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st: Is maximum likelihood lf method appropriate for lagged (time series) data?

From   "John Hund" <[email protected]>
To   <[email protected]>
Subject   st: Is maximum likelihood lf method appropriate for lagged (time series) data?
Date   Thu, 11 Mar 2004 03:05:11 -0600

I am new to Stata and especially new to the maximum likelihood command
in Stata, and I am trying to assess whether the lf method is possible
for the likelihood function I need to maximize.  The fundamental problem
is to estimate the mean and standard deviation of an (unobserved)
geometric diffusion process.  The unobserved data is a nonlinear
function of the observed data under a particular model; I can (and
already have written a Stata function to) numerically invert the
function to get observations to apply to the density.  If the data was
observed, the density would simply be the lognormal distribution; as it
is, the density is the lognormal distribution multiplied by the Jacobian
of the nonlinear transformation (which I have in closed form).

I think that the lf method would work fine given that the transformation
is a one-to-one mapping...but the likelihood of each individual
observation depends on the observation previous to it.  That is, the
likelihood contains the ratio of two successive observations, since the
diffusion process describes the change through time.  Specifically, if
the data at t is Vt, the likelihood incorporates the term: ln(Vt/Vt-1).
I can't just transform the data before I hand it to the maximizer since
I have to invert the data inside the maximization step (the
transformation depends on the estimated parameters).  I have tried to
pass two sets of data into my program...the data and the lagged data,
but haven't gotten it to work, and I'm not sure whether that has to do
with my inexperience or if this incorporates a violation of the linear
form restrictions.

Does anyone have any experience with either fitting diffusions in Stata,
or more generally, using the ml command to fit time-series models (where
this sort of issue would naturally arise)?  Is there a reference
somewhere for writing Stata likelihoods for time-series
maximization...nearly all of the examples I've seen are cross-sectional?

Thanks in advance for your consideration...

John Hund
Assistant Professor of Finance
A. B. Freeman School of Business
Tulane University
New Orleans, LA  70118-5669

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