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Long-memory GARCH via a two-dimensional Markov chain

This paper introduces a novel GARCH-type model utilizing a two-dimensional Markov chain to generate state-dependent power-law decay of past shocks, thereby achieving long-memory volatility persistence with strong theoretical stability guarantees and competitive empirical forecasting performance.

Original authors: Kyungsub Lee, Kennedy Titus Kayaki

Published 2026-07-29

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Kyungsub Lee, Kennedy Titus Kayaki

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the stock market as a giant, chaotic ocean. Sometimes the waves are gentle and predictable; other times, a sudden storm hits, and the water churns violently for days. In the world of finance, this "churning" is called volatility. For decades, scientists have tried to build mathematical maps to predict these waves. The most famous maps, called GARCH models, work like a simple echo: if a big wave hits today, the ocean stays a bit rougher tomorrow, but the memory of that wave fades away quickly, like a shout in a canyon that dies out after a few seconds.

However, real-world data suggests the ocean has a much longer memory. A massive storm today might make the waves feel "rough" for weeks or even months, a phenomenon known as long-memory. Traditional maps struggle with this because they are built to forget too fast. To fix this, some researchers tried building maps with infinite layers of memory, but those became so complicated they were hard to use or prove were stable. This paper enters the scene with a fresh idea: what if we could capture that long, lingering memory using a simple, two-part system that is easy to track and mathematically sound?

The Two-Dimensional Memory Machine

The authors, Kyungsub Lee and Kennedy Titus Kayaki, propose a new way to model financial volatility. Instead of just tracking how "rough" the water is right now (the level), their model also tracks how fast that roughness is expected to fade away (the slope).

Think of it like a magical, self-adjusting rubber band.

The Level: This is how stretched the rubber band is right now. A big shock (like a sudden market crash) stretches it out, making the "volatility" high.
The Slope: This is the rubber band's "stiffness" or how quickly it wants to snap back to normal.

In older models, the rubber band always snapped back at the same fixed speed. But in this new model, every time a shock happens, it doesn't just stretch the band; it also changes the band's stiffness. If the shock is big enough, it might make the band "looser," meaning it will take much longer to snap back to normal. This creates a feedback loop: the history of shocks changes the future speed of recovery, allowing the model to remember the past for a very long time without needing an infinite number of variables.

How They Proved It Works

The biggest worry with these complex, self-adjusting systems is that they might go crazy. If the rubber band gets too loose, the model could predict infinite volatility, which is nonsense. The authors spent a lot of time proving that their system is safe.

They used a mathematical tool called a Foster–Lyapunov condition. Imagine this as a safety net. They showed that no matter how wild the market gets, the system has a built-in mechanism that eventually pulls the rubber band back toward a stable state. They proved that the model is positive Harris recurrent, which is a fancy way of saying: "The system will always settle down into a predictable pattern and won't run off into infinity." They also showed that this stable pattern is unique, meaning the model doesn't get confused and switch between different behaviors.

Simulations: Testing the Long Memory

To see if their idea actually creates long memory, the team ran thousands of computer simulations. They fed the model random market shocks and watched how the "volatility" behaved over time.

They found that when the model's parameters were tuned close to a specific "stability boundary" (the edge of the safety net), the simulated market behaved exactly like real long-memory data. The shocks didn't fade away quickly; instead, they lingered, creating a slow, power-law decay. This confirmed that their simple two-part system could indeed mimic the stubborn, long-lasting volatility seen in real financial markets.

Real-World Test Drive

The authors then took their model for a spin using real data from six major financial markets: the S&P 500, FTSE 100, DAX, Nikkei 225, KOSPI, and Bitcoin.

The Good News:
The model was surprisingly good at capturing the "long memory" of these markets. When they compared the model's predictions to the actual data, the memory estimates matched up very well for several markets, like the Nikkei 225 and the S&P 500. It successfully recreated the slow decay of volatility that older, simpler models missed.

The Comparison:
They pitted their model against the heavyweights of the field:

GARCH(1,1): The standard, simple model.
FIGARCH: A complex model designed specifically for long memory.
HAR-RV: A model that uses past realized variance (a very direct measure of past volatility).

The Results:

Fitting the Past: The complex FIGARCH model did the best job of "fitting" the historical data (it had the highest likelihood). The new model was slightly behind FIGARCH but very close to the standard GARCH(1,1).
Predicting the Future: This is where it gets interesting. When predicting future volatility, the new model performed just as well as the standard GARCH(1,1) in most markets. It didn't beat FIGARCH overall, but it was statistically just as good for most markets, with the exception of KOSPI and Bitcoin, where FIGARCH had a slight edge.
The Real Winner: Interestingly, the HAR-RV model (which uses direct past variance data) was the best predictor of all. However, the authors note that HAR-RV is a specialized tool that uses information the new model doesn't have (it looks at realized variance directly, not just returns).

The Takeaway

The paper concludes that you don't need an infinitely complex machine to capture long memory. By adding just one extra dimension—a "memory scale" that changes based on shocks—you can build a model that is:

Mathematically safe: It won't blow up.
Simple: It only tracks two numbers at a time.
Effective: It captures the stubborn, long-lasting nature of market volatility almost as well as the most complex models.

It's a reminder that sometimes, the best way to understand a complex, churning ocean is not to map every single wave, but to understand how the water itself remembers the storm.

