Abstract
This paper introduces a novel test designed to assess the validity of time-varying smooth transition conditional covariance models. We develop a model driven by five scalar parameters in order to build the Lagrange Multiplier test within the framework of multivariate conditional heteroskedastic time series models with smooth transition functions. We detail the development of these tests, emphasizing their applicability. The methodology is scrutinized through Monte Carlo simulations, providing insights into its finite sample properties. Additionally, empirical illustrations underscore the practical relevance of the proposed tests, demonstrating their efficiency in capturing time-varying smooth transitions within financial datasets.
References
Almeida, D. de, L. K. Hotta, and E. Ruiz. 2018. “MGARCH Models: Trade-Off between Feasibility and Flexibility.” International Journal of Forecasting 34 (1): 45–63. https://doi.org/10.1016/j.ijforecast.2017.08.003.Search in Google Scholar
Amado, C., A. Silvennoinen, and T. Teräsvirta. 2019. “Models with Multiplicative Decomposition of Conditional Variances and Correlations.” In Financial Mathematics, Volatility and Covariance Modelling, 217–60. London: Routledge.10.4324/9781315162737-10Search in Google Scholar
Amado, C., and T. Teräsvirta. 2013. “Modelling Volatility by Variance Decomposition.” Journal of Econometrics 175 (2): 142–53. https://doi.org/10.1016/j.jeconom.2013.03.006.Search in Google Scholar
Barndorff-Nielsen, O. E., P. R. Hansen, A. Lunde, and N. Shephard. 2011. “Multivariate Realised Kernels: Consistent Positive Semi-definite Estimators of the Covariation of Equity Prices with Noise and Non-synchronous Trading.” Journal of Econometrics 162 (2): 149–69. https://doi.org/10.1016/j.jeconom.2010.07.009.Search in Google Scholar
Bauwens, L., S. Laurent, and J. Rombouts. 2006. “Multivariate GARCH Models: A Survey.” Journal of Applied Econometrics 21: 79–109. https://doi.org/10.1002/jae.842.Search in Google Scholar
Boudt, K., J. Danielsson, and S. Laurent. 2013. “Robust Forecasting of Dynamic Conditional Correlation GARCH Models.” International Journal of Forecasting 29 (2): 244–57. https://doi.org/10.1016/j.ijforecast.2012.06.003.Search in Google Scholar
Boudt, K., A. Galanos, S. Payseur, and E. Zivot. 2019. “Multivariate GARCH Models for Large-Scale Applications: A Survey.” In Handbook of Statistics, Vol. 41, 193–242. Elsevier. https://doi.org/10.1016/bs.host.2019.01.001.Search in Google Scholar
Campos-Martins, S., and C. Amado. 2021. “Modelling Time-Varying Volatility Interactions.” Working Paper, SSRN: https://ssrn.com/abstract=4573593.Search in Google Scholar
Caporin, M., and M. McAleer. 2008. “Scalar BEKK and Indirect DCC.” Journal of Forecasting 27 (6): 537–49. https://doi.org/10.1002/for.1074.Search in Google Scholar
Caporin, M., and M. McAleer. 2012. “Do We Really Need Both BEKK and DCC? A Tale of Two Multivariate GARCH Models.” Journal of Economic Surveys 26 (4): 736–51. https://doi.org/10.1111/j.1467-6419.2011.00683.x.Search in Google Scholar
Chang, C.-L., C.-P. Liu, and M. McAleer. 2019. “Volatility Spillovers for Spot, Futures, and ETF Prices in Agriculture and Energy.” Energy Economics 81: 779–92. https://doi.org/10.1016/j.eneco.2019.04.017.Search in Google Scholar
Chen, B., and Y. Hong. 2016. “Detecting for Smooth Structural Changes in GARCH Models.” Econometric Theory 32 (3): 740–91. https://doi.org/10.1017/s0266466614000942.Search in Google Scholar
Christoffersen, P., V. Errunza, K. Jacobs, and X. Jin. 2014. “Correlation Dynamics and International Diversification Benefits.” International Journal of Forecasting 30 (3): 807–24. https://doi.org/10.1016/j.ijforecast.2014.01.001.Search in Google Scholar
Comte, F., and O. Lieberman. 2003. “Asymptotic Theory for Multivariate GARCH Processes.” Journal of Multivariate Analysis 84: 61–84. https://doi.org/10.1016/s0047-259x(02)00009-x.Search in Google Scholar
Ding, Z., and R. Engle. 2001. “Large Scale Conditional Covariance Matrix Modeling, Estimation and Testing.” Working Paper FIN-01-029, NYU Stern School of Business.Search in Google Scholar
Engle, R. 2002. “Dynamic Conditional Correlation – a Simple Class of Multivariate GARCH Models.” Journal of Business & Economic Statistics 20: 339–50. https://doi.org/10.1198/073500102288618487.Search in Google Scholar
Engle, R., and F. Kroner. 1995. “Multivariate Simultaneous Generalized ARCH.” Econometric Theory 11: 122–50. https://doi.org/10.1017/s0266466600009063.Search in Google Scholar
