# Sum of two GARCH(1,1) Models

I have two GARCH(1,1) processes ($q=1,2$)

$$\sigma_{q,t} = \gamma_q + \alpha_q \, \sigma^2_{q,t-1} + \beta_q \, \epsilon^2_{q,t-1}$$

that have a constant correlation $\sigma_{12,t} = \rho \, \sigma_{1,t} \, \sigma_{2,t}$. This is sometimes called a CC-GARCH(1,1).

Is a (weighted?) sum of these two processes a GARCH process? If so, say my weight on the second process is $w_2$ ($w_1=1$) is it possible to calculate $\gamma$, $\alpha$ and $\beta$ for this new process?

• why donot you use multivariate garch model... – Neeraj Oct 6 '15 at 11:33
• CC-GARCH is a multivariate GARCH model. So let's say I've fit that and have the alphas, betas and gammas for both. What does the sum of the two variables behave like? – rhaskett Oct 6 '15 at 17:52

Let me use a notation that I am more used to:

$$\sigma^2_{i,t} = \omega_i + \alpha_i\varepsilon^2_{i,t-1} + \beta_i\sigma^2_{i,t-1}$$

where $i=1,2$. Since

$$\text{Var}(X+Y) = \text{Var}(X) + \text{Var}(Y) + \text{Corr}(X,Y)\sqrt{\text{Var}(X)}\sqrt{\text{Var}(Y)}$$

and

$$\text{Var}(x_{1,t})=\sigma_{1,t}^2, \ \ \ \text{Var}(x_{2,t})=\sigma_{2,t}^2 \ \ \ \text{and} \ \ \ \text{Corr}(x_{1,t},x_{2,t})=\rho,$$

we have

\begin{align} \text{Var}(x_{1,t}+x_{2,t}) &= \sigma_{1,t}^2 + \sigma_{2,t}^2 + \rho \sigma_{1,t} \sigma_{2,t} \\ &= (\omega_1 + \alpha_1\varepsilon^2_{1,t-1} + \beta_1\sigma^2_{1,t-1}) + (\omega_2 + \alpha_2\varepsilon^2_{2,t-1} + \beta_2\sigma^2_{2,t-1}) \\ &+ (\rho\sqrt{\omega_1 + \alpha_1\varepsilon^2_{1,t-1} + \beta_1\sigma^2_{1,t-1}} \sqrt{\omega_2 + \alpha_2\varepsilon^2_{2,t-1} + \beta_2\sigma^2_{2,t-1}}) \end{align}

which does not seem coercible to the shape of

$$\sigma^2_{t} = \omega + \alpha\varepsilon^2_{t-1} + \beta\sigma^2_{t-1}$$

for any $(\omega,\alpha,\beta)$. Therefore, generally a sum of two GARCH(1,1) processes is not a GARCH(1,1) process. (I say this without a formal proof.)

A very special case that is coercible is when $\alpha_1=\alpha_2, \beta_1=\beta_2$ and $\rho=0$; then $\omega=\omega_1+\omega_2,\alpha=\alpha_1=\alpha_2,\beta=\beta_1=\beta_2$. This is the case when the conditional variance dynamics is the same for both series and the only potential difference in the two GARCH models is the potentially different base level $\omega_1$ versus $\omega_2$.

• Makes sense. I ended up finding this empirically but this is much cleaner, thanks. – rhaskett Mar 1 '16 at 18:27
• If sum of tho GARCH processes isn't GARCH, how could we model Index and its components at once ? (say small index with up to 20 stocks), in intraday case for example we would like to quantive how sudden jumps in volatility of components could rise volatility of Index (to opposite jumps could be not visible when we look at Index) – Qbik Apr 20 '16 at 20:06