Modeling Volatility and GARCH Family › Volatility Forecasting

The primary application of GARCH models is forecasting future volatility. For GARCH(1,1), the one-step-ahead forecast is straightforward:

$$\hat{\sigma}_{t+1}^2 = \omega + \alpha \epsilon_t^2 + \beta \sigma_t^2$$

where:

• $\hat{\sigma}_{t+1}^2$: forecasted variance for the next period
• $\omega, \alpha, \beta$: estimated model parameters
• $\epsilon_t^2$: today's squared return shock
• $\sigma_t^2$: today's conditional variance

Given today's shock $\epsilon_t$ and variance $\sigma_t^2$, tomorrow's variance forecast is simply the GARCH equation evaluated at current values. This forecast incorporates two sources of information: the specific return shock that just occurred and the overall volatility state. The simplicity of this calculation makes GARCH particularly attractive for real-time applications where you must update forecasts daily.

For longer horizons, you iterate the GARCH equation forward. Since the expected squared shock equals the variance ($E_T[\epsilon_{T+k}^2] = E_T[\sigma_{T+k}^2]$), the forecast follows a recursive pattern:

$$E_T[\sigma_{T+k}^2] = \omega + (\alpha + \beta) E_T[\sigma_{T+k-1}^2]$$

where:

• $E_T[\sigma_{T+k}^2]$: expected variance $k$ steps ahead
• $\omega, \alpha, \beta$: model parameters
• $E_T[\sigma_{T+k-1}^2]$: expected variance at the previous step

Subtracting the unconditional variance $\bar{\sigma}^2$ from both sides reveals the mean-reverting dynamic:

$$E_T[\sigma_{T+k}^2] - \bar{\sigma}^2 = (\alpha + \beta)(E_T[\sigma_{T+k-1}^2] - \bar{\sigma}^2)$$

where:

• $E_T[\sigma_{T+k}^2] - \bar{\sigma}^2$: expected deviation of variance from the long-run average at horizon $k$
• $\alpha + \beta$: persistence parameter determining decay rate
• $E_T[\sigma_{T+k-1}^2] - \bar{\sigma}^2$: expected deviation at the previous step

This relationship shows that the expected deviation from unconditional variance shrinks by a factor of $(\alpha + \beta)$ each period. Starting from any initial variance, forecasts converge geometrically toward the long-run level. Iterating this relationship $h-1$ times yields the $h$-step-ahead forecast from time $T$:

$$E_T[\sigma_{T+h}^2] = \bar{\sigma}^2 + (\alpha + \beta)^{h-1}(\hat{\sigma}_{T+1}^2 - \bar{\sigma}^2)$$

where:

• $E_T[\sigma_{T+h}^2]$: expected variance $h$ days ahead given info at $T$
• $\bar{\sigma}^2$: unconditional long-run variance
• $\alpha, \beta$: persistence parameters
• $h$: forecast horizon
• $\hat{\sigma}_{T+1}^2$: next day's variance forecast

As the horizon $h$ increases, the forecast converges to the unconditional variance. This is mean reversion in action. As you become increasingly uncertain about future volatility, your best guess converges to the long-run average. At very long horizons, your specific knowledge of current market conditions becomes irrelevant, and you fall back on the unconditional variance as your forecast.