Volatility clustering is not evidence that an ARMA mean model has failed; it is evidence that the model’s squared innovations may remain predictable. Engle’s ARCH LM diagnostic tests exactly that residual feature. Once it is present, the ARCH/GARCH mechanics specify the variance recursion; this note establishes what likelihood-based estimation and inference mean when financial innovations are heavy-tailed rather than conditionally Gaussian.
Reading the figure. The return level need not be autocorrelated for its magnitude to cluster. The highlighted episodes motivate a conditional-variance test, but visual clustering alone neither selects a lag order nor distinguishes ARCH from structural breaks.
Learning objectives. By the end of this note you should be able to:
- run Engle’s ARCH LM test on fitted mean-model residuals;
- formulate Gaussian quasi-likelihood without claiming normality;
- state the consistency and asymptotic-normality reading of GARCH QMLE;
- use a robust sandwich covariance matrix for parameter inference.
In this lesson
- Test the residual variance, not the raw chart
- Gaussian likelihood as a quasi-likelihood
- A disciplined estimation sequence
Before you start
- Box–Jenkins Methodology: Identification, Estimation, and Diagnostic Checking
- ARCH and GARCH Volatility Models
The full lesson, with worked examples and exercise solutions, is for members.
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