Technical Summary: Long-memory GARCH via a two-dimensional Markov chain

Problem Statement
Standard GARCH models capture volatility clustering through a simple recursive feedback of past squared innovations and conditional variances. However, empirical evidence suggests that financial volatility exhibits "long memory," characterized by slowly decaying dependence in volatility proxies (e.g., squared returns) that conventional GARCH specifications fail to replicate. Existing solutions, such as Fractional GARCH (FIGARCH), HYGARCH, or infinite-order ARCH models, introduce persistence through fractional filters, hyperbolic decay, or infinite lag structures. While effective, these approaches often lack a finite-dimensional Markov representation, complicating stability analysis and computational tractability. Conversely, stochastic volatility models with latent factors or rough volatility models address persistence but often struggle with empirical distinguishability or lack a direct conditional variance recursion.

Methodology
The authors propose a new GARCH-type model that generates long-memory behavior while retaining a finite-dimensional (two-dimensional) Markov state. The core mechanism adapts the "level-and-slope matching" construction (Lee, 2026b) to conditional volatility modeling.

Latent Power-Law Kernel: Instead of a fixed autoregressive decay, the conditional variance is derived from a latent power-law kernel $h_n(t) = \mu + a_n(t+c_n)^{-p}$ .
State Update Mechanism: Upon observing a return shock $\eta_n$ $η_{n}$ , the kernel is updated to the next period by matching both the level and the slope at the new origin.
- The state vector is $(X_n, c_n)$ , where $X_n$ represents the excess variance level and $c_n$ represents an endogenous memory scale.
- The update rules (Equations 6–7) ensure that the new state $(X_n, c_n)$ is a deterministic function of the previous state and the squared innovation $\eta_n^2$ .
Markov Representation: The resulting process is a time-homogeneous Markov chain on the state space $S = [0, \infty) \times [\gamma, \infty)$ . The model admits a random-coefficient ARCH( $\infty$ ) representation where the decay factors are endogenous and state-dependent, rather than fixed deterministic coefficients.

Key Contributions

Stability Theory: The primary mathematical contribution is a rigorous stability analysis for this non-standard Markov chain. Unlike standard GARCH models where stability reduces to a one-dimensional log-moment condition, this model requires a joint Foster–Lyapunov condition.
- The authors construct a Lyapunov function $V(x, c) = \log(1+x) + \lambda c$ that couples the variance level and the memory scale.
- They derive a joint drift condition (Assumption 1) requiring the existence of a $\lambda > 0$ such that the supremum of the combined drift of the log-variance and the memory scale is negative.
- Under this condition and regularity assumptions on the innovation distribution, the paper proves the chain is positive Harris recurrent and admits a unique invariant probability measure.
Diagnostic Condition: The paper derives a necessary diagnostic condition for stability (Equation 35), analogous to the standard GARCH log-moment condition but evaluated at a critical endogenous scale $c^*$ . Numerical experiments show this diagnostic boundary is extremely close to the true joint stability boundary.
Regularity Verification: Using control-theoretic arguments (Meyn and Tweedie framework), the authors establish $\psi$ -irreducibility and the petiteness of compact sets by demonstrating that the deterministic control model can reach a boundary anchor $(0, \gamma)$ and that the Jacobian rank condition holds locally.

Results

Simulation Studies:
- Using the local Whittle estimator on log-squared innovations, simulations demonstrate that the model generates substantial low-frequency persistence (long memory).
- Persistence is strongest when the parameters place the system near the diagnostic stability boundary. As the memory scale $c_n$ becomes large, the endogenous decay factors remain close to unity, producing slowly decaying dependence.
- The estimated memory parameter $\hat{d}$ increases monotonically as the system approaches the stability frontier.
Empirical Analysis:
- The model was fitted to daily returns of five international equity indices (S&P 500, FTSE 100, DAX, Nikkei 225, KOSPI) and Bitcoin.
- In-Sample Fit: The LM-GARCH model captures significant long-memory behavior, with model-implied memory estimates closely matching data-based estimates for the Nikkei 225, S&P 500, and FTSE 100. However, its in-sample log-likelihood is generally lower than that of FIGARCH and asymmetric GARCH models (GJR, EGARCH) for equity indices, though it outperforms them for Bitcoin.
- Out-of-Sample Forecasting: Using QLIKE loss on realized variance, LM-GARCH performs competitively with GARCH(1,1). While FIGARCH yields lower average losses across all markets, the difference is statistically significant only for KOSPI and Bitcoin. HAR-RV (using realized variance directly) outperforms all return-based models, as expected.

Significance and Claims
The paper claims to offer a parsimonious framework that reconciles the generation of long-memory volatility with a low-dimensional Markov structure.

Mechanism: It demonstrates that long memory can emerge from the dynamics of the state itself (specifically the evolution of the memory scale $c_n$ ) rather than being imposed via fractional integration or infinite lags.
Tractability: By maintaining a two-dimensional state, the model allows for standard stability analysis and computationally efficient estimation, unlike infinite-order or fractional models.
Empirical Utility: The model successfully reproduces the persistence observed in financial data and offers forecast accuracy comparable to standard benchmarks, suggesting that the "long-memory" phenomenon can be effectively modeled without abandoning the Markovian framework.

The authors remain modest regarding the model's dominance, acknowledging that while it captures substantial persistence, it does not outperform FIGARCH in-sample across all markets, and its out-of-sample advantage over standard GARCH is limited to specific markets. The primary contribution is the theoretical demonstration that a finite-dimensional Markov process can generate complex, long-memory volatility dynamics through endogenous state evolution.