A dynamic multivariate heavy-tailed model for time-varying volatilities and correlations ⁄

TLDR: In this paper, a new class of observation-driven time-varying parameter models for dynamic volatilities and correlations is proposed to handle time series from heavy-tailed distributions.

Abstract: We propose a new class of observation-driven time-varying parameter models for dynamic volatilities and correlations to handle time series from heavy-tailed distributions. The model adopts generalized autoregressive score dynamics to obtain a time-varying covariance matrix of the multivariate Student t distribution. The key novelty of our proposed model concerns the weighting of lagged squared innovations for the estimation of future correlations and volatilities. When we account for heavy tails of distributions, we obtain estimates that are more robust to large innovations. We provide an empirical illustration for a panel of daily equity returns.

Figures

Table 2: Mean absolute error and mean squared error results: out-of-sample

Table 3: Parameter estimates, Likelihoods, and Information Criteria

Figure 1: Estimated correlations

Table 1: Mean absolute error and mean squared error results: in-sample

Full text

TI 2010-032/2
Tinbergen Institute Discussion Paper
A Dynamic Multivariate Heavy-Tailed
Model for Time-Varying Volatilities
and Correlations
Drew Creal
a
Siem Jan Koopman
b,d
André Lucas
c,d
a
Booth School of Business, University of Chicago;
b
Dept. of
Econometrics, VU University Amsterdam;
c
Dept. of Finance, VU University Amsterdam, Duisenberg school of finance;
d
Tinbergen Institute.

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A dynamic multivariate heavy-tailed model for
time-varying volatilities and correlations
∗
Drew Creal
a
, Siem Jan Koopman
b,d
, Andr´e Lucas
c,d
(a)
Booth School of Business, University of Chicago
(b)
Department of Econometrics, VU University Amsterdam
(c)
Department of Finance, VU University Amsterdam, Duisenberg school of finance
(d)
Tinbergen Institute, Amsterdam
March 15, 2010
Abstract
We propose a new class of observation-driven time-varying parameter models for
dynamic volatilities and correlations to handle time series from heavy-tailed distributions.
The model adopts generalized autoregressive score dynamics to obtain a time-varying
covariance matrix of the multivariate Student’s t distribution. The key novelty of our
proposed model concerns the weighting of lagged squared innovations for the estimation
of future correlations and volatilities. When we account for heavy tails of distributions,
we obtain estimates that are more robust to large innovations. The model also admits
a representation as a time-varying heavy-tailed copula which is particularly useful if the
interest focuses on dependence structures. We provide an empirical illustration for a panel
of daily global equity returns.
Keywords: dynamic dependence, multivariate Student’s t distribution, copula.
JEL classification codes: C10, C22, C32, C51.
∗
We would like to thank Ruey Tsay for his comments on an earlier draft of the paper. Corresponding author:
Drew Creal, University of Chicago, Booth School of Business, 5807 S. Woodlawn Ave, Chicago, IL 60637, phone:
773.834.5249 , email: dcreal@chicagobooth.edu.

1 Introduction
We contribute to the literature on multivariate modeling of volatilities and correlations by
introducing a class of observation-driven time-varying parameter models with heavy tailed
distributions. In particular, we consider a multivariate Student’s t model with time-varying
volatilities and correlations for which the multivariate Gaussian model is a special case. To
anticipate the needs of different users, we introduce two levels of flexibility in the model.
First, we propose a copula version of the model which treats the marginal distributions of
each of the series separately from the dependence structure. The copula model enables the
optimization problem to b e broken into more manageable pieces. Secondly, we mo dify the model
to accommodate alternative covariance matrix specifications. For example, we can consider the
square root of the correlation matrix in terms of hyperspherical coordinates. The general
model formulation enable us to impose a factor structure on either or both of the time-varying
volatilities and correlations.
Modeling the conditional distribution of a large group of assets is an important challenge in
modern financial time series analysis. Empirical evidence indicates that both the conditional
volatilities and correlations of assets change over time. Time-varying volatilities and correlations
among assets have practical implications for risk management and asset pricing. To capture
these features of the data, two classes of models are generally considered in the literature. The
first class comprises observation-driven models which include multivariate extensions of the
univariate generalized autoregressive conditional heteroskedastic (GARCH) family of models
introduced by Engle (1982) and Bollerslev (1986). The second class are parameter-driven
models such as the multivariate stochastic volatility models of Chib, Nardari, and Shephard
(2006) and Gourieroux, Jasiak, and Sufana (2009). This paper focuses on observation-driven
models for time-varying correlations. Time-varying correlation GARCH models were originally
developed by Ding and Engle (2001), Engle (2002), Engle and Sheppard (2001), and Tse and
Tsui (2002). Bauwens, Laurent, and Rombouts (2006) present a survey on multivariate GARCH
models covering time-varying correlation models as well as models for time-varying covariances.
When modeling the time-varying covariance matrix of a multivariate time series, it is well-
known that the number of static as well as time-varying parameters grows quickly as more series
are added. The increasing dimensionality of the parameter space creates challenging numerical
2

