A Maximum-Entropy Markov-Switching GARCH Framework: Information-Theoretic Bounds for Cryptocurrency Volatility Regime Detection

The distributional specification in Markov-switching GARCH (MS-GARCH) models has historically been driven by empirical convention. This paper derives the regime-conditional Student-t innovation distribution from the Tsallis Maximum Entropy Principle, providing an information-theoretic foundation for the choice of heavy-tailed innovations. The GARCH variance dynamics and Markov-switching structure are standard modelling choices adopted independently of the MaxEnt derivation. The framework is applied to five major cryptocurrencies over January 2017 to March 2026, comprising 15,824 daily observations. Three principal findings emerge. First, Tsallis entropy maximisation under a variance constraint yields the q-Gaussian density, which coincides with the Student-tνk distribution for qk=(νk+3)/(νk+1), with degrees of freedom determined endogenously from the empirical excess kurtosis. Second, calm-regime half-lives τC∈[1.21,2.37] days and stationary turbulent probabilities πT∈[0.254,0.437] confirm that both regimes are economically active across all assets; a Francq–Zakoïan stationarity verification confirms global ergodicity. Third, near-unity turbulent GARCH persistence suppresses the point-forecast advantage of regime-switching, consistent with a Fano-type Forecasting Irreversibility Bound; HAR-RV achieves the lowest QLIKE loss for three of five assets. Value-at-Risk backtests confirm adequate tail-risk calibration for four of five assets at the 1% and 5% levels, outperforming single-regime benchmarks. An empirical assessment of the VolShock extension identifies asset-class boundary conditions, motivating a proportional specification for future work.