# Is a linear combination of GARCH processes also a GARCH process?

If two time series follow a GARCH process, and a third is a linear combination of them, is the third also GARCH process?

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The answer probably depends on the type of GARCH process. Combining $\mathrm{GARCH}(1, 1)$ and $\mathrm{eGARCH}(1, 1)$ probably won't work. Do you specifically mean standard GARCH so that your question becomes: is $\mathrm{GARCH}(p, q)$ closed under linear combination? –  Bob Jansen Jun 19 '12 at 18:16

I think there are a lot of different ways to specify this problem. For simplicity, consider independent Garch processes $$r_{1,t} \sim N\left(0,\sigma_{1,t}^{2}\right)$$ $$\sigma_{1,t}^{2} = \beta_{1,1}+\beta_{1,2}\varepsilon_{1,t-1}^{2}+\beta_{1,3}\sigma_{1,t-1}^{2}$$ and $$r_{2,t} \sim N\left(0,\sigma_{2,t}^{2}\right)$$ $$\sigma_{2,t}^{2} = \beta_{2,1}+\beta_{2,2}\varepsilon_{2,t-1}^{2}+\beta_{2,3}\sigma_{2,t-1}^{2}$$ where $\left[\begin{array}{cc} \varepsilon_{1,t} & \varepsilon_{2,t}\end{array}\right]\sim N\left(0,\left[\begin{array}{cc} 1 & 0\\ 0 & 1 \end{array}\right]\right)$.
In this case, the linear combination equals $$r_{3,t} = \alpha_{1}r_{1,t}+\alpha_{2}r_{2,t} \sim N\left(0,\alpha_{1}^{2}\sigma_{1,t}^{2}+\alpha_{2}^{2}\sigma_{2,t}^{2}\right)$$
Assuming the coefficients in the Garch equations are constrained to be positive and sum to less than or equal to one on the lagged values, then $r_{3,t}$ will also follow a Garch process as a result of inheriting the Garch variances of the other variables.