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

Question

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

Answer 1

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.

Answer 2

No, a sum of two GARCH processes is generally not a GARCH process.

(I am not even sure whether there exists a nontrivial special case where the opposite holds.)

By GARCH I mean the classic definition of GARCH due to Bollerslev (1986), not an arbitrary variation like EGARCH, IGARCH, FIGARCH or whatever else.

Let me provide an example. Take two independent zero-conditional-mean processes $e_{1,t}$ and $e_{2,t}$. Let their conditional variances follow GARCH(1,1). Then the conditional variance equations of $e_{1,t}$ and $e_{2,t}$ are

$$ \begin{aligned} \sigma_{1,t}^2 = \omega_1 + a_1 e_{1,t-1}^2 + b_1 \sigma_{1,t-1}^2; \\ \sigma_{2,t}^2 = \omega_2 + a_2 e_{2,t-1}^2 + b_2 \sigma_{2,t-1}^2. \\ \end{aligned} $$

Take $e_t$ to be the simplest possible linear combination of $e_{1,t}$ and $e_{2,t}$, namely, their sum:

$$ e_t := e_{1,t} + e_{2,t}. $$

Will its conditional variance follow a GARCH process? If it would, we could express the conditional variance of $e_t$ as

$$ \sigma_t^2 = \omega + \sum_{i=1}^s \alpha_i e_{t-i}^2 + \sum_{i=1}^r \beta_i \sigma_{t-i}^2 $$

(a GARCH($s$,$r$) equation). To show the conditional variance of $e_t$ follows GARCH($s$,$r$) we need to find the appropriate $\omega$, $\alpha$s, $\beta$s, $s$ and $r$. Can this be done?

Let us start by writing the conditional variance of $e_t$ explicitly based on the fact that $e_t = e_{1,t} + e_{2,t}$ and the properties of $e_{1,t}$ and $e_{2,t}$. The conditional variance of $e_t$ will be the sum of the conditional variances of $e_{1,t}$ and $e_{2,t}$ (there are no covariances due to the assumed independence):

$$ \begin{aligned} \sigma_t^2 &= \sigma_{1,t}^2 + \sigma_{2,t}^2 \\ &= \omega_1 + a_1 e_{1,t-1}^2 + b_1 \sigma_{1,t-1}^2 \\ &+ \omega_2 + a_2 e_{2,t-1}^2 + b_2 \sigma_{2,t-1}^2 \\ &= (\omega_1+\omega_2) + (a_1 e_{1,t-1}^2+a_2 e_{2,t-1}^2) + (b_1 \sigma_{1,t-1}^2+b_2 \sigma_{2,t-1}^2). \\ \end{aligned} $$

It does not seem possible to express this in terms of $\sigma_t^2 = \omega + \sum_{i=1}^s \alpha_i e_{t-i}^2 + \sum_{i=1}^r \beta_i \sigma_{t-i}^2$ (but how to prove it formally?). And this is the simple example where $e_{1,t}$ and $e_{2,t}$ are independent (so we spare any covariances that would otherwise appear in the above expressions) and the lag orders of their respective GARCH processes coincide.

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Why do I arrive at a different conclusion than @John? His claim

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

is unfounded, i.e. there is no proof or derivation supporting it. On the contrary, the above expressions illustrate (admittedly, without a formal proof) that the inheritence from the two component processes does not add up to fit in the form of a GARCH model.

References:

• Bollerslev, Tim. "Generalized autoregressive conditional heteroskedasticity." Journal of Econometrics 31.3 (1986): 307-327.