GARCH Volatility Forecasting
Fit GARCH(1,1) models to components and residuals for volatility forecasts.
GARCH Volatility Forecasting
GARCH(1,1) models are fit to each principal component and residual series to forecast volatility over the investment horizon. These forecasts are combined to produce a forward-looking covariance matrix that incorporates current market conditions.
Key Steps:- Calculate residual indices from factor model
- Fit GARCH(1,1) to each component
- Fit GARCH(1,1) to each residual
- Generate volatility forecasts
- Construct GARCH covariance matrix
Garch Model:
$$
\sigma_t^2 = \omega + \alpha \epsilon_{t-1}^2 + \beta \sigma_{t-1}^2
$$
Garch Forecast:
$$
\sigma_{t+h}^2 = \omega \cdot \frac{1 - (\alpha + \beta)^h}{1 - (\alpha + \beta)} + (\alpha + \beta)^h \sigma_t^2
$$
Garch Covariance:
$$
Q_{\text{garch}} = \beta^\top \text{diag}(\sigma_{\text{comp}}^2) \beta + \text{diag}(\sigma_{\text{resid}}^2)
$$