | 6 | 20 | 6 | 4 | 3 | 13 | 3 | 2 | 2 | 7 | 40 | 2 | 2 | 5 | 9 | 4 | 1 | 3 | 8 | 5 | 1 | 3 | 9 | 26 |

99% Level | 99.5% Level | |||||||||||||||||||||||||

LR_{uc} | S_{99,uc} | LR_{cc} | S_{99,cc} | DQ | S_{99,dq} | S_{99} | LR_{uc} | S_{995,uc} | LR_{cc} | S_{995,cc} | DQ | S_{995,dq} | S_{995} | |||||||||||||

S-CCC | 2 | 1 | 0 | 3 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 5 | 2 | 1 | 0 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 3 |

NS-CCC | 1 | 1 | 0 | 2 | 1 | 0 | 1 | 2 | 1 | 0 | 1 | 2 | 6 | 1 | 1 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 |

S-DCC | 4 | 5 | 4 | 13 | 3 | 2 | 2 | 7 | 2 | 2 | 2 | 6 | 26 | 5 | 5 | 5 | 15 | 1 | 0 | 0 | 1 | 1 | 0 | 3 | 4 | 20 |

NS-DCC | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 2 | 1 | 0 | 1 | 2 | 5 | 2 | 3 | 0 | 5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 5 |

S-BEKK | 2 | 3 | 0 | 5 | 2 | 2 | 0 | 4 | 1 | 1 | 1 | 3 | 12 | 2 | 4 | 1 | 7 | 0 | 0 | 1 | 1 | 1 | 0 | 2 | 2 | 10 |

NS-BEKK | 1 | 1 | 1 | 3 | 2 | 1 | 1 | 4 | 1 | 0 | 1 | 2 | 9 | 1 | 2 | 0 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 3 |

NS-ADCC | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 4 | 0 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 7 |

_{uc}, LR

_{cc}, and DQ underneath the 95% level at the first panel are respectively summarized from the numbers in the column Sum of Panels A, B, and C in Table 6. (3) The numbers in the columns S

_{95,uc}(S

_{99,uc}), S

_{95,cc}(S

_{99,cc}), and S

_{95,dq}(S

_{99,dq}) underneath the 95% (99%) level at the first (second) panel denote the total number of portfolios that pass the LR

_{uc}, LR

_{cc}, and DQ tests at the specified model for all 21 bi-component stock portfolios, respectively. (4) The numbers in column S

_{95}(S

_{99}) underneath the 95% (99%) level at the first (second) panel denote the total sum of three corresponding numbers in columns S

_{95,uc}(S

_{99,uc}), S

_{95,cc}(S

_{99,cc}), and S

_{95,dq}(S

_{99,dq}) representing the total number of portfolios that pass the LR

_{uc}, LR

_{cc}, or DQ tests at the specified model for the 95% (99%) level. (5) In the same inference process, the numbers in column S

_{90}(S

_{995}) underneath the 90% (99.5%) level at the first (second) panel denote the total sum of three corresponding numbers in columns S

_{90,uc}(S

_{995,uc}), S

_{90,cc}(S

_{995,cc}), and S

_{90,dq}(S

_{995,dq}). (6) The bold font in columns S

_{90}, S

_{95}, S

_{99}, and S

_{995}denotes the greater number when two numbers corresponding to two models are compared with each other and these two models have the same bivariate variance-covariance specification but a different parameter estimate approach (i.e., the S-CCC vs. NS-CCC; the S-DCC vs. NS-DCC; and the S-BEKK vs. NS-BEKK). (7) The underlined font in columns S

_{90}, S

_{95}, S

_{99}, and S

_{995}denotes the greatest number when three numbers corresponding to the three models are compared with each other and these three models have the same parameter estimate approach but a different bivariate variance-covariance specification (i.e., the S-CCC, S-DCC, and S-BEKK; the NS-CCC, NS-DCC, and NS-BEKK). (8) The shade font in columns S

_{90}, S

_{95}, S

_{99}, and S

_{995}denotes the greatest number when six numbers corresponding to the six models are compared with each other and these six models are the S-CCC, NS-CCC, S-DCC, NS-DCC, S-BEKK, and NS-BEKK models. (9) The italic font in columns S

_{90}, S

_{95}, S

_{99}, and S

_{995}denotes the greater number when two numbers corresponding to two models are compared with each other and these two models are the NS-DCC and NS-ADCC.

