In honour of the DaysOfMLCode challenge, some of my colleagues and I have decided to partake, pushing ourselves to expand our knowledge and capabilities. I will endeavour to post at least once a week. This is week 1. In my dissertationI have been exploring the application of hidden Markov models HMMs to a particular financial dataset. HMMs also called Markov-switching models are great because they allow one to uncover different states or regimes of the world, from the data.
They have been applied to fields as diverse as speech-recognition and financial data. But, as my Masters supervisor has taken great pains to emphasize thanks! After all you, the modeller, set the number of states a prioriand the model picks parameter estimates by maximizing the likelihood of those parameters given the data. There is really nothing divine happening here — the model will, in some sense, find what you tell it to find.
This is why good validation and model checking is key. Below is the equation for the simple HMM we will be considering for the rest of this post. In addition, the error at time t is independent of any other error at another time index. One can see that other HMMs could have richer specifications — lagged Y values also known as a Markov-switching Autoregressive model or even differently distributed errors.
In plain English — an HMM with 2 states, has a different set of parameters for each state. Hamilton devised the Hamilton Filter to estimate these models, and provide probabilities of the process being in a particular state at time t.
I refer the reader to this excellent lecture note by Ching-Ming Kuan on the mathematical equations for the estimation, which proceeds via a recursive set of equations. The optimization of the log-likelihood function described above is done using the algorithm of your choice. This Hessian is useful because it can be used to calculate standard errors of the parameter estimates. The only snag with this algorithm is that it is unconstrained, which means one must use a monotonic function to transform the parameters lest you end up with, for example, a negative variance!
This is not based on an analysis of the code, but rather the output.
I find I almost always get the following message from Scipy when running the optimization: Desired error not necessarily achieved due to precision loss. What I have noticed is that when I get this message, the Hessian matrix returned by Scipy the function actually returns the inverse Hessian is considerably different to the inverse of the Hessian returned by R.
I hope that this provides a nice, simple introduction to these models. Please let me know in the comments if something was explained badly, or if something was just plain wrong! Your email address will not be published. Skip to content In honour of the DaysOfMLCode challenge, some of my colleagues and I have decided to partake, pushing ourselves to expand our knowledge and capabilities.
The dark mode beta is finally here. Change your preferences any time. Stack Overflow for Teams is a private, secure spot for you and your coworkers to find and share information. I have been using statsmodel. MarkovAutoregressio to replicate Hamilton's markov switching model published in However, when I used current available real GNP or GDP data in dollar and took their log difference quarterly as input, the model doesn't give satisfactory results.
I plotted the log difference of Hamilton gnp and that's from the current available real GNP. They are quite close with slight differences. Can anyone enlighten me why it is the case? Does it have anything to do with the seasonality adjustment of current GNP data? If so, is there is any way to counter it?
Result using current available GNP. Result using paper provided GNP. I'll also note that I double-checked these results using E-views, and it agrees with Statsmodels' output almost exactly. In this case, the model is an AR 4 on the growth rate of real GNP, with a regime-specific intercept; the model allows two regimes.Markow switching model application
The idea is that "recessions" should correspond to a low or negative average growth rate and expansions should correspond to a higher average growth rate.
Now, as you saw, we can instead fit the model using the "updated" dataset which looks pretty much like the original datasetto get the following parameters and regime probabilities:. To understand what the model is doing, look at the intercepts in the two regimes.
In Hamilton's model, the "low" regime has an intercept of What that tells us is that with the updated dataset, the model is doing a "better job" fitting the data in terms of a higher likelihood by choosing the "low" regime to be much deeper recessions.
In particular, looking back at the GNP data series, it's apparent that it's using the "low" regime to fit the very low growth in the late 's and early 's.
In contrast, the fitted parameters from Hamilton's model allow the "low" regime to fit "moderately low" growth rates that cover a wider range of recessions. We can't compare these two models' outcomes using e. One thing we could try, though is to use the fitted parameters from Hamilton's dataset on the updated GNP data. Doing that, we get the following result:. But notice that the log-likelihood here is Note again that you can't compare these log-likelihoods to the Learn more.A Markov chain is a mathematical system usually defined as a collection of random variables, that transition from one state to another according to certain probabilistic rules.
