Forecasting with GARCH(m,s)

Procedure

Computing multi-step ahead volatility forecasts for GARCH models, with special focus on the GARCH(1,1) case.

Setup

Consider a GARCH(1,1) model at forecast origin $h$:

$${\sigma }_{h+\ell }^2={\alpha }_0+{\alpha }_1{a}_{h+\ell -1}^2+{\beta }_1{\sigma }_{h+\ell -1}^2$$

The general GARCH(m,s) follows similar recursive logic.

1-Step-Ahead Forecast

All quantities are known at time $h$:

$$\boxed{{\sigma }_h^2(1)={\alpha }_0+{\alpha }_1{a}_h^2+{\beta }_1{\sigma }_h^2}$$

Note: ${\sigma }_h^2$ is the estimated conditional variance at time $h$.

2-Step-Ahead Forecast

At time $h+2$, both $a_{h+1}^2$ and ${\sigma }_{h+1}^2$ are unknown. We use:

$$a_{h+1}^2={\sigma }_{h+1}^2{ϵ}_{h+1}^2$$

Since $E({ϵ}_{h+1}^2|F_h)=1$:

$$E(a_{h+1}^2|F_h)={\sigma }_h^2(1)$$

Therefore:

$${\sigma }_h^2(2)={\alpha }_0+{\alpha }_1{\sigma }_h^2(1)+{\beta }_1{\sigma }_h^2(1)={\alpha }_0+({\alpha }_1+{\beta }_1){\sigma }_h^2(1)$$

ℓ-Step-Ahead Forecast (General Formula)

For $\ell >1$:

$$\boxed{{\sigma }_h^2(\ell )={\alpha }_0+({\alpha }_1+{\beta }_1){\sigma }_h^2(\ell -1)}$$

This recursive formula shows the key role of $({\alpha }_1+{\beta }_1)$ in forecast dynamics.

Explicit Formula

By repeated substitution:

$${\sigma }_h^2(\ell )=\frac{{\alpha }_0[1-({\alpha }_1+{\beta }_1{)}^{\ell -1}]}{1-({\alpha }_1+{\beta }_1)}+({\alpha }_1+{\beta }_1{)}^{\ell -1}{\sigma }_h^2(1)$$

Long-Run Behavior

As $\ell \to \infty$:

$$\boxed{{\sigma }_h^2(\ell )\to \frac{{\alpha }_0}{1-{\alpha }_1-{\beta }_1}}$$

provided that ${\alpha }_1+{\beta }_1<1$ (stationarity condition).

Interpretation

The term $({\alpha }_1+{\beta }_1)$ determines volatility persistence:

• Values close to 1 → slow mean reversion, forecasts stay elevated longer
• Values close to 0 → rapid mean reversion

Half-Life of Shocks

The “half-life” of a volatility shock is:

$$\text{Half-life}=\frac{\ln (0.5)}{\ln ({\alpha }_1+{\beta }_1)}$$

This measures how many periods it takes for a shock to half-decay.

Comparison: ARCH vs GARCH Forecasting

Feature	ARCH(m)	GARCH(1,1)
1-step	${\alpha }_0+\sum {\alpha }_i{a}_{h+1-i}^2$	${\alpha }_0+{\alpha }_1{a}_h^2+{\beta }_1{\sigma }_h^2$
ℓ-step recursive	${\sigma }_h^2(\ell )={\alpha }_0+\sum {\alpha }_i{\sigma }_h^2(\ell -i)$	${\sigma }_h^2(\ell )={\alpha }_0+({\alpha }_1+{\beta }_1){\sigma }_h^2(\ell -1)$
Long-run	$\frac{{\alpha }_0}{1-\sum {\alpha }_i}$	$\frac{{\alpha }_0}{1-{\alpha }_1-{\beta }_1}$
Persistence	Determined by $\sum {\alpha }_i$	Determined by ${\alpha }_1+{\beta }_1$

Forecasting Returns

Return forecasts:

$$E(r_{h+\ell }|F_h)=\mu \text{(if constant mean)}$$

Forecast variance increases with horizon due to accumulated uncertainty:

$$\text{Var}(r_{h+\ell }|F_h)>{\sigma }_h^2(\ell )$$

Related

• GARCH(m,s) Model
• Forecasting with ARCH(m)
• Property: Volatility Persistence