Flexible copula models with dynamic dependence and application to financial data
Introduction
The main goal of this article is to develop a dependence model that (a) can handle arbitrary continuous univariate time series models, (b) has dependence that can mostly be explained by one latent variable, (c) has dynamic or time-varying dependence, (d) has asymmetric tail dependence, and (e) is computationally tractable for maximum likelihood estimation.
Because of (a), (b) and (d), we consider factor copula models; and because of (b) and (e), we allow for conditional dependence given the latent variable. By choosing copulas linking observed variables to the latent variable having asymmetric tail dependence, and specifying the conditional distribution (i.e., conditional dependence) of the observed variables given the latent variable having a Gaussian copula, we can get the joint distribution of all observed variables with tail dependence. By having one latent variable and a Gaussian copula for the conditional dependence, the copula density of the observed variable involves only one-dimensional integration and the dependence structure can be quite flexible to allow for between group and within group dependence when the variables are in non-overlapping groups. The use of the Gaussian copula for conditional dependence also means that we can specify different parsimonious structural time-varying correlation matrices.
Our main application is for time series over many years of multiple financial assets such as stock returns, bond yields or CDS spreads. Our models are specified to match empirical and dependence behavior seen in initial data analysis of asset return data. Financial asset log returns typically show some positive serial correlations in the absolute log returns, and this is typically handled with GARCH filtering.
Suppose the data are denoted as . To see some dependence properties, it is useful to obtain summaries based on the rank transforms to normal scores; the jth variable vector in the period with ma < i ≤ mb, is ranked in the increasing order to get ranks and then is obtained using the normal scores transform of the jth variable in this time interval. Different intervals (ma, mb] can be used to assess time-varying patterns when the data correspond to many years (say over 3 years) so that stationarity over time might not be a good assumption.
The following patterns tend to hold with or without GARCH filtering.
- 1.
Consider moving window averages and standard deviations (SD) of separately by asset j, and moving window Spearman’s rank correlations, lower/upper semi-correlations (Section 2.17 of Joe (2014)) and lower/upper tail-weighted dependence measures (Krupskii and Joe (2015b)) for pairs from ma < i ≤ mb, j ≠ k. The quantities with the most variability are the SDs of returns. In contrast, Spearman’s rank correlations, semi-correlations and tail-weighted dependence measures change slowly over time.
- 2.
Bivariate normal scores plots of ma < i ≤ mb, j ≠ k, often show lower and upper tail dependence (with clouds of points that are sharper than ellipses in the lower and upper quadrant).
- 3.
The empirical correlation matrix of normal scores () often has close to a low-dimensional factor dependence structure.
For high-dimensional financial asset return data, the best fitting models for the short term have been vine copulas (Dißmann et al. (2013)) and factor copulas based on vines (Krupskii and Joe (2013), Krupskii and Joe (2015a)). Advantages of factor copulas over vine copulas include closure under margins, and possibly simpler interpretation with time-varying dependence.
An assumption of unchanging dependence can be too restrictive when modeling data over a long period of time. One example of such data consists of financial returns that have stronger dependence during market downturns. In this case, one can fit a static model to data for several non-overlapping time intervals and analyze the change in dependence in these intervals (Tamakoshi and Hamori, 2014) or use a normal or Student-t copula with time varying correlations proposed by C. Ausin and F. Lopes (2010). Silva et al. (2014) used this model to analyze the co-movement of credit default swaps (CDS) spreads and stock prices and Atil et al. (2016) analyzed the pairwise dependence in the sovereign CDS spreads between the US and Europe. This model cannot handle data with asymmetric dependence, however, and it can have many parameters to estimate if the dimension is very high.
Examples of copula models with time-varying dependence include bivariate observation-driven models in Patton (2006) and Creal et al. (2013), bivariate stochastic autoregressive copula models in Hafner and Manner (2012), semiparametric dynamic bivariate copula model in Hafner and Reznikova (2010), regime switching copulas in Okimoto (2008) (bivariate copulas), Chollette et al. (2009) (multivariate Gaussian and canonical vine copulas), time-varying high-dimensional D-vine copulas in Almeida et al. (2016).