problems and some of the additional parameters may be statistically insignificant. Various
trade-offs have been recognized between building models that lead to a better fit of the data
statistically and balancing practical considerations. Many alternative models and estimation
procedures are proposed to address these challenges.
A popular device is the use of time-varying multivariate Gaussian copulas in which the
variances/standard deviations are modeled separately from the correlations. The numerical
optimization problem is then separated into more manageable pieces. Another key device is to
impose restrictions on the parameter space and to limit the number of parameters that control
the dynamics of the correlation matrix. Both of these strategies are taken by Engle (2002)
in the successful dynamic conditional correlation (DCC) model. The modeling approach is
motivated by pragmatic reasons as the DCC is intended to scale well when the cross-sectional
dimension increases. In related models a factor structure is imposed on the volatilities and
correlations; see, for example, Tsay (2005) and Fan, Wang, and Yao (2008). A factor structure
reduces the number of time-varying parameters and potentially allows the user to extract more
information from the data. Factor structures also allow us to pose interesting questions such as
which series share common features and what economic factors drive correlations. For example,
common macroeconomic shocks as well as arbitrage opportunities generally force some common
dynamics on groups of assets. Ultimately, the appropriateness of a model and its associated
estimation procedure depends upon the application.
This paper presents the details of how time-varying volatilities and correlations can be
incorporated in the multivariate Student’s t density using the generalized autoregressive score
(GAS) framework of Creal, Koopman, and Lucas (2010). The resulting model is shown to be
effective in treating different dynamic features simultaneously in a unified way. In our empirical
illustration, we analyze daily return data for a group of global equity indices over a period of
15 years. We show that our Student’s t GAS model accounts for outliers when updating the
correlations and volatilities over time.
The remainder of the paper is organized as follows. In Sections 2 and 3 we present the
basic model specification and updating equation. Section 4 proposes alternative specifications
and factor model structures. Section 5 discusses maximum likelihood estimation and carries
out a Monte Carlo study to compare its performance with other dynamic correlation models.
Section 6 contains our illustration. Section 7 concludes.
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Frequently asked questions

Q1. What contributions have the authors mentioned in the paper "A dynamic multivariate heavy-tailed model for time-varying volatilities and correlations" ?

The authors propose a new class of observation-driven time-varying parameter models for dynamic volatilities and correlations to handle time series from heavy-tailed distributions. The authors provide an empirical illustration for a panel of daily global equity returns. 

Q2. What are the future works in "A dynamic multivariate heavy-tailed model for time-varying volatilities and correlations" ?

Such and other extensions provide interesting avenues for further research. 

Q3. What is the key device in the dynamic conditional correlation model?

Another key device is to impose restrictions on the parameter space and to limit the number of parameters that control the dynamics of the correlation matrix. 

Q4. What is the generalized autoregressive score (GAS) model?

The generalized autoregressive score (GAS) model is an observation-driven model that allows parameters to change over time using information from the score of the observation density. 

Q5. What is the main reason for the problem with a correlation matrix?

The difficulty with specifying a correlation matrix is that three necessary conditions are needed: (i) the matrix Rt has to be positive (semi) definite; (ii) the off-diagonal elements of Rt all lie in the interval [−1, 1]; and (iii) the diagonal elements of Rt are equal to one for all values of t. 

Q6. What is the way to model the time-varying covariance matrix?

When modeling the time-varying covariance matrix of a multivariate time series, it is wellknown that the number of static as well as time-varying parameters grows quickly as more series are added. 

Q7. What is the popular method of analyzing the correlations?

A popular device is the use of time-varying multivariate Gaussian copulas in which the variances/standard deviations are modeled separately from the correlations. 

Q8. How many observations are used for the forecasting?

The authors extend the simulation of a bivariate series to 1005 observations from which the authors use the first n = 1000 for parameter estimation and the last 5 for out-of-sample forecasting. 

Q9. What is the simplest way to reduce the number of parameters in the model?

In addition, the number of parameters can be reduced by having coefficient matrices in (4) as scaled identity matrices or diagonal matrices. 

Q10. What is the reason why the density of the observations yt is heavy-tailed?

If the density of the observations yt is heavy-tailed (ν −1 > 0), large values in yty′t (in absolute terms) do not automatically lead to dramatic changes in the elements of Σt. 

Q11. What is the effect of the weight wt on the t-GAS?

Since incidental large squared errors enter the Gaussian factor recursion without the weight wt in Theorem 1, the maximum likelihood procedure downplays their impact on future volatilities and correlations by reducing the persistence parameters b1, . . . , b7. By contrast, the weight wt plays a role in the Student’s t-based factor recursion and, hence, it takes care of these observations in a natural way. 

Q12. What is the driving mechanism of a multivariate GARCH model?

For the normal distribution, the weights wt collapse to 1 and the authors obtain the familiar driving mechanism of a multivariate GARCH model. 

Q13. What is the driving mechanism of the density function at time t?

The GAS framework is developed by Creal, Koopman, and Lucas (2010) who let the driving mechanism st be the scaled derivative of the density function at time t with respect to the parameter vector ft, that isst = 

Q14. How can the authors generalize the model to the case 0 2?

It is straightforward, however, to generalize the model below to the case 0 < ν ≤ 2 by taking the scaling matrix of the Student’s t distribution, rather than its variance matrix as the key parameter. 

Q15. What is the common practice for estimating the constant vectors a and b?

In the multivariate GARCH cases, it is common practice to estimate the constant vectors a and b initially using sample variances and sample correlations. 

Q16. What is the significance of the variance matrix specification for the daily returns?

The authors therefore conclude that for their data set of daily returns, the variance matrix specification is less important for the formulation of volatility and correlation dynamics.