[R] fitting Markov chains

[kjetil brinchmann halvorsen]

On 1 Oct 2003 at 17:14, Tamas Papp wrote:

> I need to find a computationally simple process for the movement of
> interest rates. In this simplified model, an interest rate can have
> 3--5 possible values, and its movement is characterized by a matrix of
> transition probabilities (ie, it is a Markov process).
> 
> I would like to estimate this process from a given set of data. 
> 
> For example, let the interest rate time series be:
> 
> 7 3 8 2 5 9 6
> 
> Assume that the discretized variable can take the following values:
> (3, 5, 8), then we find the nearest discrete point and give its index:
> 
> 3 1 3 1 2 3 2
> 
> Then estimate the transition probabilities.
> 
> I have the following questions:
> 
> - how should I select the discrete set of values that the variable can
> assume? Eg simply get the maximum and minimum, and divide this
> interval into, say, three pieces? Or estimate the mean, and make the
> other two values mean plus-minus one standard deviation?
> 

?Try with simulation what is best, after yuou have solved the 
estimation problem?

> - once the variable is discretized, how do I transform each data point
> to its discretized value (its index)?
> 
> - the most important: how should I estimate the transition
> probabilities?
> 
> References to introductory literature on estimating Markov chains like
> this would be welcome. 

A good reference covering this is U. Narayan Bhat: "Elements of 
Applied Stochastic Processes". For an almost ML solution, condition 
on the  first observation, then take each transition i -> somewhere 
as an observation from the multinomial distribution given by the i'th 
row of the transition matrix, and use the multinomial estimates. 

Kjetil Halvorsen


Most importantly, I need to know how robust an
> estimation is to selecting the discrete points, or is there a simple
> "goodness of fit" estimation.
> 
> Thanks,
> 
> Tamas Papp
> 
> -- 
> Tamás K. Papp
> E-mail: tpapp at axelero.hu (preferred, especially for large messages)
>         tpapp at westel900.net
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> 
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