How to cite this WIREs title:
WIREs Comp Stat

Computational finance: correlation, volatility, and markets

Advanced Review

Katherine Bennett Ensor, Ginger M. Koev

Published Online: Jul 25 2014

DOI: 10.1002/wics.1323

Full article on Wiley Online Library: HTML PDF

Can't access this content? Tell your librarian.

Financial data by nature are inter‐related and should be analyzed using multivariate methods. Many models exist for the joint analysis of multiple financial instruments. Early models often assumed some type of constant behavior between the instruments over the time period of analysis. But today, time‐varying covariance models are a key component of financial time series analysis leading to a deeper understanding of changing market conditions. Models for covolatility of financial data quickly grow in their complexity and parameters, and 20 years of research offers a variety of solutions to this complexity. After a short introduction of univariate volatility models, this article begins with the basic multivariate formulation for time series covariance modeling and moves to leading time series tools that address this complexity. Coupling these models with regime switching via a Markov process extends the features that can be understood from market behavior. We ground this review in an example of modeling the covariance of securities within sectors and sectors within markets, with dynamics that allow for two different market regimes. Specifically, we simultaneously model individual daily stock data that belong to one of three market sectors and examine the behavior of the market as a whole as well as the behavior of the market sectors over time. A motivation for this characterization concerns portfolio diversification and stock anomalies, and we capture the changing comovement of stocks within and between sectors as market conditions change. For example, some of these market conditions include market crashes or collapses and common external influences. WIREs Comput Stat 2014, 6:326–340. doi: 10.1002/wics.1323 This article is categorized under: Applications of Computational Statistics > Computational Finance Statistical and Graphical Methods of Data Analysis > Multivariate Analysis Data: Types and Structure > Time Series, Stochastic Processes, and Functional Data

Biplot for covariance matrix of October 4, 2001.

[ Normal View | Magnified View ]

One step ahead forecasts for average correlations between sectors. The scale for each plot is 0 to 0.8.

[ Normal View | Magnified View ]

Daily stock returns of 63 series spanning three sectors. High‐ and low‐correlation regimes are depicted. The estimated regime probabilities, rounded to 1 decimal place, are given at the top of each graph.

[ Normal View | Magnified View ]

Daily stock returns of 63 series spanning three sectors.

[ Normal View | Magnified View ]

Biplot for covariance matrix of June 2, 2000.

[ Normal View | Magnified View ]

