SignalTrace

First-Order Memory Assumption

John Montgomery4 min read

First-Order Memory Assumption

The first-order memory assumption is the defining — and limiting — property of a Markov Chain Trading Model: the probability of the next price state depends only on the current state, so the price path that produced that state is discarded. Do Carmo 2017 names this “memorylessness.” Stated as a modelling choice it is innocuous; stated as a claim about markets it is a strong empirical hypothesis, and the failure mode is that the hypothesis is not well supported by the data. The assumption rules out, by construction, both trend persistence and longer-range dependence — exactly the effects that momentum and trend-following strategies trade on.

The econometric evidence cuts in two directions, and both are bad for a stationary first-order chain. On one side, raw returns themselves contain little linear autocorrelation — consistent with weak-form efficiency — so for the mean return a low-order model is not obviously wrong, and semiparametric tests find little long memory in returns. On the other side, the absolute and squared returns show strong, slow-decaying autocorrelation: volatility clustering. Ding, Granger & Engle (1993) is the canonical result — “substantially more correlation between absolute returns than returns themselves,” with the power transformation |r|^d displaying a long-memory property whose dependence decays hyperbolically, far more slowly than the geometric decay a low-order Markov model implies. Cont’s (2001) catalogue of stylised facts lists this persistence as a basic feature of asset returns. A memoryless first-order chain on discretised returns cannot encode it: its transition law conditions only on the current state.

First-Order Memory Assumption [part-of] Markov Chain Trading Model Ding Granger Engle 1993 [contradicts] First-Order Memory Assumption

There is a live debate over whether the slow-decaying dependence is “true” long memory or the statistical artefact of structural breaks and level shifts (Mikosch & Starica; Perron & Qu 2010 argue S&P 500 volatility is better described by a stationary model with level shifts). For this vault the debate is moot in a useful way: either reading is fatal to a stationary first-order chain. If the dependence is true long memory, the chain discards it; if it is structural breaks, the chain’s time-homogeneity assumption is also violated (the Non-Stationary Transition Matrix problem). The first-order chain only survives if returns are both short-memory and stationary, which the evidence does not support.

First-Order Memory Assumption [relates] Non-Stationary Transition Matrix

The obvious fix — higher-order chains that condition on the last k states — does not rescue the approach, and the vault’s own evidence shows why. A k-th order chain over n states has n^k rows in its transition matrix, so each row’s probabilities are estimated from k-fold sparser data: the extra memory is bought at the cost of severe estimation noise. Aronsson Folkesson 2023 tested first- and second-order chains on OMXS30 and found second-order chains did not beat first-order — and the second-order up/down model actually scored below random chance (42.7%). Wilinski 2019 likewise tested both orders and reported no decisive gain. The extra memory spreads thin data thinner faster than it adds predictive content. The net verdict for this vault: the first-order memory assumption is a confirmed structural limitation — a modelling convenience that discards empirically documented persistence — and the standard remedy of adding memory fails the out-of-sample test because of the data-sparsity trade-off it triggers.

Aronsson Folkesson 2023 [supports] First-Order Memory Assumption First-Order Memory Assumption [relates] State-Count Selection

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