**Table 8.**All the summary results of the out-of-sample VaR forecast performances for the three weight combinations stock portfolios.

Panel A.${\mathbf{w}}_{\mathbf{1}}\mathbf{=}\mathbf{50}\mathbf{\%}\mathbf{,}\mathrm{and}{\mathbf{w}}_{\mathbf{2}}\mathbf{=}\mathbf{50}\mathbf{\%}$bi-component portfolios | |||||||||||||||||

90% Level | 95% Level | 99% Level | 99.5% Level | SUM | |||||||||||||

LR_{uc} | LR_{cc} | DQ | Sum | LR_{uc} | LR_{cc} | DQ | Sum | LR_{uc} | LR_{cc} | DQ | Sum | LR_{uc} | LR_{cc} | DQ | Sum | ||

S-CCC | 18 | 8 | 7 | 33 | 5 | 6 | 7 | 18 | 3 | 1 | 1 | 5 | 3 | 0 | 0 | 3 | 59 |

NS-CCC | 19 | 11 | 8 | 38 | 9 | 8 | 10 | 27 | 2 | 2 | 2 | 6 | 2 | 0 | 0 | 2 | 73 |

S-DCC | 21 | 18 | 13 | 52 | 20 | 18 | 17 | 55 | 13 | 7 | 6 | 26 | 15 | 1 | 4 | 20 | 153 |

NS-DCC | 20 | 14 | 8 | 42 | 7 | 7 | 9 | 23 | 1 | 2 | 2 | 5 | 5 | 0 | 0 | 5 | 75 |

S-BEKK | 21 | 14 | 8 | 43 | 9 | 8 | 9 | 26 | 5 | 4 | 3 | 12 | 7 | 1 | 2 | 10 | 91 |

NS-BEKK | 18 | 13 | 7 | 38 | 7 | 7 | 11 | 25 | 3 | 4 | 2 | 9 | 3 | 0 | 0 | 3 | 75 |

NS-ADCC | 20 | 13 | 7 | 40 | 9 | 8 | 9 | 26 | 0 | 0 | 0 | 0 | 6 | 0 | 1 | 7 | 73 |

Panel B.${\mathbf{w}}_{\mathbf{1}}\mathbf{=}\mathbf{25}\mathbf{\%}\mathbf{,}\mathrm{and}{\mathbf{w}}_{\mathbf{2}}\mathbf{=}\mathbf{75}\mathbf{\%}$bi-component portfolios | |||||||||||||||||

90% Level | 95% Level | 99% Level | 99.5% Level | SUM | |||||||||||||

LR_{uc} | LR_{cc} | DQ | Sum | LR_{uc} | LR_{cc} | DQ | Sum | LR_{uc} | LR_{cc} | DQ | Sum | LR_{uc} | LR_{cc} | DQ | Sum | ||

