Extending Hidden Semi-Markov Models with Cumulative Returns
Summary
A hidden semi-Markov model allows the probability of leaving a hidden state to depend on time spent in that state, unlike a standard hidden Markov model’s memoryless transition structure. The document asks whether a trading model could also use cumulative return since state entry as an input to state transitions. The motivation is that traders may interpret an extended rise or decline differently from a brief move, even when the current one-period return is similar.
The proposed direction is to combine state-duration information with cumulative returns, alongside familiar inputs such as short-horizon returns and volatility. This is a modeling question rather than a specified algorithm: no transition equations, estimation procedure, data, or trading results are supplied. Any implementation would need to define how state-conditioned cumulative returns affect transitions and assess whether the extra structure improves out-of-sample decisions over a conventional HSMM. The market example motivates the idea but does not establish that cumulative return is predictive or that the approach is profitable.
Key ideas
- An HSMM allows hidden-state transition probabilities to depend on time spent in the current state.
- Cumulative return since state entry could be considered alongside duration when modeling state transitions.
- The proposed motivation is that extended rises and declines may shape traders’ interpretation of market conditions.
- The document poses a modeling question but does not specify an estimation method or report empirical results.
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Full text
# Generalizing a hidden semi-Markov model for trading # Generalizing a hidden semi-Markov model for trading Taken from Wikipedia: > A hidden semi-Markov model (HSMM) is a statistical model with the same structure as a hidden Markov model except that the unobservable process is semi-Markov rather than Markov. This means that the probability of there being a change in the hidden state depends on the amount of time that has elapsed since entry into the current state. This is in contrast to hidden Markov models where there is a constant probability of changing state given survival in the state up to that time. My high level understanding of these models is that the duration in a given state is modeled by its own matrix (i.e., discrete probabilities). I'm wondering if the basic idea can be generalized for any continuous value. For example, the S&P has moved down since the start of 2022, but has done so in a very oscillatory fashion: Rather than using duration to inform state probabilities, I'm wondering if cumulative return within a given state could be used, which I think reflects the way actual traders read charts. In plain English, traders look to take profits after an extended uptrend or look for buying opportunities after an extended downtrend. Time is probably a factor here (i.e., a vanilla HSMM might have some value on its own), but the more important factor is cumulative returns in a given state. Or maybe both time and cumulative returns could be considered. The math on this is a little over my head at the moment, so I'm looking for any suggestions on how this type of model could be formulated. The goal would be to take advantage of standard inputs that might be used for an HMM for trading (e.g., single period or short-duration returns, volatility, etc.) while also capturing cumulative state information. Any suggestions on how to build such a model, reading material, etc. are appreciated.
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