Regime Discovery and Intra-Regime Return Dynamics in Global Equity Markets

TL;DR

This paper addresses regime-dependent return dynamics in global equity markets by marrying a data-driven Hilbert--Huang Transform (HHT) regime identification with Holo--Hilbert Spectral Analysis (HHSA) for volatility profiling and Variable-Length Markov Chains (VLMC) to model intra-regime return transitions. Regimes are identified from the instantaneous energy $E(t)$ of the Hilbert spectrum, with thresholds defined by $\mu$ and $\sigma$ to yield Normal, High, and Extreme states; HHSA then provides cross-frequency volatility signatures via carrier frequency $\omega_c$, amplitude-modulation frequency $\omega_{am}$, and amplitude-modulation energy (AME or AME) to profile each regime. Within each regime, returns discretized into quintiles $R_1$ to $R_5$ are analyzed through VLMC context trees, yielding metrics on self-persistence, tail reversals, continuation, exhaustion, zigzag alternation, and burst-from-calm across orders $k=1,2,3$. The study finds that Extreme regimes concentrate tail risk and exhibit strong nonlinear dynamics, High regimes maximize entropy, and Normal regimes show distinct differences between developed and developing markets, with developing markets retaining residual tail dependence and burst risk even in Normal conditions. The results have practical implications for regime-aware risk management and highlight systematic maturity-related differences in regime stabilization and tail dynamics.

Abstract

Financial markets alternate between tranquil periods and episodes of stress, and return dynamics can change substantially across these regimes. We study regime-dependent dynamics in developed and developing equity indices using a data-driven Hilbert--Huang-based regime identification and profiling pipeline, followed by variable-length Markov modeling of categorized returns. Market regimes are identified using an Empirical Mode Decomposition-based Hilbert--Huang Transform, where instantaneous energy from the Hilbert spectrum separates Normal, High, and Extreme regimes. We then profile each regime using Holo--Hilbert Spectral Analysis, which jointly resolves carrier frequencies, amplitude-modulation frequencies, and amplitude-modulation energy (AME). AME, interpreted as volatility intensity, declines monotonically from Extreme to High to Normal regimes. This decline is markedly sharper in developed markets, while developing markets retain higher baseline volatility intensity even in Normal regimes. Building on these regime-specific volatility signatures, we discretize daily returns into five quintile states $\mathtt{R}_1$ to $\mathtt{R}_5$ and estimate Variable-Length Markov Chains via context trees within each regime. Unconditional state probabilities show tail states dominate in Extreme regimes and recede as regimes stabilize, alongside persistent downside asymmetry. Entropy peaks in High regimes, indicating maximum unpredictability during moderate-volatility periods. Conditional transition dynamics, evaluated over contexts of length up to three days from the context-tree estimates, indicate that developed markets normalize more effectively as stress subsides, whereas developing markets retain residual tail dependence and downside persistence even in Normal regimes, consistent with a coexistence of continuation and burst-like shifts.

Figures (7)

• Figure 1: Methodological flowchart illustrating the pipeline of the study: Daily returns data, checking non-linearity via BDS test, identifying and profiling market regimes via Empirical mode decomposition (EMD)-based Hilbert--Huang Transform (HHT) and Holo-Hilbert Spectral Analysis (HHSA), followed by intra-regime return dynamics modeling using Variable-Length Markov Chains (VLMC) and analysis with metrics.
• Figure 2: Regime classification for the NYSE Composite index NYA. Panel (a) shows the daily log-returns of the closing price. Panel (b) shows the 2D Hilbert spectrum from the Hilbert--Huang Transform, with carrier frequency on the vertical axis and time on the horizontal axis, and color indicating amplitude. Panel (c) shows the normalized instantaneous energy $E(t)$ computed from the instantaneous amplitudes associated with the Hilbert spectrum. Points are color-coded using energy thresholds, with green denoting Normal for $E(t)\le \mu+\sigma$, orange denoting High for $\mu+\sigma < E(t)\le \mu+6\sigma$, and red denoting Extreme for $E(t)>\mu+6\sigma$. Dashed horizontal lines mark $\mu+\sigma$ and $\mu+6\sigma$, where $\mu$ and $\sigma$ are the sample mean and standard deviation of the normalized energy series.
• Figure 3: Holo--Hilbert spectra (HHS) for the NYSE Composite index (NYA) over one-year windows selected to represent the three regimes identified from the instantaneous energy series in Fig. \ref{['fig:HHT_IE']}: (a) Extreme regime year 2008, corresponding to the red-coded energy points, (b) High regime year 2011, corresponding to the orange-coded energy points, and (c) Normal regime year 2005, corresponding to the green-coded energy points. In each panel, the vertical axis is the carrier frequency $\omega_c$ and the horizontal axis is the amplitude-modulation frequency $\omega_{am}$, while the color scale indicates amplitude-modulation energy (volatility intensity).
• Figure 4: Extreme (2008) regime context tree for NYSE Composite (NYA) index. The root node (*) with a rectangular box in bold border shows unconditional probabilities of $\mathtt{R}_1, \mathtt{R}_2, \mathtt{R}_3, \mathtt{R}_4 \space \text{and}\space \mathtt{R}_5$. First-level nodes represent conditioning on the most recent day (Day$-1$). Deeper nodes represent longer context sequences by adding older lags (Day$-2$, Day$-3$, etc.); e.g., the child node $\mathtt{R}_5$ under $\mathtt{R}_1$ corresponds to the two-day context [Day$-2=\mathtt{R}_5$] and [Day$-1=\mathtt{R}_1$].
• Figure 5: Holo--Hilbert spectra (HHS) for the Bovespa index (BVSP) over one-year windows selected to represent the three regimes identified from the instantaneous energy series: (a) Extreme regime year 2008, corresponding to the red-coded energy points, (b) High regime year 2011, corresponding to the orange-coded energy points, and (c) Normal regime year 2005, corresponding to the green-coded energy points. In each panel, the vertical axis is the carrier frequency $\omega_c$ and the horizontal axis is the amplitude-modulation frequency $\omega_{am}$, while the color scale indicates amplitude-modulation energy, used here as a measure of volatility intensity.