S-CCC | 10 | 5 | 3 | 18 | 7 | 6 | 7 | 20 | 1 | 0 | 0 | 1 | 2 | 0 | 0 | 2 | 41 |

NS-CCC | 20 | 14 | 11 | 45 | 13 | 11 | 11 | 35 | 2 | 1 | 1 | 4 | 3 | 0 | 3 | 6 | 90 |

S-DCC | 21 | 16 | 14 | 51 | 18 | 12 | 14 | 44 | 11 | 5 | 4 | 20 | 12 | 2 | 6 | 20 | 135 |

NS-DCC | 20 | 14 | 11 | 45 | 13 | 11 | 11 | 35 | 1 | 1 | 1 | 3 | 2 | 0 | 3 | 5 | 88 |

S-BEKK | 19 | 11 | 11 | 41 | 15 | 8 | 9 | 32 | 0 | 0 | 0 | 0 | 5 | 1 | 4 | 10 | 83 |

NS-BEKK | 17 | 14 | 11 | 42 | 11 | 11 | 12 | 34 | 0 | 0 | 1 | 1 | 1 | 0 | 1 | 2 | 79 |

NS-ADCC | 19 | 14 | 11 | 44 | 13 | 11 | 11 | 35 | 1 | 1 | 1 | 3 | 2 | 0 | 3 | 5 | 87 |

Panel C.${\mathbf{w}}_{\mathbf{1}}\mathbf{=}\mathbf{75}\mathbf{\%}\mathbf{,}\mathrm{and}{\mathbf{w}}_{\mathbf{2}}\mathbf{=}\mathbf{25}\mathbf{\%}$bi-component portfolios | |||||||||||||||||

90% Level | 95% Level | 99% Level | 99.5% Level | SUM | |||||||||||||

LR_{uc} | LR_{cc} | DQ | Sum | LR_{uc} | LR_{cc} | DQ | Sum | LR_{uc} | LR_{cc} | DQ | Sum | LR_{uc} | LR_{cc} | DQ | Sum | ||

S-CCC | 16 | 15 | 10 | 41 | 12 | 11 | 10 | 33 | 7 | 2 | 0 | 9 | 8 | 0 | 0 | 8 | 91 |

NS-CCC | 20 | 17 | 14 | 51 | 16 | 16 | 13 | 45 | 1 | 2 | 1 | 4 | 4 | 0 | 1 | 5 | 105 |

S-DCC | 21 | 19 | 17 | 57 | 20 | 19 | 17 | 56 | 13 | 5 | 2 | 20 | 11 | 3 | 4 | 18 | 151 |

NS-DCC | 21 | 16 | 14 | 51 | 17 | 16 | 14 | 47 | 3 | 2 | 1 | 6 | 4 | 0 | 0 | 4 | 108 |

S-BEKK | 21 | 17 | 14 | 52 | 19 | 18 | 16 | 53 | 6 | 3 | 2 | 11 | 7 | 1 | 1 | 9 | 125 |

NS-BEKK | 20 | 16 | 12 | 48 | 18 | 15 | 12 | 45 | 2 | 1 | 1 | 4 | 3 | 0 | 0 | 3 | 100 |

NS-ADCC | 21 | 16 | 14 | 51 | 18 | 15 | 14 | 47 | 2 | 2 | 1 | 5 | 3 | 0 | 0 | 3 | 106 |

_{uc}, LR

_{cc}, and DQ underneath the 90% level at panel A are the numbers in columns S

_{90,uc}, S

_{90,cc}, and S

_{90,dq}underneath the 90% level in Table 7. Similarly, the numbers in column Sum underneath the 90% level at panel A are the numbers in columns

**S**underneath the 90% level case in Table 7. As to the numbers in columns LR

_{90}_{uc}, LR

_{cc}, DQ, and Sum underneath the 95%, 99%, 99.5% levels in panel A, they are defined as the 90% level. (3) The numbers in the column Sum at each panel denote the total sum of three corresponding numbers in columns LR

_{uc}, LR

_{cc}, and DQ. (4) The numbers in the column SUM at each panel denote the total sum of the four corresponding numbers in the columns Sum of the 90%, 95%, 99%, and 99.5% levels. (5) The bold font in the columns Sum, and SUM denotes the greater number when two numbers corresponding to two models are compared with each other and these two models have the same bivariate variance-covariance specification but a different parameter estimate approach (i.e., the S-CCC vs. NS-CCC; the S-DCC vs. NS-DCC; and the S-BEKK vs. NS-BEKK). (6) The underlined font in the columns Sum, and SUM denotes the greatest number when three numbers corresponding to three models are compared with each other and these three models have the same parameter estimate approach but with different bivariate variance-covariance specifications (i.e., the S-CCC, S-DCC, and S-BEKK; NS-CCC, NS-DCC, and NS-BEKK). (7) The shaded font in the columns Sum, and SUM denotes the greatest number when seven numbers corresponding to even models are compared with each other. (8) The italic font in the columns Sum, and SUM denotes the greater number when two numbers corresponding to two models are compared with each other, and these two models are the NS-DCC and NS-ADCC.

