When a Weighted Sum of GARCH(1,1) Processes Is Not GARCH
Summary
The document asks whether a weighted sum of two conditionally correlated GARCH(1,1) series can itself be represented as a GARCH(1,1) process. It applies the variance formula for a sum, incorporating both component variances and their correlation. The resulting conditional variance includes a cross term involving the product of the components’ conditional standard deviations. In general, that expression does not have the standard single-series GARCH form, so the sum cannot generally be assigned parameters by simply combining the component parameters.
A special case is identified: when both component models share the same ARCH and GARCH coefficients and their correlation is zero, the sum has the same dynamic coefficients, while its constant term is the sum of the component constants. The answer presents the general conclusion informally rather than proving it, and it does not provide a corresponding result for arbitrary weights or nonzero correlation. It is therefore a useful structural warning, not a full characterization of all possible cases.
Key ideas
- The variance of a sum includes a term determined by the correlation between its components.
- That cross term generally prevents the sum from having the standard GARCH(1,1) form.
- Component parameters cannot generally be combined to obtain a single GARCH specification.
- With matching dynamic coefficients and zero correlation, the summed process retains those coefficients and adds the constant terms.
- The general non-equivalence is argued informally rather than formally proved.
Tags
Full text
# Sum of two GARCH(1,1) Models
# 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?
## Answer by Richard Hardy (score 2, accepted)
https://quant.stackexchange.com/a/24644
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 $\omega_2$.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.