Forecasting with ARCH(m)

Procedure

Computing multi-step ahead volatility forecasts for ARCH(m) models.

Setup

Consider an ARCH(m) model at forecast origin $h$:

$${\sigma }_{h+\ell }^2={\alpha }_0+\sum _{i=1}^m{\alpha }_i{a}_{h+\ell -i}^2$$

We want to compute ${\sigma }_h^2(\ell )=E({\sigma }_{h+\ell }^2|F_h)$, the $\ell$-step ahead forecast.

1-Step-Ahead Forecast

At time $h+1$, all lagged values are known:

$$\boxed{{\sigma }_h^2(1)={\alpha }_0+{\alpha }_1{a}_h^2+{\alpha }_2{a}_{h-1}^2+...+{\alpha }_m{a}_{h+1-m}^2}$$

All quantities on the right-hand side are known at time $h$.

2-Step-Ahead Forecast

At time $h+2$, $a_{h+1}^2$ is unknown, so we substitute its expectation:

$${\sigma }_h^2(2)={\alpha }_0+{\alpha }_1{\sigma }_h^2(1)+{\alpha }_2{a}_h^2+...+{\alpha }_m{a}_{h+2-m}^2$$

ℓ-Step-Ahead Forecast (General Formula)

For $\ell >1$:

$$\boxed{{\sigma }_h^2(\ell )={\alpha }_0+\sum _{i=1}^m{\alpha }_i{\sigma }_h^2(\ell -i)}$$

where:

• If $\ell -i\le 0$, use $a_{h+\ell -i}^2$ (observed value)
• If $\ell -i>0$, use ${\sigma }_h^2(\ell -i)$ (previous forecast)

This is a recursive formula that builds forecasts step by step.

Long-Run Behavior

As $\ell \to \infty$, forecasts converge to the unconditional variance:

$${\sigma }_h^2(\ell )\to \frac{{\alpha }_0}{1-{\sum }_{i=1}^m{\alpha }_i}=E(a_t^2)$$

provided that ${\sum }_{i=1}^m{\alpha }_i<1$ (stationarity condition).

Interpretation

Volatility forecasts “mean-revert” toward the long-run average volatility. The speed of mean reversion depends on the sum of ARCH coefficients.

Example: ARCH(1)

For an ARCH(1) model ${\sigma }_t^2={\alpha }_0+{\alpha }_1{a}_{t-1}^2$:

Step	Formula	Notes
1	${\sigma }_h^2(1)={\alpha }_0+{\alpha }_1{a}_h^2$	Known $a_h^2$
2	${\sigma }_h^2(2)={\alpha }_0+{\alpha }_1{\sigma }_h^2(1)$	Substitute forecast
3	${\sigma }_h^2(3)={\alpha }_0+{\alpha }_1{\sigma }_h^2(2)$	Continue recursively
$\ell$	${\sigma }_h^2(\ell )={\alpha }_0+{\alpha }_1{\sigma }_h^2(\ell -1)$	General formula

Long-run: ${\sigma }_h^2(\ell )\to \frac{{\alpha }_0}{1-{\alpha }_1}$ as $\ell \to \infty$

Forecast Intervals

Forecast intervals for returns incorporate the conditional variance:

$$r_{h+\ell }|F_h\sim N\left(0,{\sigma }_h^2(\ell )\right)$$

A $(1-\alpha )$ forecast interval is:

$$\pm z_{\alpha /2}\cdot \sqrt{{\sigma }_h^2(\ell )}$$

where $z_{\alpha /2}$ is the standard normal critical value.

Related

• ARCH(m) Model
• Forecasting with GARCH(m,s)