Modeling Mean-Reversion with Markov Regimes and Structural Breaks
Summary
The document considers a mean-reverting process whose long-run level can shift after an unexpected event. It describes extending the process with a latent state variable, assigning a different mean-reversion level to each state and modeling the probabilities of transitions between states. This is the structure of a Markov regime-switching model and allows the process to retain mean-reverting dynamics while adapting to distinct regimes.
For estimation, the response suggests a regime-switching autoregression, with coefficients and variance allowed to change by state, and identifies maximum likelihood with the Hamilton filter as a basic approach. Bayesian Markov chain Monte Carlo is offered as another method. A second answer distinguishes sudden stochastic jumps, which can be represented with a Poisson jump term, from structural-break analysis, which tests for parameter changes. These are modeling directions rather than a worked implementation: the document provides no data, estimation results, or guidance on selecting states, validating a fit, or forecasting regime changes.
Key ideas
- A latent state can select among different mean-reversion levels in a regime-switching process.
- State transition dynamics are needed to describe how the process moves between regimes.
- Regime-switching autoregressions can be estimated with maximum likelihood and the Hamilton filter.
- Bayesian MCMC provides an alternative estimation approach.
- Poisson jump-diffusion models and structural-break tests address related but distinct forms of change.
Tags
Full text
# Regime switching in mean reverting stochastic process
# Regime switching in mean reverting stochastic process
Let you have a mean reverting stochastic process with a statistically significant autocorrelation coefficient; let it looks like you can well model it using an $ARMA(p,q)$.
This time series could be described by a mean reverting stochastic process like
$dS=k(\theta-S_{t})dt+\sigma S_{t}^{\beta}dz$
where $\theta$ is the mean reversion level, $k$ is the speed of mean reversion and $\beta$ determines the structural form of diffusion term (so $\beta=0$ yields the normally distributed mean reversion model, aka the Ornstein-Uhlenbeck process).
Regardless of the actual value of $S_{0}$, we know $S_{t}$ will go $\theta$ in the long run, right?
Now let there's an unlikely event which can drastically change $\theta$'s value: e.g. let $\theta=100$, you model the process, ok, then... bang! Starting from $t=\tau$ it happens that $\theta=30$ and you will have to deal with this new scenario.
My question: is there any model which can deal with such a situation?
## Answer by John (score 2, accepted)
https://quant.stackexchange.com/a/4949
As far as I can tell, you've essentially written the model that you are concerned with. The only difference is that you would instead have $\theta_{i}$ when $s_{t}=i$ where $s_{t}$ is a latent variable that reflects the probability of being in state $i$. You would also need to include the dynamics that drive the probability transitions as another part of the model. You could set them up as standard Markov Regime-Switching models are set up, though there are other options.
So the question becomes what do you want to do with the model?
If you are concerned with estimating the parameters of such a model, you would begin by setting this up as a regime-switching AR(p) model (these are more popular to use than ARMA models). You could set it up in levels and allow the coefficients on all the variables (and the variance) to switch between states. You could also set it up in differences and include the lag of the level as an independent variable.
To estimate the parameters, the simplest approach is to apply maximum likelihood using the Hamilton filter. There is a Matlab implementation that I have used to implement this approach. You could also estimate the regime-switching model by Bayesian MCMC.
## Answer by Jay (score 0)
https://quant.stackexchange.com/a/21517
Just read some papers recently. Hope it is helpful.
Some model I know with such Bang phenomena is called: Jump-Diffusion Model. The idea is a little different. We add Poisson counter term into the equation.
On the other side, there is lots of work on test of such Bang phenomena, which is called "structural break". One early reference is the following: "Estimating and Testing Linear Models with Multiple Structural Changes." Jushan Bai and Pierre Perron, 1998.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.