Why DCC-GARCH Does Not Model Direct Volatility Spillovers
Summary
The document explains why a standard dynamic conditional correlation (DCC) GARCH model does not allow unrestricted cross effects in the conditional variance equations. In the described two asset setup, each asset's conditional variance is fitted with a univariate GARCH process. Standardized residuals from those fits are then used to estimate changing correlations. This differs from a multivariate variance specification where one asset's lagged squared shock or variance directly affects another asset's variance.
The answer says the requested full ARCH and GARCH matrices cannot be specified within the stated DCC model, and notes that the equation in the question resembles a restricted VECH model, with squared errors required. It recommends considering BEKK-GARCH to represent spillovers, while noting that the answer's author had not seen it implemented in Stata. The discussion is conceptual and points to the original model paper and software manual; it does not provide implementation details or empirical comparisons.
Key ideas
- DCC-GARCH models each series' conditional variance separately before estimating dynamic correlations.
- Standard DCC does not include direct cross-asset shock or variance terms in those variance equations.
- A full cross-effect variance model is outside the DCC specification described.
- BEKK-GARCH is suggested as an alternative for explicit volatility spillovers.
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# DCC GARCH: specifying ARCH and GARCH parameter matrices in STATA
# DCC GARCH: specifying ARCH and GARCH parameter matrices in STATA
The command in STATA to estimate the DCC model of two variables is:
`mgarch dcc ( x1 x2=, noconstant) , arch(1) garch(1) distribution(t)`
$$ \begin{bmatrix} h_1{t} \\ h_2{t} \end{bmatrix} = \begin{bmatrix} w_{10} \\ w_{20} \end{bmatrix} + \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix} \begin{bmatrix} \epsilon_{1t-1} \\ \epsilon_{2t-1} \end{bmatrix} + \begin{bmatrix} g_{11} & g_{12} \\ g_{21} & g_{22} \end{bmatrix} \begin{bmatrix} h_{1t-1} \\ h_{2t-1} \end{bmatrix} $$
When I give this command, STATA understands that the ARCH and GARCH matrices are diagonal, i.e. $a_{21}=a_{12}=g_{21}=g_{12}=0$. How can I change this to implement a FULL ARCH and GARCH parameter matrices, to capture the spillover effects?
## Answer by Richard Hardy (score 2)
https://quant.stackexchange.com/a/24534
> How can I change this to implement FULL ARCH and GARCH parameter matrices, to capture the spillover effects?
You cannot.
The original paper by Engle (2002) as well as the Stata manual for the DCC-GARCH model reveal that the model admits a different form than the one represented in the equation in your question. (What you have there is a special case of a restricted VECH-GARCH model -- but the error terms in your formula should be squared.)
A DCC-GARCH model starts out by modelling the conditional variances of the individual assets as univariate GARCH processes. The fitted cond. variances are used to scale the residuals from the cond. mean model (if any; otherwise the residuals coincide with the raw data). Then the scaled residuals are used for modelling the cond. correlation matrices; the model used in this step is sort of a GARCH model but this time it considers cond. correlation matrices instead of scalar cond. variances.
This is roughly the logic of the DCC model. For more details and formulas you may refer to the original paper or the Stata manual. The takeaway in your case is that the spillover effects cannot be modelled explicitly using the DCC-GARCH model -- because there is no explicit dependence of the cond. variance $h_{1,t}$ of the component series $x_{1,t}$ on the lagged cond. variance $h_{2,t-1}$ or the lagged squared error $\varepsilon^2_{2,t-1}$ from the component series $x_{2,t}$.
For spillover effects you could use BEKK-GARCH model, but I have not seen it implemented in Stata.
References
- Engle, Robert. "Dynamic conditional correlation: A simple class of multivariate generalized autoregressive conditional heteroskedasticity models." Journal of Business & Economic Statistics 20.3 (2002): 339-350.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.