Gao, J., B. Peng, W. B. Wu, and Y. Yan. 2024. “Time-varying Multivariate Causal Processes.” Journal of Econometrics 240 (1): 105671. https://doi.org/10.1016/j.jeconom.2024.105671.Search in Google Scholar
Hafner, C., and A. Preminger. 2009. “On Asymptotic Theory for Multivariate GARCH Models.” Journal of Multivariate Analysis 100 (9): 2044–54. https://doi.org/10.1016/j.jmva.2009.03.011.Search in Google Scholar
Hall, A. D., A. Silvennoinen, and T. Teräsvirta. 2023. “Building Multivariate Time-Varying Smooth Transition Correlation GARCH Models, with an Application to the Four Largest Australian Banks.” Econometrics 11 (1). https://doi.org/10.3390/econometrics11010005.Search in Google Scholar
Lee, T.-H., and X. Long. 2009. “Copula-based Multivariate GARCH Model with Uncorrelated Dependent Errors.” Journal of Econometrics 150 (2): 207–18. https://doi.org/10.1016/j.jeconom.2008.12.008.Search in Google Scholar
Luukkonen, R., P. Saikkonen, and T. Teräsvirta. 1988. “Testing Linearity against Smooth Transition Autoregressive Models.” Biometrika 75 (3): 491–9. https://doi.org/10.1093/biomet/75.3.491.Search in Google Scholar
Noureldin, D., N. Shephard, and K. Sheppard. 2014. “Multivariate Rotated ARCH Models.” Journal of Econometrics 179 (1): 16–30. https://doi.org/10.1016/j.jeconom.2013.10.003.Search in Google Scholar
Pakel, C., N. Shephard, K. Sheppard, and R. F. Engle. 2021. “Fitting Vast Dimensional Time-Varying Covariance Models.” Journal of Business & Economic Statistics 39 (3): 652–68. https://doi.org/10.1080/07350015.2020.1713795.Search in Google Scholar
Pedersen, R. S., and A. Rahbek. 2014. “Multivariate Variance Targeting in the BEKK-GARCH Model.” The Econometrics Journal 17 (1): 24–55. https://doi.org/10.1111/ectj.12019.Search in Google Scholar
Péguin-Feissolle, A., and B. Sanhaji. 2016. “Tests of the Constancy of Conditional Correlations of Unknown Functional Form in Multivariate GARCH Models.” Annals of Economics and Statistics 123/124: 77–101. https://doi.org/10.15609/annaeconstat2009.123-124.0077.Search in Google Scholar
Phillips, P. C., D. Li, and J. Gao. 2017. “Estimating Smooth Structural Change in Cointegration Models.” Journal of Econometrics 196 (1): 180–95. https://doi.org/10.1016/j.jeconom.2016.09.013.Search in Google Scholar
Sanhaji, B. 2017. “Testing for Nonlinearity in Conditional Covariances.” Journal of Time Series Econometrics 9 (2): 20160010. https://doi.org/10.1515/jtse-2016-0010.Search in Google Scholar
Silvennoinen, A., and T. Teräsvirta. 2009a. “Modeling Multivariate Autoregressive Conditional Heteroskedasticity with the Double Smooth Transition Conditional Correlation GARCH Model.” Journal of Financial Econometrics 7 (4): 373–411. https://doi.org/10.1093/jjfinec/nbp013.Search in Google Scholar
Silvennoinen, A., and T. Teräsvirta. 2009b. “Multivariate GARCH Models.” In Handbook of Financial Time Series, 201–29. Springer.10.1007/978-3-540-71297-8_9Search in Google Scholar
Silvennoinen, A., and T. Teräsvirta. 2015. “Modeling Conditional Correlations of Asset Returns: A Smooth Transition Approach.” Econometric Reviews 34 (1–2): 174–97. https://doi.org/10.1080/07474938.2014.945336.Search in Google Scholar
Supplementary Material
This article contains supplementary material (https://doi.org/10.1515/snde-2023-0109).
© 2025 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- Interview
- From Model Misspecification to Multidimensional Welfare: A Conversation with Professor Esfandiar Maasoumi
- Research Articles
- A Test for Time-Varying Smooth Transition Conditional Covariance Models in Multivariate Time Series
- Quasi-Maximum Likelihood for Estimating Structural Models
- Monetary Policy Uncertainty in the United States and Investment Sentiment in Advanced Economies
- Inflation: Demand Pull or Cost Push? A Markov Switching Approach
- Divisia Monetary Aggregates for India
- Introducing sspaneltvp: A Code to Estimating State-Space Time-Varying Parameter Models in Panels. An Application to Okun’s Law
Articles in the same Issue
- Frontmatter
- Interview
- From Model Misspecification to Multidimensional Welfare: A Conversation with Professor Esfandiar Maasoumi
- Research Articles
- A Test for Time-Varying Smooth Transition Conditional Covariance Models in Multivariate Time Series
- Quasi-Maximum Likelihood for Estimating Structural Models
- Monetary Policy Uncertainty in the United States and Investment Sentiment in Advanced Economies
- Inflation: Demand Pull or Cost Push? A Markov Switching Approach
- Divisia Monetary Aggregates for India
- Introducing sspaneltvp: A Code to Estimating State-Space Time-Varying Parameter Models in Panels. An Application to Okun’s Law