HMC

Fitting GARCH(1,1) models via NUTS on a 1,004-day market returns window isn’t just about crunching likelihoods—it’s navigating a brutal, multi-modal posterior landscape where standard samplers get permanently trapped. This is the third part of the volatility modeling series—the machinery behind everything here is part 1 and part 2. Here is what it found on long windows. When modeling extensive return histories, the posterior features a wide, high plateau alongside a lower mode tucked away at the $\omega \to 0$ corner of the persistence ridge, separated by a steep 12-nat drop. A standard MLE seed only finds a local shoulder, and naive samplers explore at best one corner. The lower mode’s peak sits $e^{-12} \approx 1.6 \times 10^{-6}$ times the plateau’s density—on the full history it contributes a sliver of the posterior mass. But the gap scales with how much data sits on the ridge (the window-size study below varies exactly that), and the two modes carry different $(\omega, \alpha+\beta)$ estimates: what a downstream model reads depends on which basin the chains landed in, and on long histories that must be found deliberately, not left to luck. ...