References

Engle, R. Autoregressive conditional heteroscedasticity with estimates of the variance of UK inflation. Econometrica 1982, 50:987–1008.
Bollerslev, T. Generalized autoregressive conditional heteroskedasticity. J Econom 1986, 31:307–327.
Bauwens, L, Laurent, S, Rombouts, JVK. Multivariate GARCH models: a survey. J Appl Econom 2006 CORE Discussion Paper 2003/31, 21:79–109.
Alexander, CO. Multivariate orthogonal factor GARCH, 1997. University of Sussex, Mimeo.
Bollerslev, T, Engle, RF, Wooldridge, JM. A capital asset pricing model with time varying covariances. J Polit Econ 1988, 96:116–131.
Bollerslev, T. Modeling the coherence in short‐run nominal exchange rates: a multivariate generalized arch model. Rev Econ Stat 1990, 72:498–505.
Christodoulakis, GA, Satchett, SE. Correlated arch: modelling the time‐varying correlation between financial asset returns. Eur J Multivariate Anal 2002, 139:351–370.
Engle, RF, Ng, VK, Rothschild, M. Asset pricing with a factor‐ARCH covariance structure: empirical estimates for treasury bills. J Econom 1990, 45:213–238.
Engle, R, Kroner, FK. Multivariate simultaneous generalized ARCH. Econom Theory 1995, 11:122–150.
Engle, RF. Dynamic conditional correlation—a simple class of multivariate GARCH models. J Bus Econ Stat 2002, 20:339–350.
Kariya, T. Mtv model and its application to the prediction of stock prices. In: Pullila, T, Puntanen, S, eds. Proceedings of the Second International Tampere Conference in Statistics, University of Tempere Finland, 1988.
Klaassen, F. Have exchange rates become more closely tied? Evidence from a new multivariate GARCH model, 1999. CentER Discussion Paper Series No. 1999‐10, Tilburg University.
Tse, YK, Tsui, AKC. A multivariate generalized autoregressive conditional heteroscedasticity model with time‐varying correlations. J Bus Econ Stat 2002, 20:351–362.
Vrontos, ID, Dellaportas, P, Politis, DN. A full‐factor multivariate GARCH model. Econom J 2003, 6:311–333.
Fleming, J, Kirby, C. Component‐driven regime‐switching volatility. J Financ Econom 2013, 2:263–301.
Klaassen, F. Improving GARCH volatility forecasts with regime‐switching GARCH. Empir Econ 2002, 27:363–394.
Davis, GM. An examination of some open problems in time series analysis. Ph.D. Dissertation directed by K. B. ensor, Rice University, http://hdl.handle.net/1911/18887, 2005.
Davis, GM, Ensor, KB. Risk: Diversification and performance precursors for stocks. Am J Math Manage Sci 2008, 28:255–273.
Pelletier, D. Regime switching for dynamic correlations. J Econom 2006, 131:445–473.
da Liva Filho, OC, Ziegelmann, FA, Dueker, MJ. Modeling dependence dynamics through copulas with regime switching. Insur: Math Econ 2012, 50:346–356.
Arruda, BP, Valls Pereira, PL. Analysis of the volatility`s dependency structure during the subprime crisis. Appl Econ 2013, 45:5031–5045.
Audrino, F. Forecasting correlations during the late‐2000s financial crisis: the short‐run component, the long‐run component, and structural breaks. Comput Stat Data Anal 2013, 76:43–60.
Kritzman, M, Li, Y, Page, S, Rigobon, R. Principal components as a measure of systemic risk. Technical Report, MIT Sloan Research Paper No. 4785‐10, 2010.
At‐Sahalia, Y, Fan, J, Xiu, D. High‐frequency covariance estimates with noisy and asynchronous financial data. J Am Stat Assoc 2010, 105:1504–1517.
Barndorff‐Nielsen, OE, Hansen, PR, Lunde, A, Shephard, N. Multivariate realised kernels: consistent positive semi‐definite estimators of the covariation of equity prices with noise and non‐synchronous trading. J Econom 2011, 162:149–169.
Hautsch, N, Kyj, LM, Oomen, RCA. A blocking and regularization approach to high‐dimensional realized covariance estimation. J Appl Econom 2012, 27:625–645.
Kyj, LM, Ostdiek, B, Ensor, KB. Realized covariance estimation in dynamic portfolio optimization. Technical Report, Working Paper, Humboldt‐Universit, 2009.
Kyj, LM. Estimating realized covariance using high frequency data. PhD Dissertation directed by K. B. Ensor and B. Ostdiek, Rice University. Available at: http://hdl.handle.net/1911/22195, 2008.
Kyj, LM, Ostdiek, B, Ensor, KB. Covariance estimation in dynamic portfolio optimization: a realized single factor model. In: American Finance Association 2010 Atlanta Meetings Paper, 2010.
Wang, Y, Zou, J. Vast volatility matrix estimation for high‐frequency financial data. Ann Stat 2010, 38:943–978.
Zhang, L. Estimating covariation: Epps effect, microstructure noise. J Econom 2011, 160:33–47.
Peña, D, Tiao, GC, Tsay, RS. A Course in Time Series Analysis. New York: John Wiley %26 Sons, Inc.; 2001.
Tsay, RS. Analsis of Financial Time Series. 3rd ed. New York: Wiley; 2010.
Ensor, KB. Time series factor models. WIREs Comp Stat 2013, 5:97–104. doi: 10.1002/wics.1245.
Lanne, M, Saikkonen, P. A multivariate generalized orthogonal factor GARCH model. J Bus Econ Stat 2007, 25:61–75.
Billio, M, Caporin, M, Gobbo, M. Flexible dynamic conditional correlation multivariate GARCH for asset allocation. Appl Financ Econom Lett 2006, 2:123–130.
Aielli, GP, Caporin, M. Variance clustering improved dynamic conditional correlation MGARCH estimators. Comput Stat Data Anal 2014, 76:556–576.
Bauwens, L, Hafner, CM, Pierret, D. Multivariate volatility modeling of electricity futures. J Appl Econom 2013, 28:743–761.
Yu, Y. Multivariate GARCH modeling and comparison to real kernel estimates. Doctoral Disertation directed by D. Pelletier and D. Pearce, North Carolina State University, 2013.
Hafner, CM, Reznikova, O. On the estimation of dynamic conditional correlation models. Comput Stat Data Anal 2012 1st issue of the Annals of Computational and Financial Econometrics; Sixth Special Issue on Computational Econometrics, 56:3533–3545.
Aielli, GP. Dynamic conditional correlation: on properties and estimation. J Bus Econ Stat 2013, 31:282–299.
Boudt, K, Danelsson, J, Laurent, S. Robust forecasting of dynamic conditional correlation GARCH models. Int J Forecast 2013, 29:244–257.
Muler, N, Yohai, V. Robust estimates for GARCH models. J Stat Plann Inference 2008, 138:2918–2940.
Taylor, S. Modelling Financial Time Series. Chichester: John Wiley %26 Sons, Inc.; 1986.
A. Ghalanos,. rugarch: Univariate garch models. R package version 1.2‐9, 2013. Available at: http://cran.r-project.org/web/packages/rugarch/.
Hamilton, JD. Time Series Analysis. Princeton, NJ: Princeton University Press; 1994.
Chan, T, Golub, G, LeVeque, R. Algorithms for computing the sample variance: analysis and recommendations. Am Stat 1983, 37:242–247.
Thomas, SJ, Ray, BK, Ensor, KB. Understanding the volatility and liquidity of corporate bonds through multivariate time series of counts with zero‐inflation. Technical Report, Rice University, Department of Statistics, 2013.
Nakatani, T. ccgarch: An R Package for modeling multivariate GARCH models with conditional correlations, r package version 0.2.1. Available at: http://cran.r-project.org/web/packages/ccgarch/ edition, 2013.
Ghalanos, A. rmgarch: Multivariate GARCH models. r package version 1.2‐0, 2013. Available at: http://cran.r-project.org/web/packages/rmgarch/ edition.
SAS Institute Inc.. Sas Onlinedoc, Version 8. Cary, NC: SAS Institute Inc.; 2000.
Stata Corp. Stata Statistical Software: Release 13. StataCorp LP: College Station, TX; 2013.

Access to this WIREs title is by subscription only.

Recommend to Your
Librarian Now!

The latest WIREs articles in your inbox

Sign Up for Article Alerts