**Table 9.**The out-of-sample VaR forecasts performance of the equal weight currency-stock portfolios based on alternative accuracy tests for the 90% level.

Udi-Ny | Udi-Sp | Udi-Na | Udi-Ca | Udi-Da | Udi-Ft | Udi-Sm | Sum | |
---|---|---|---|---|---|---|---|---|

Panel A. The LR_{uc} test | ||||||||

S-CCC | 0.3643[0.5461] | 0.0899[0.7642] | 0.0000[1.0000] | 1.7119[0.1907] | 5.8681[0.0154] | 0.0000[1.0000] | 1.0466[0.3062] | 6 |

NS-CCC | 0.0220[0.8818] | 0.0223[0.8811] | 0.0220[0.8818] | 1.3597[0.2435] | 5.8681[0.0154] | 0.0878[0.7669] | 0.3474[0.5555] | 6 |

S-DCC | 0.0899[0.7642] | 1.1375[0.2861] | 0.8303[0.3621] | 0.0000[1.0000] | 1.3597[0.2435] | 0.0223[0.8811] | 0.8303[0.3621] | 7 |

NS-DCC | 0.2036[0.6517] | 1.9058[0.1674] | 0.0899[0.7642] | 0.0220[0.8818] | 1.3597[0.2435] | 0.0223[0.8811] | 0.5729[0.4491] | 7 |

S-BEKK | 0.0878[0.7669] | 0.0000[1.0000] | 0.0220[0.8818] | 2.5309[0.1116] | 7.2612[0.0070] | 0.1965[0.6575] | 1.3597[0.2435] | 6 |

NS-BEKK | 0.0899[0.7642] | 1.9058[0.1674] | 0.0000[1.0000] | 0.1965[0.6575] | 1.7119[0.1907] | 0.0878[0.7669] | 0.2036[0.6517] | 7 |

NS-ADCC | 0.2036[0.6517] | 1.9058[0.1674] | 0.0899[0.7642] | 0.0220[0.8818] | 1.3597[0.2435] | 0.0223[0.8811] | 0.3643[0.5461] | 7 |

Panel B. The LR_{cc} test | ||||||||

S-CCC | 0.5221[0.7702] | 0.1279[0.9380] | 2.4 × 10[0.9999]^{−5} | 2.4195[0.2982] | 6.0139[0.0494] | 5.0545[0.0798] | 6.0584[0.0483] | 5 |

NS-CCC | 0.1643[0.9211] | 0.0313[0.9844] | 0.0329[0.9836] | 2.2588[0.3232] | 6.0139[0.0494] | 4.1640[0.1246] | 7.0446[0.0295] | 5 |

S-DCC | 0.5617[0.7551] | 1.1650[0.5584] | 0.8347[0.6587] | 1.9366[0.3797] | 3.1793[0.2039] | 3.7804[0.1510] | 7.1652[0.0278] | 6 |

NS-DCC | 0.2911[0.8645] | 2.0396[0.3606] | 0.1279[0.9380] | 2.9583[0.2278] | 2.2588[0.3232] | 3.7804[0.1510] | 6.3292[0.0422] | 6 |

S-BEKK | 0.1634[0.9215] | 2 × 10[0.9999]^{−5} | 0.0329[0.9836] | 2.9262[0.2315] | 7.2906[0.0261] | 5.5319[0.0629] | 5.8702[0.0531] | 6 |

NS-BEKK | 0.1279[0.9380] | 2.0396[0.3606] | 2 × 10[0.9999]^{−5} | 1.3339[0.5132] | 2.4195[0.2982] | 2.6549[0.2651] | 9.3299[0.0094] | 6 |

NS-ADCC | 0.2911[0.8645] | 2.0396[0.3606] | 0.1279 [0.9380] | 2.9583[0.2278] | 2.2588[0.3232] | 3.7804[0.1510] | 5.5747[0.0615] | 7 |

Panel C. The DQ test | ||||||||

S-CCC | 1.9277[0.9637] | 1.1787[0.9914] | 2.3263[0.9395] | 5.2358[0.6312] | 9.1635[0.2411] | 8.9207[0.2583] | 14.5328[0.0424] | 6 |

