An Application of Copula-based DCC GARCH Model

Multivariate GARCH model: relationship of volatility between different financial data.

Example:

• 1 Spill-over effect between different markets
• 2 Co-persistence of volatility (Cointegration)

Basic Model: DCC-GARCH (Robert Engle 2002)

\[ \begin{split} &r_t|I_{t-1}\sim N(0,D_tR_tD_t)\\ D_t&=diag(\sigma_{1,t},\sigma_{2,t},...,\sigma_{N,t})\\ \sigma_{i,t}^2&=\omega+\sum_{p=1}^{P}a_pr_{i,t-p}^2+\sum_{q=1}^{Q}\beta_q\sigma_{i,t-q}^{2}\\ R_t&=diag(Q_t)^{-1/2}Q_t diag(Q_t)^{-1/2} \\ Q_t&=(1-a-b)\bar{Q}+az_{t-1}z'_{t-1}+bQ_{t-1} \\ \end{split} \]

Drawbacks:

• 1 Complicated Correlation (e.g. Extreme value)

• 2 Different Data type (e.g. Operational risk)

\( \rightarrow \) Copula-based DCC-GARCH

Sklar Theorem (1959): Let \( F \) be a d-dimensional distribution function, with d marginal distributions \( F_1,F_2,...,F_d \). Let \( A_j \) denote the range of \( F_j,A_j:=F_j(\mathbb{\bar{R}}) \). Then there exists a copula \( C \) such that for all \( (x_1,x_2,...,x_d)\in \mathbb{\bar{R}}^{d} \),

\[ F(x_{1},x_{2},x_{3},...,x_{d})=C(F_{1}(x_{1}),F_{2}(x_{2}),F_{3}(x_{3}),...,F_{d}(x_{d});\theta) \]

The Copula function is:

\[ C(u_1,u_2,...,u_d)=F(F_1^{-1}(u_2),F_1^{-1}(u_2),...,F_d^{-1}(u_d)) \]

where \( F_i^{-1} \) is pseudo-inverse function of \( F_i \).

Drawback 1: A variety of measures of correlations

• 1 Spearman's \( \rho \)

\[ \rho(C)=12\int_{0}^{1}\int_{0}^{1}C(u_1,u_2)du_1du_2-3 \]

• 2 Quadrant Dependent \( P[X_1\leq x_1,X_2\leq x_2]\leq(\geq)P[X_1\leq x_1]\times P[X_2\leq x_2] \)

• 3 Tail Dependence \( lim_{u\rightarrow 1}P(U_1>u|U_2>u) \)

Drawback 2: Transformation by marginal distribution function

\[ \begin{split} X_1&\rightarrow Poisson(\lambda)\rightarrow U_1\in[0,1] \\ X_2&\rightarrow N(\mu,\sigma^2)\rightarrow U_2\in[0,1] \\ &\rightarrow C(U_1,U_2) \\ \end{split} \]

Examples:

• 1 Elliptical copulas: normal-Copula, t-Copula
• 2 Archimedean copulas: Gumbel, Clayton and Frank Copula

Copula based DCC-GARCH

\[ \begin{split} &r_t|I_{t-1}\sim N(0,D_tR_tD_t)\\ D_t&=diag(\sigma_{1t},\sigma_{2t},...,\sigma_{Nt})\\ \sigma_{it}^2&=\omega+\sum_{p=1}^{P}a_pr_{t-p}^2+\sum_{q=1}^{Q}\beta_q\sigma_{t-q}^{2}\\ F(z_{1t},z_{2t},z_{3t},...,z_{dt})&=C(F_{1}(z_{1t}),F_{2}(z_{2t}),F_{3}(z_{3t}),...,F_{d}(z_{dt});R_t) \\ R_t&=diag(Q_t)^{-1/2}Q_t diag(Q_t)^{-1/2} \\ Q_t&=(1-a-b)\bar{Q}+az_{t-1}z'_{t-1}+bQ_{t-1} \\ \end{split} \]

2-step estimation:

• 1 Estimate marginal distribution (GARCH(1,1))

• 2 Estimate Copula function (MLE)

In this paper we choose daily price data of two stocks, Facebook (FB), Google (Go), from 2012/10/16 to 2016/10/16.

Conclusion: two data have ARCH effect.

Marginal Model 1: \( GARCH(1,1)-t(\nu) \)

Marginal Model 2: \( GARCH(1,1)-t(\nu) \)

Copula:\( t-Copula DCC(1,1) \)

• Mean Models:

\[ \begin{split} r_{1,t}&=\sigma_{1,t}z_{1,t},z_{1,t}\sim t_1(z;\nu=8.54) \\ r_{2,t}&=\sigma_{2,t}z_{2,t},z_{2,t}\sim t_2(z;\nu=6.65) \\ \end{split} \]

• Variance Models:

\[ \begin{split} \sigma_{1,t-1}^2&=0.00+0.06r_{1,t-1}^2+0.93\sigma_{1,t-1}^2 \\ \sigma_{2,t-1}^2&=0.00+0.14r_{1,t-1}^2+0.77\sigma_{1,t-1}^2 \\ \end{split} \]

Copula-DCC model:

\[ Q_t=(1-0.01-0.97)\bar{Q}+0.01z_{t-1}z'_{t-1}+0.97Q_{t-1} \]

t-Copula shape: \( \nu=9.28 \)

• 1 Long-term memory in volatility and correlation (\( \alpha+\beta \))

• 2 Fat tail of the residual distribution (\( \nu \))

• Marginal model: Parametric (SV, GAS), Realized Volatility

• Copula: Mixed Copula (Archimedean Copulas), Non-parametric (Kernel, MRF)

• Bauwens, L., S. Laurent, and J.V.K. Rombouts (2008), “Multivariate GARCH models: A survey”, Journal of Applied Econometrics.

• Drew Creal, Siem Jan Koopman and Andre Lucas (2013), “Generalized Autoregressive Score Models with Applications”, Journal of Applied Econometrics.

• Taras Bodnar, Nikolaus Hautsch (2012), “Copula-Based Dynamic Conditional Correlation Multiplicative Error Processes”.

• Luc Bauwens, Christian Hafner, Sebatien Laurent (2012), “Handbook of Volatility Models and Their Applications”.

• Jianqing Fan, Han Liu, Yang Ning, Hui Zou (2014), “High Dimensional Semiparametric Latent Graphical Model for Mixed Data”.