Estimating Conditional Variance in a GARCH Model

Summary

The document introduces the GARCH model’s decomposition of a return innovation into a time varying conditional scale and a standardized shock. Its variance equation combines a constant, lagged squared innovations, and lagged conditional variances. The author notes that squared residuals serve as estimates of the squared innovations, then asks how the conditional variance itself is estimated from a time series during GARCH fitting.

No answer, estimation procedure, or fitted example is included. The note therefore functions as a focused conceptual question about the recursive variance sequence produced by model estimation, rather than as a practical guide to fitting or forecasting volatility. It identifies the distinction between observed residual based inputs and latent conditional variance estimates, but leaves initialization, parameter estimation, and diagnostic checks unspecified.

Key ideas

• A GARCH model writes the innovation as a conditional scale times a standardized shock.
• Conditional variance depends on a constant, lagged squared innovations, and lagged variances.
• The document asks how the conditional variance sequence is estimated from observed data.
• It does not provide a fitting method, initialization rule, or worked example.

Tags

# Questions on the concept of GARCH model


# Questions on the concept of GARCH model












As we all know, the GARCH model is stated as $\epsilon_t = \sigma_tz_t$ $\sigma_t^2 = w + \sum^q_{i=1}\alpha_i\epsilon_{t-i}^2 + \sum^q_{i=1}\beta_i\sigma_{t-i}^2$

In application, the estimate $\hat{\epsilon}_t^2$ is just the squared of the residuals. My question is what is the estimate of $\hat{\sigma}_t^2$ and how can it be obtained from the time series when using the GARCH estimation?

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.