NS-CCC | 2.5911[0.9200] | 1.3865[0.9859] | 2.4799[0.9286] | 6.1659[0.5205] | 9.1240[0.2438] | 7.4978[0.3789] | 13.0912[0.0699] | 7 |

S-DCC | 3.5911[0.8254] | 2.1972[0.9481] | 4.8508[0.6781] | 8.3832[0.3000] | 13.7697[0.0554] | 7.4171[0.3867] | 11.3058[0.1258] | 7 |

NS-DCC | 2.7042[0.9109] | 2.7454[0.9075] | 2.4149[0.9333] | 8.0607[0.3272] | 10.8364[0.1459] | 6.9173[0.4375] | 9.9127[0.1935] | 7 |

S-BEKK | 2.4411[0.9314] | 2.3166[0.9402] | 2.5963[0.9196] | 7.0117[0.4276] | 11.5043[0.1180] | 9.6540[0.2090] | 12.9049[0.0744] | 7 |

NS-BEKK | 2.3700[0.9365] | 2.3697[0.9365] | 2.2142[0.9470] | 6.5538[0.4767] | 9.2576[0.2346] | 6.5970[0.4720] | 12.8241[0.0765] | 7 |

NS-ADCC | 2.6933[0.9118] | 3.0184[0.8832] | 2.5772[0.9211] | 8.0607[0.3272] | 10.8363[0.1459] | 6.9173[0.4375] | 11.0342[0.1371] | 7 |

_{uc}, LR

_{cc}, and DQ) at the 90% level for equal weight bi-component currency-stock portfolios. On the contrary, the numbers in the bracket beside the preceding numbers denote the corresponding p-values of those test statistics. (3) The bold font indicates that the null hypotheses of the LR

_{uc}, LR

_{cc}, and DQ tests statistics are accepted at the 5% significance level. (4) The LR

_{uc}(respectively, LR

_{cc}) test statistic is asymptotically distributed χ

^{2}(1) (respectively, χ

^{2}(2)) and its corresponding critical value at the 5% significance level is 3.841 (respectively, 5.991). Conversely, the DQ test statistic is asymptotically distributed χ

^{2}(7) and its corresponding critical value at the 5% significance level is 14.067. (5) The numbers in column Sum denote the total number of portfolios passing the above the three accuracy tests at the 5% significance level.

**Table 10.**All the summary results of the out-of-sample VaR forecast performance for the three weight combinations of currency-stock-based portfolios.

Panel A.${\mathbf{w}}_{\mathbf{1}}\mathbf{=}\mathbf{50}\mathbf{\%}\mathbf{,}\mathrm{and}{\mathbf{w}}_{\mathbf{2}}\mathbf{=}\mathbf{50}\mathbf{\%}$bi-component portfolios | |||||||||||||||||

90% Level | 95% Level | 99% Level | 99.5% Level | SUM | |||||||||||||

LR_{uc} | LR_{cc} | DQ | Sum | LR_{uc} | LR_{cc} | DQ | Sum | LR_{uc} | LR_{cc} | DQ | Sum | LR_{uc} | LR_{cc} | DQ | Sum | ||

S-CCC | 6 | 5 | 6 | 17 | 5 | 3 | 4 | 12 | 1 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 30 |

NS-CCC | 6 | 5 | 7 | 18 | 5 | 3 | 4 | 12 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 30 |

S-DCC | 7 | 6 | 7 | 20 | 6 | 7 | 5 | 18 | 1 | 2 | 0 | 3 | 0 | 1 | 1 | 2 | 43 |

NS-DCC | 7 | 6 | 7 | 20 | 6 | 7 | 5 | 18 | 1 | 2 | 1 | 4 | 0 | 1 | 2 | 3 | 45 |

S-BEKK | 6 | 6 | 7 | 19 | 3 | 5 | 4 | 12 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 31 |

