ARCH and GARCH

A return model can leave uncorrelated residuals whose squares remain predictable. That pattern calls for a variance model, not automatically a new mean model. ARCH testing checks for dependence in squared residuals. GARCH uses the latest shock and variance to forecast the next variance for risk estimation and position sizing. This lesson connects those mechanics to quasi-likelihood estimation and robust uncertainty estimates.

• run Engle's $TR^2$ ARCH LM test on mean-model residuals;
• write the GARCH(1,1) recursion, state positivity and $\alpha+\beta<1$, and read the news-impact curve;
• formulate Gaussian quasi-likelihood without claiming normality;
• use sandwich (robust QML) standard errors for inference.

1. Test residual variance, not the raw chart
2. ARCH and GARCH(1,1): standard model statements
3. Gaussian likelihood as quasi-likelihood
4. Estimation sequence