Overview
GARCH (Generalized Autoregressive Conditional Heteroskedasticity) models the conditional variance of a return series, capturing volatility clustering: the tendency for large price moves to be followed by more large moves, and calm periods to be followed by more calm periods. Within this framework, GARCH is paired with ARIMA: ARIMA addresses the conditional mean (direction), GARCH addresses the conditional variance (dispersion around that mean).
Mathematical Formulation
The conditional variance is modeled as sigma_t^2 = omega + alpha * epsilon_(t-1)^2 + beta * sigma_(t-1)^2, where epsilon_(t-1) is the prior period’s shock and alpha + beta represents the persistence of volatility shocks. The model’s direct output is the conditional standard deviation, sigma_t, not a symmetric band by itself. Bands such as mu_t ± z * sigma_t are constructed afterward for a chosen purpose: z = 1 corresponds to roughly a 68% interval, z = 1.645 to roughly 90%, and z in the 1.96 to 2.33 range corresponds to standard 95 to 99% Value-at-Risk thresholds.
Empirical Notes
In internal testing on BTCUSD, realized coverage within a ±1 sigma GARCH band was approximately 67%, close to the theoretical ~68.3% for a normal distribution, a reasonable result for the first-moment band. At ±2 sigma and ±3 sigma, however, realized coverage diverged sharply from the theoretical ~95.4% and ~99.7%, with excursions well beyond the band. This is the standard signature of fat tails in crypto return distributions, and motivated evaluating a Student-t innovation distribution (GARCH-t) rather than a fixed-regime Gaussian assumption.
Regime-Switching Extensions (MS-GARCH)
Markov-Switching GARCH extends the single-regime model by allowing volatility parameters to differ across latent regimes (e.g. calm vs. crisis), estimated via the Baum-Welch algorithm with regime paths recovered via the Viterbi algorithm. This is a natural fit for FX, where central bank policy shifts, interventions, and peg breaks create identifiable regime boundaries; 2-regime (normal/crisis) models often fit emerging-market FX pairs well. It is a poor fit for BTC-style crypto markets, where market structure itself changes too irregularly for the time-invariant regime parameters assumption to hold, and where fat-tailed jumps are common. Reported classification accuracy figures for MS-GARCH regimes (sometimes 70 to 90% in published studies) are frequently full-sample smoothing results (Viterbi applied with hindsight) rather than genuine causal filtering results, and should not be taken at face value as representative of real-time performance.
Limitations
GARCH parameter estimates are unstable with limited sample sizes, particularly with 3+ regimes in the Markov-Switching case. Coverage should always be checked at multiple sigma levels, not just ±1 sigma, before drawing conclusions about tail risk, since fat tails can be masked by a good first-moment fit.
Related
Math Models · ARIMA · Kalman Filter · Wavelet Transform · Methodology