NS-BEKK | 7 | 6 | 7 | 20 | 6 | 7 | 6 | 19 | 1 | 2 | 1 | 4 | 0 | 0 | 1 | 1 | 44 |

NS-ADCC | 7 | 7 | 7 | 21 | 6 | 6 | 5 | 17 | 1 | 1 | 1 | 3 | 0 | 1 | 2 | 3 | 44 |

Panel B.${\mathbf{w}}_{\mathbf{1}}\mathbf{=}\mathbf{25}\mathbf{\%}\mathbf{,}\mathrm{and}{\mathbf{w}}_{\mathbf{2}}\mathbf{=}\mathbf{75}\mathbf{\%}$bi-component portfolios | |||||||||||||||||

90% Level | 95% Level | 99% Level | 99.5% Level | SUM | |||||||||||||

LR_{uc} | LR_{cc} | DQ | Sum | LR_{uc} | LR_{cc} | DQ | Sum | LR_{uc} | LR_{cc} | DQ | Sum | LR_{uc} | LR_{cc} | DQ | Sum | ||

S-CCC | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

NS-CCC | 7 | 6 | 6 | 19 | 6 | 6 | 6 | 18 | 0 | 1 | 1 | 2 | 0 | 0 | 0 | 0 | 39 |

S-DCC | 7 | 6 | 7 | 20 | 7 | 5 | 5 | 17 | 2 | 1 | 1 | 4 | 3 | 0 | 0 | 3 | 44 |

NS-DCC | 7 | 6 | 7 | 20 | 7 | 5 | 5 | 17 | 0 | 2 | 1 | 3 | 0 | 0 | 1 | 1 | 41 |

S-BEKK | 7 | 6 | 6 | 19 | 5 | 4 | 4 | 13 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 33 |

NS-BEKK | 7 | 6 | 6 | 19 | 7 | 5 | 4 | 16 | 0 | 1 | 1 | 2 | 0 | 0 | 1 | 1 | 38 |

NS-ADCC | 7 | 6 | 7 | 20 | 7 | 5 | 5 | 17 | 0 | 2 | 1 | 3 | 0 | 0 | 1 | 1 | 41 |

Panel C.${\mathbf{w}}_{\mathbf{1}}\mathbf{=}\mathbf{75}\mathbf{\%}\mathbf{,}\mathrm{and}{\mathbf{w}}_{\mathbf{2}}\mathbf{=}\mathbf{25}\mathbf{\%}$bi-component portfolios | |||||||||||||||||

90% Level | 95% Level | 99% Level | 99.5% Level | SUM | |||||||||||||

LR_{uc} | LR_{cc} | DQ | Sum | LR_{uc} | LR_{cc} | DQ | Sum | LR_{uc} | LR_{cc} | DQ | Sum | LR_{uc} | LR_{cc} | DQ | Sum | ||

S-CCC | 0 | 1 | 4 | 5 | 2 | 2 | 6 | 10 | 7 | 4 | 4 | 15 | 7 | 3 | 4 | 14 | 44 |

NS-CCC | 7 | 7 | 7 | 21 | 7 | 7 | 7 | 21 | 3 | 2 | 0 | 5 | 3 | 0 | 0 | 3 | 50 |

S-DCC | 7 | 5 | 6 | 18 | 7 | 6 | 7 | 20 | 5 | 2 | 2 | 9 | 4 | 0 | 2 | 6 | 53 |

NS-DCC | 7 | 5 | 6 | 18 | 7 | 7 | 7 | 21 | 7 | 3 | 3 | 13 | 6 | 0 | 2 | 8 | 60 |

S-BEKK | 7 | 7 | 7 | 21 | 7 | 7 | 6 | 20 | 3 | 2 | 0 | 5 | 3 | 0 | 0 | 3 | 49 |

NS-BEKK | 7 | 5 | 6 | 18 | 7 | 7 | 7 | 21 | 5 | 2 | 3 | 10 | 5 | 0 | 2 | 7 | 56 |

NS-ADCC | 7 | 6 | 6 | 19 | 7 | 7 | 7 | 21 | 7 | 3 | 3 | 13 | 6 | 0 | 2 | 8 | 61 |

_{uc}, LR

_{cc}, and DQ underneath the 90% level at panel A are the numbers in the columns Sum at Panels A, B, and C in Table 9. The numbers in the column Sum underneath the 90% level at panel A are the total sum of three corresponding numbers in columns LR

_{uc}, LR

_{cc}, and DQ. As to the numbers in columns LR

_{uc}, LR

_{cc}, DQ, and Sum underneath the 95%, 99%, 99.5% levels at panel A, they are defined as the 90% level case. (3) As to the other notes, please refer to the notes 3–8 in Table 8.

