# 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?

• I think this question is off topic. You should post it on the cross validated or stackoverflow sites – Quantopik Mar 21 '15 at 1:33
• In my opinion it's on-topic here and less so over there. GARCH main area of application seems to be Quantitative Finance / Risk Management. – Bob Jansen Mar 21 '15 at 10:48
• Have I answered your question? – Richard Hardy Aug 14 '16 at 11:38
• @BobJansen, there seems to be no finance-specific aspect in this question, except that GARCH and DCC models are usually used in finance. But is that sufficient? GARCH is a statistical time series model and as such should belong to Cross Validated. There are just over 300 threads on Cross Validated tagged with ARCH and GARCH (compare to under 170 here on QF), and more threads on volatilty forecasting. (However, the software-implementation aspect of the question would be off topic on Cross Validated.) – Richard Hardy Aug 14 '16 at 12:14
• On Stata I do not know how you can capture the spillover effect. On R there is the package dccgarch, in which you can fit an extended dccgarch model. – Konstantinos Gk Apr 29 '17 at 15:06

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