Non-homogeneous volatility correlations in the bivariate multifractal model

Abstract

In this paper, we consider an extension of the recently proposed bivariate Markov-switching multifractal model of Calvet, Fisher, and Thompson [2006. “Volatility Comovement: A Multifrequency Approach.” Journal of Econometrics 131: 179–215]. In particular, we allow correlations between volatility components to be non-homogeneous with two different parameters governing the volatility correlations at high and low frequencies. Specification tests confirm the added explanatory value of this specification. In order to explore its practical performance, we apply the model for computing value-at-risk statistics for different classes of financial assets and compare the results with the baseline, homogeneous bivariate multifractal model and the bivariate DCC-GARCH of Engle [2002. “Dynamic Conditional Correlation: A Simple Class of Multivariate Generalized Autoregressive Conditional Heteroskedasticity Models.” Journal of Business & Economic Statistics 20 (3): 339–350]. As it turns out, the multifractal model with heterogeneous volatility correlations provides more reliable results than both the homogeneous benchmark and the DCC-GARCH model.

Keywords:

• long memory
• multifractal models
• simulation-based inference
• value-at-risk

JEL Classification:

• C11
• C13
• G15

Acknowledgements

The authors of this manuscript are grateful for helpful comments by Roman Liesenfeld, Friedrich Wagner and two anonymous referees. Financial support by the German Science Foundation (Deutsche Forschungsgemeinschaft) is gratefully acknowledged.

Notes

1. The U.S. 1- and 2-year treasury constant maturity rates have been converted to equivalent bond prices before calculating returns.

2. Since we are conducting multiple tests for comparison of a range of heterogeneous correlation models against the homogeneous benchmarks, the p-values should be interpreted with some caution. All test statistics shown in the last two rows of Tables 5–7 are certainly highly correlated, e.g. if, say, the heterogeneous model with j=4 were the ‘true’ model, it would also be likely that a LR test of the ‘neighboring’ models with j=3 or 5 against the homogeneous model would lead to a rejection of the later. A standard approach that has been found to provide a satisfactory control on the false discovery rate is the modified Bonferroni procedure proposed by Simes (1986). This adaptation consists in adjusting the rejection probabilities for multiple tests: Let be the ordered p-vales for a set of n null hypothesis . Then a null hypothesis H₀ is rejected if for any j=1, … , n. A glance at Tables 5–7 shows that the underlying joint null hypotheses of the LR test (the null hypothesis that the homogeneous model cannot be rejected against any of the alternatives) can easily be rejected at standard significance levels of 5 or 1%.

3. We also used the simple constant correlation CC-GARCH model of Bollerslev (1990) which, however, performed mostly worse than DCC-GARCH.

4. Christoffersen's test covers only linear first-order dependence, but has no power against, for example, higher-order dependency in hit rates. Engle and Manganelli (2004) have proposed a test that captures dependency structures of higher order and other than linear form.