**Table 11.**The efficiency evaluation test based on the market risk capital for equal weight stock and currency-stock portfolios.

Panel A. The bi-component stock portfolios | ||||||||

Ny-Sp | Ny-Na | Ny-Ca | Ny-Da | Ny-Ft | Ny-Sm | Sp-Na | Sum | |

S-DCC | 7.8910(1.724) | 8.0044(2.008) | 8.2003(2.348) | 8.4113(1.788) | 6.8843(1.994) | 6.9209(2.080) | 8.0728(2.027) | 1 |

[0.000] | [0.000] | [0.000] | [0.000] | [0.510] | [0.010] | [0.000] | ||

NS-DCC | 7.2111(2.455) | 7.5252(2.501) | 7.9801(2.340) | 8.0803(2.160) | 7.0083(2.383) | 6.8579(2.166) | 7.6419(2.577) | 0 |

[0.000] | [0.004] | [0.000] | [0.000] | [0.000] | [0.000] | [0.007] | ||

NS-BEKK | 7.0659(2.337) | 7.3937(2.269) | 7.8118(2.146) | 7.8791(1.923) | 6.8751(2.120) | 6.6864(1.885) | 7.4977(2.295) | 7 |

[0.498] | [0.528] | [0.568] | [0.619] | [0.557] | [0.523] | [0.535] | ||

NS-ADCC | 7.2111(2.455) | 7.5250(2.501) | 7.9720(2.367) | 8.0878(2.163) | 7.0207(2.425) | 6.8287(2.112) | 7.6419(2.577) | 0 |

[0.000] | [0.001] | [0.000] | [0.000] | [0.001] | [0.000] | [0.004] | ||

Sp-Ca | Sp-Da | Sp-Ft | Sp-Sm | Na-Ca | Na-Da | Na-Ft | ||

S-DCC | 7.8820(2.256) | 8.2991(1.942) | 6.8906(2.186) | 6.9046(2.072) | 8.3522(1.901) | 8.3967(1.829) | 7.0895(1.999) | 2 |

[0.037] | [0.000] | [0.104] | [0.000] | [0.019] | [0.073] | [0.634] | ||

NS-DCC | 7.8908(2.339) | 8.0216(2.170) | 6.9040(2.405) | 6.7965(2.182) | 8.1983(2.369) | 8.3949(2.230) | 7.2283(2.402) | 0 |

[0.000] | [0.000] | [0.003] | [0.000] | [0.031] | [0.000] | [0.011] | ||

NS-BEKK | 7.7161(2.116) | 7.8271(1.902) | 6.7619(2.113) | 6.6260(1.868) | 8.1093(2.122) | 8.2155(1.924) | 7.1130(2.097) | 7 |

[0.520] | [0.665] | [0.917] | [0.523] | [0.733] | [0.945] | [0.442] | ||

NS-ADCC | 7.8820(2.368) | 8.0500(2.172) | 6.8929(2.407) | 6.7560(2.148) | 8.2153(2.413) | 8.4142(2.221) | 7.2187(2.453) | 0 |

[0.000] | [0.000] | [0.006] | [0.000] | [0.013] | [0.000] | [0.066] | ||

Na-Sm | Ca-Da | Ca-Ft | Ca-Sm | Da-Ft | Da-Sm | Ft-Sm | ||

S-DCC | 7.5970(1.947) | 10.2190(2.242) | 9.2129(1.436) | 8.8354(1.232) | 8.5388(0.915) | 9.3045(1.837) | 8.1860(2.395) | 5(8) |

[0.000] | [0.822] | [0.140] | [0.465] | [0.889] | [0.548] | [0.000] | ||

NS-DCC | 7.0929(2.193) | 10.6129(2.697) | 9.1373(2.960) | 9.1406(2.709) | 9.1602(2.746) | 9.1858(2.414) | 7.9692(2.715) | 2(2) |

