Non-Stationary Transition Matrix

A discrete Markov Chain Trading Model rests on time-homogeneity — the assumption that the transition probability matrix P, which gives the probability of moving from each price state to each other, is constant through time. The matrix is estimated by maximum-likelihood counting on a training window (P_ij = n_ij / n_i) and then used to forecast the next period. The failure mode is that in real markets P is not constant: it drifts with volatility regimes, news flow, liquidity conditions and structural breaks, so a matrix estimated on past data systematically misforecasts the next period. This is the model-specific, mechanical instance of the general Non-Stationarity failure mode, and it is the dominant reason simple Markov-chain price prediction fails out-of-sample.

The vault’s two discrete-Markov price-prediction papers both run straight into it. Wilinski 2019 — the only peer-reviewed paper in the family — treats non-stationarity as the headline problem and responds by abandoning the fixed matrix entirely: his “heterogeneous” (non-homogeneous) chain re-estimates P over a sequence of fixed-length sliding windows. That design choice is, in effect, an admission that the classic fixed-matrix chain is not viable on its own. Aronsson Folkesson 2023 encounter the same problem from the data side: on a sliding window, recent history sometimes contains no observed transitions out of a given state, producing zero-probability rows that force the model into a pure-chance fallback. Their remedy — a voting ensemble of ten chains with random window sizes — is another non-stationarity workaround, and even so the model barely beats random chance.

Non-Stationary Transition Matrix [part-of] Non-Stationarity Non-Stationary Transition Matrix [causes] Markov Chain Trading Model Wilinski 2019 [opposes] Non-Stationary Transition Matrix

The econometrics literature has formalised the response. The standard fix is the time-varying transition probability (TVTP) model: rather than holding P constant, the transition probabilities are made to depend on observable covariates or to evolve under their own dynamic law. Diebold, Lee & Weinbach (1994) and Filardo (1994) pioneered covariate-driven transition probabilities; Bazzi Blasques Koopman Lucas 2017 (Journal of Time Series Analysis) let the probabilities evolve via a score-driven observation model. The very existence of this body of work confirms that, in serious econometric practice, a constant transition matrix is treated as a misspecification to be corrected, not a safe default. Empirically, Ravn & Sola (1999) document that a single structural change — a redefinition of the US M2 money stock — had “a dramatic impact on the separation of regimes” a Markov-switching model implied, direct evidence that estimated transition structure is not stable across time.

Non-Stationary Transition Matrix [relates] State-Count Selection Bazzi Blasques Koopman Lucas 2017 [opposes] Non-Stationary Transition Matrix

Crucially, the standard mitigations do not eliminate the risk — they trade it for a different one. Rolling-window re-estimation tracks recent dynamics but introduces window length as a free parameter: too short and the matrix is estimated from too few transitions (high estimation noise); too long and it averages over regimes it was meant to separate. If that window length is tuned against the same data used to report performance, the non-stationarity fix becomes a fresh channel for Overfitting in Quantitative Trading and Data-Snooping Bias. TVTP models add covariates whose predictive value must itself be established out-of-sample. The honest reading for this vault’s research goal: non-stationarity of the transition matrix is a confirmed structural fragility of Markov-chain trading models, it is why fixed-matrix backtests do not generalise, and the available remedies improve robustness without delivering demonstrated tradeable alpha.

Non-Stationary Transition Matrix [causes] Overfitting in Quantitative Trading

Connections

• Non-Stationarity — part-of, model-specific instance of the general failure mode, source: https://arxiv.org/html/2410.14504v2
• Markov Chain Trading Model — suffers_overfitting_risk, source: https://www.sciencedirect.com/science/article/abs/pii/S0957417419303033
• Wilinski 2019 — contradicts, motivates the heterogeneous rolling-window chain as a direct response, source: https://www.sciencedirect.com/science/article/abs/pii/S0957417419303033
• Aronsson Folkesson 2023 — relates, zero-probability rows on sliding windows force a pure-chance fallback, source: https://kth.diva-portal.org/smash/get/diva2:1823899/FULLTEXT01.pdf
• Bazzi Blasques Koopman Lucas 2017 — relates, time-varying transition probability model as the econometric remedy, source: https://onlinelibrary.wiley.com/doi/10.1111/jtsa.12211
• State-Count Selection — relates, window length and state count are coupled degrees of freedom, source: https://www.sciencedirect.com/science/article/abs/pii/S0957417419303033
• Overfitting in Quantitative Trading — suffers_overfitting_risk, window-length tuning is a data-snooping channel, source: https://www.sciencedirect.com/science/article/abs/pii/S0957417419303033

Sources

• Wiliński (2019), Time series modeling and forecasting based on a Markov chain with changing transition matrices, Expert Systems with Applications Vol. 133
• Aronsson & Folkesson (2023), Stock market analysis with a Markovian approach, KTH
• Bazzi, Blasques, Koopman & Lucas (2017), Time-Varying Transition Probabilities for Markov Regime-Switching Models, Journal of Time Series Analysis 38:458-478
• Markov-Switching Models with State-Dependent Time-Varying Transition Probabilities, ScienceDirect