These set of transition satisfies the Markov Propertywhich states that the probability of transitioning to any particular state is dependent solely on the current state and time elapsed, and not on the sequence of state that preceded it. This unique characteristic of Markov processes render them memoryless. In this tutorial, you will discover when you can use markov chains, what the Discrete Time Markov chain is.
You'll also learn about the components that are needed to build a Discrete-time Markov chain model and some of its common properties. Next, you'll implement one such simple model with Python using its numpy and random libraries.
You will also learn some of the ways to represent a Markov chain like a state diagram and transition matrix. Want to tackle more statistics topics with Python? Markov Chains have prolific usage in mathematics. They are widely employed in economics, game theory, communication theory, genetics and finance. They arise broadly in statistical specially Bayesian statistics and information-theoretical contexts.
When it comes real-world problems, they are used to postulate solutions to study cruise control systems in motor vehicles, queues or lines of customers arriving at an airport, exchange rates of currencies, etc. The algorithm known as PageRank, which was originally proposed for the internet search engine Google, is based on a Markov process. Reddit's Subreddit Simulator is a fully-automated subreddit that generates random submissions and comments using markov chains, so cool!
A Markov chain is a random process with the Markov property. A random process or often called stochastic property is a mathematical object defined as a collection of random variables. A Markov chain has either discrete state space set of possible values of the random variables or discrete index set often representing time - given the fact, many variations for a Markov chain exists.
A discrete-time Markov chain involves a system which is in a certain state at each step, with the state changing randomly between steps. The steps are often thought of as moments in time But you might as well refer to physical distance or any other discrete measurement. A discrete time Markov chain is a sequence of random variables X 1X 2X 3Putting this is mathematical probabilistic formula:.
Which means the knowledge of the previous state is all that is necessary to determine the probability distribution of the current state, satisfying the rule of conditional independence or said other way: you only need to know the current state to determine the next state.
The possible values of X i form a countable set S called the state space of the chain. The state space can be anything: letters, numbers, basketball scores or weather conditions.
While the time parameter is usually discrete, the state space of a discrete time Markov chain does not have any widely agreed upon restrictions, and rather refers to a process on an arbitrary state space. However, many applications of Markov chains employ finite or countably infinite state spaces, because they have a more straightforward statistical analysis. A Markov chain is represented using a probabilistic automaton It only sounds complicated! The changes of state of the system are called transitions.
The probabilities associated with various state changes are called transition probabilities. A probabilistic automaton includes the probability of a given transition into the transition function, turning it into a transition matrix. Every state in the state space is included once as a row and again as a column, and each cell in the matrix tells you the probability of transitioning from its row's state to its column's state.
If the Markov chain has N possible states, the matrix will be an N x N matrix, such that entry I, J is the probability of transitioning from state I to state J. Additionally, the transition matrix must be a stochastic matrix, a matrix whose entries in each row must add up to exactly 1.
Since each row represents its own probability distribution. So, the model is characterized by a state space, a transition matrix describing the probabilities of particular transitions, and an initial state across the state space, given in the initial distribution. When Cj is sad, which isn't very usual: she either goes for a run, goobles down icecream or takes a nap.
From historic data, if she spent sleeping a sad day away.Why Stata? Supported platforms. Stata Press books Books on Stata Books on statistics. Policy Contact. Bookstore Stata Journal Stata News. Contact us Hours of operation. Advanced search. Think of economic recessions and expansions.
At the onset of a recession, output and employment fall and stay low, and then, later, output and employment increase. Think of bipolar disorders in which there are manic periods followed by depressive periods, and the process repeats. Statistically, means, variances, and other parameters are changing across episodes regimes.
Our problem is to estimate when regimes change and the values of the parameters associated with each regime. Asking when regimes change is equivalent to asking how long regimes persist.
In Markov-transition models, in addition to estimating the means, variances, etc. The estimated transition probabilities for some problem might be, the following:. Start in state 1. The probability of transiting from state 1 to state 1 is 0. Said differently, once in state 1, the process tends to stay there. With probability 0. State 2 is not as persistent. Markov-switching models are not limited to two regimes, although two-regime models are common.
In the example above, we described the switching as being abrupt; the probability instantly changed.
Such Markov models are called dynamic models. Markov models can also accommodate smoother changes by modeling the transition probabilities as an autoregressive process. Let's look at mean changes across regimes. In particular, we will analyze the Federal Funds Rate. The Federal Funds Rate is the interest rate that the central bank of the U. We are going to look at changes in the federal funds rate from to the end of Here are the data:. We have quarterly data. High interest rates seem to characterize the seventies and eighties.