Recently, H. Oh and Patton (2017) introduced a flexible linear factor model for modeling multivariate financial data and H. Oh and Patton (2018) proposed an extension of this model with time-varying coefficients which can be used for modeling non-Gaussian data with dynamic dependence. The authors adopted the generalized autoregressive score (GAS) model of Creal et al. (2011) to model time-varying coefficients in their model. A similar approach was used by Lucas et al. (2014) to model dynamics in coefficients of the skew-t distribution. The driving variable in these models is hard to interpret and some simplifying assumptions about the coefficients in these models are required to make them computationally tractable in high dimensions.
In this paper, we propose a new class of dynamic copula models which is an extension of factor copula models proposed by Krupskii, Joe, 2013, Krupskii, Joe, 2015a. The variables have a multivariate Gaussian distribution with time-varying correlations after conditioning on an unobserved latent factor. The copulas linking observed variables and the unobserved factor can be selected to handle tail dependence and asymmetry, and some exogenous variables can be included to explain time-varying correlations for the conditional Gaussian distribution. Flexible and computationally tractable models can be obtained with some parsimonious parameterizations of the correlation matrix.
The rest of this paper is organized as follows. In Section 2 we introduce a new class of one-factor conditional Gaussian copulas and discuss their dependence properties; we also consider different parsimonious parameterizations for the correlation matrix in this section. We provide more details on parameter estimation in Section 3. Simulations studies are given in Section 4. The proposed models are applied to analyze government bond yield data, CDS data and several sectors of stock return data in Section 5. The estimated models can be used for stress testing and we estimate the expected maximum number of companies or governments in distress under different scenarios for the first two data sets. Section 6 concludes with a discussion.
Section snippets
One latent factor combined with Gaussian copula for conditional dependence
In this section, we derive a copula model with one latent variable and conditional dependence given the latent variable specified with a Gaussian copula. In Section 2.1 we show through dependence properties that this model can handle joint upper and joint lower tail dependence with tail asymmetry, as well as having flexible dependence structure. In Section 2.2 we consider different parsimonious parameterizations of the correlation matrix of the conditional Gaussian copula.
Let d be the number of
Parameter estimation
In this section, some details are given in order that maximum likelihood estimation of model (1) can proceed efficiently when there is a parametric family for each Cj,0.
The details are provided for the model with Σt which has a nested structure, as the other cases in Section 2 are simpler. We assume that there are G > 1 non-overlapping groups of variables of size . Let the observed data . We assume that are realizations of independent random
Simulation results
In this section we show that the models in Section 2 can be used to handle heterogeneous dependence for data with group structures and check the performance of the estimation algorithm for the two simulated data sets generated from the model (1) with parameters corresponding to slowly and quickly changing dependence structure. We obtained similar results for other sets of copula parameters.
Dependence properties for bivariate marginals
We consider the nested structure for Σt as defined in Section
Empirical studies
In this section, we apply the model (1) with a nested structure of the correlation matrix Σt to analyze three financial data sets. The data sets used in the first two subsections were downloaded via a UBC Bloomberg terminal.
In Section 5.1 the first data set includes 10-year government bond yields for 11 European countries: Belgium, Denmark, France, Germany, Greece, Italy, Netherlands, Portugal, Spain and Sweden and United Kingdom. We use daily yields from March 2008 to December 2017, with 2265
Discussion
We proposed a copula-based model for modeling non-Gaussian data with dynamic dependence. The model allows greater flexibility when modeling data with heterogeneous dynamic dependence, in particular when dependence changes differently for different groups of variables. Parameters in the model can be estimated using the maximum likelihood approach even for high-dimensional data sets. The driving observed variables are used to model dynamic dependence and this allows interpretability. One example
Declaration of Competing Interest
We have no conflicts of interest.
Acknowledgments
The authors would like to thank the associate editor and external referee for their constructive comments that helped to improve this paper. The research was supported by funding from the Scotiabank-UBC Risk Analytics Initiative, and the Natural Sciences and Engineering Research Council of Canada (NSERC) Discovery Grant 8698.
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2022, Econometrics and StatisticsCitation Excerpt :Copulas provide a modular parameterization of multivariate distributions that decouples the modeling of marginals from the dependencies between them (Joe and Xu, 1996). Copulas have been extensively used in finance (Cherubini et al., 2004), and development of copula-based models is an active area of research, e.g., (Czado et al., 2019; Krupskii and Joe, 2020; Bladt and McNeil, 2021). They have also been used to model component distributions within mixture models, e.g., by Fujimaki et al. (2011); Kosmidis and Karlis (2016); Zhuang et al. (2021).