Impact of Likelihood Approximation Choices on the Performance of Markov-Switching GARCH Models: Evidence from Thirty Global Financial Assets

Markov-Switching GARCH (MS-GARCH) models are widely used to capture changing volatility regimes in financial returns, but their exact likelihood is computationally intractable because it requires summing over an exponentially growing number of possible regime histories. Several approximation methods have been developed to solve this problem, yet there is little systematic evidence on how these methods compare when applied to real financial data with diverse distributional properties. This gap is important: choosing the wrong approximation method can lead to materially different parameter estimates, convergence failures, and unreliable risk assessments. This study addresses the gap by comparing four likelihood approximation methods for MS-GARCH(1,1) models across thirty financial assets from five market types: equities, bonds, cryptocurrencies, derivatives, and foreign exchange. The four methods are Gray’s moment-matching collapsing procedure, the Plug-in approximation, five-point Gauss-Hermite Quadrature, and a Particle Filter using sequential importance resampling with 1,000 particles. Models are estimated under two specifications: a baseline (internal volatility dynamics only) and an exogenous (cross-market spillover conditioning) version, giving 220 experimental cases in total. Assets are grouped into four kurtosis categories - low (K<10), medium (10≤K<20), high (20≤K<50), and extreme (K≥50) - to examine how heavy-tailed distributions affect method performance. The Plug-in method produced the highest log-likelihood values across all kurtosis groups and market types, often exceeding Gray and Quadrature by more than 2,000 log-likelihood units for equity-index futures. Gray and Quadrature converged more reliably for cryptocurrency assets under exogenous conditioning. The Particle Filter converged most often but always gave the lowest log-likelihood values, falling two to four thousand units below the analytical methods for all asset classes and with parameters stuck near their starting values. The CBOE Volatility Index (ˆVIX) was the only asset for which all analytical methods failed to converge in every specification, marking a hard estimation limit for the adopted framework. Exogenous conditioning improved model fit for some assets but not all, with bonds and equity markets benefiting most. A practical, kurtosis-based method-selection guide is proposed for practitioners.