[0.000] | [0.000] | [0.005] | [0.000] | [0.889] | [0.548] | [0.002] | ||

NS-BEKK | 6.9765(1.852) | 10.3884(2.538) | 8.9687(2.588) | 8.7994(2.439) | 9.0521(2.563) | 8.7474(2.143) | 7.7521(2.266) | 7(21) |

[0.856] | [0.178] | [0.889] | [0.548] | [0.889] | [0.548] | [0.521] | ||

NS-ADCC | 7.0177(2.155) | 10.6069(2.707) | 9.1373(2.960) | 9.1406(2.709) | 9.1602(2.746) | 9.1864(2.415) | 7.9690(2.715) | 2(2) |

[0.144] | [0.000] | [0.008] | [0.000] | [0.889] | [0.548] | [0.001] | ||

Panel B. The bi-component currency-stock portfolios | ||||||||

Udi-Ny | Udi-Sp | Udi-Na | Udi-Ca | Udi-Da | Udi-Ft | Udi-Sm | Sum | |

S-DCC | 4.1895(1.214) | 4.2479(1.338) | 4.7965(1.348) | 6.2143(1.543) | 6.3642(1.361) | 4.8727(1.547) | 5.1948(1.394) | 5 |

[0.002] | [0.433] | [0.028] | [0.115] | [0.388] | [0.315] | [0.297] | ||

NS-DCC | 4.1493(1.220) | 4.2775(1.272) | 4.7987(1.332) | 6.2952(1.528) | 6.4516(1.314) | 4.9358(1.567) | 5.2512(1.328) | 0 |

[0.005] | [0.000] | [0.000] | [0.000] | [0.000] | [0.000] | [0.001] | ||

NS-BEKK | 4.1102(1.214) | 4.2392(1.233) | 4.7541(1.236) | 6.1904(1.469) | 6.3584(1.270) | 4.8605(1.536) | 5.1756(1.149) | 7 |

[0.649] | [0.634] | [0.927] | [0.885] | [0.612] | [0.685] | [0.703] | ||

NS-ADCC | 4.1486(1.223) | 4.2577(1.276) | 4.7742(1.341) | 6.2952(1.528) | 6.4516(1.315) | 4.9358(1.567) | 5.2512(1.328) | 1 |

[0.002] | [0.049] | [0.129] | [0.000] | [0.000] | [0.000] | [0.001] |

**Table 12.**The summary results of the efficiency evaluation test based on the market risk capital (MRC) for the three weight combinations of the stock and currency-stock portfolios.

Stock Portfolios | Currency-Stock Portfolios | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

w_{1} | w_{2} | w_{1} | w_{2} | w_{1} | w_{2} | w_{1} | w_{2} | w_{1} | w_{2} | w_{1} | w_{2} | |

0.5 | 0.5 | 0.25 | 0.75 | 0.75 | 0.25 | 0.5 | 0.5 | 0.25 | 0.75 | 0.75 | 0.25 | |

S-DCC | 8 | 8 | 7 | 5 | 4 | 0 | ||||||

NS-DCC | 2 | 0 | 1 | 0 | 1 | 0 | ||||||

NS-BEKK | 21 | 19 | 20 | 7 | 7 | 7 | ||||||

NS-ADCC | 2 | 0 | 1 | 1 | 1 | 0 |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Su, J.-B.; Hung, J.-C.
The Value-At-Risk Estimate of Stock and Currency-Stock Portfolios’ Returns. *Risks* **2018**, *6*, 133.
https://doi.org/10.3390/risks6040133

**AMA Style**

Su J-B, Hung J-C.
The Value-At-Risk Estimate of Stock and Currency-Stock Portfolios’ Returns. *Risks*. 2018; 6(4):133.
https://doi.org/10.3390/risks6040133

**Chicago/Turabian Style**

Su, Jung-Bin, and Jui-Cheng Hung.
2018. "The Value-At-Risk Estimate of Stock and Currency-Stock Portfolios’ Returns" *Risks* 6, no. 4: 133.
https://doi.org/10.3390/risks6040133