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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
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up vote 10 down vote accepted

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.

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good answer, but usually we work with logarithmic returns, if 'x1' and'x2' are returns and x3=a1*x1+a2*x2, and r1=ln(x1),r2=ln(x2), r3=ln(x3) then r3 isn't equal to r1+r2 –  Qbik Jun 20 '12 at 19:42
    
True, but the answer I provided was mainly for illustrative purposes since you weren't particularly clear about under what conditions, what kind of Garch, etc. –  John Jun 20 '12 at 21:06
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