We will assume there is another regime for lower interest rates that seem to characterize the other decades. Among the things you can predict after estimation is the probability of being in the various states. We have only two states, and thus the probability of being in say state 2 tells us the probability for both states. We can obtain the predicted probability and graph it along with the original data:. The model has little uncertainty as to regime at every point in time.
We see three periods of high-rate states and four periods of moderate-rate states. Let's look at an example of disease outbreak, namely mumps per 10, residents in New York City between and You might think that outbreaks correspond to mean changes, but what we see in the data is an even greater change in variance:. We graphed variable SWhether or not to include a trend.
Default is an intercept.
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The order of the model describes the dependence of the likelihood on previous regimes. This depends on the model in question and should be set appropriately by subclasses. Array of exogenous or lagged variables to use in calculating time-varying transition probabilities TVTP. TVTP is only used if this variable is provided. If an intercept is desired, a column of ones must be explicitly included in this array. If a boolean, sets whether or not all trend coefficients are switching across regimes.
If an iterable, should be of length equal to the number of trend variables, where each element is a boolean describing whether the corresponding coefficient is switching.
Default is True. If a boolean, sets whether or not all regression coefficients are switching across regimes.
If an iterable, should be of length equal to the number of exogenous variables, where each element is a boolean describing whether the corresponding coefficient is switching. Whether or not there is regime-specific heteroskedasticity, i. Default is False. This model is new and API stability is not guaranteed, although changes will be made in a backwards compatible way if possible.
The trend is accommodated by prepending columns to the exog array. Kim, Chang-Jin, and Charles R. MIT Press Books. The MIT Press. Transform unconstrained parameters used by the optimizer to constrained parameters used in likelihood evaluation. Transform constrained parameters used in likelihood evaluation to unconstrained parameters used by the optimizer.
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Did you care to read the disclaimer on the website you linked? Let me paste it here, for your convenience: "Note: this code is now obsolete - it has been superseded by a new version which can be found here. Thanks, I didn't notice it. However, if I now run from statsmodels. Active Oldest Votes. It is possible that a final release of v0. Sign up or log in Sign up using Google. Sign up using Facebook. Sign up using Email and Password. Post as a guest Name.
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It can be written:. Each period, the regime transitions according to the following matrix of transition probabilities:. The model class is MarkovAutoregression in the time-series part of statsmodels.
After creation, the model is fit via maximum likelihood estimation. Under the hood, good starting parameters are found using a number of steps of the expectation maximization EM algorithm, and a quasi-Newton BFGS algorithm is applied to quickly find the maximum.
We plot the filtered and smoothed probabilities of a recession. From the estimated transition matrix we can calculate the expected duration of a recession versus an expansion. In this case, it is expected that a recession will last about one year 4 quarters and an expansion about two and a half years.
This model demonstrates estimation with regime heteroskedasticity switching of variances and no mean effect. Since there is no autoregressive component, this model can be fit using the MarkovRegression class. Below we plot the probabilities of being in each of the regimes; only in a few periods is a high-variance regime probable.
This model demonstrates estimation with time-varying transition probabilities. In the above models we have assumed that the transition probabilities are constant across time. Here we allow the probabilities to change with the state of the economy. Otherwise, the model is the same Markov autoregression of Hamilton Each period, the regime now transitions according to the following matrix of time-varying transition probabilities:.
Here we demonstrate another feature of model fitting - the use of a random search for MLE starting parameters. Because Markov switching models are often characterized by many local maxima of the likelihood function, performing an initial optimization step can be helpful to find the best parameters. Below, we specify that 20 random perturbations from the starting parameter vector are examined and the best one used as the actual starting parameters.
Because of the random nature of the search, we seed the random number generator beforehand to allow replication of the result. Below we plot the smoothed probability of the economy operating in a low-production state, and again include the NBER recessions for comparison.
Using the time-varying transition probabilities, we can see how the expected duration of a low-production state changes over time:.
During recessions, the expected duration of a low-production state is much higher than in an expansion. Use the functions in the public API at pandas.
DatetimeIndex dta. Markov Switching Model Results Dep. Variable: rgnp No. L2 L3