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comtech
science forum Guru Wannabe

Joined: 08 Oct 2005
Posts: 150

Posted: Thu Jul 20, 2006 12:49 am    Post subject: possible to use Generalized Method of Moments for this problem?

Hi all,

In this problem, I try to use Generalized Method of Moments(GMM) to
obtain estimate of some parameters.

For a time series of data, I will be able to find its sample moments,
1st, 2nd, 3rd, 4th order, etc.

However, the theoretical moments are very complicated:

The 1st moment of the data:

E(St)=integrate( (x - c(y))* f(x, y), dxdy),

where f(x, y) is a very complicated joint density function of x and y.
c(y) is a function which involves integral of some other complicated
form from 0 to y. So in c(y), y appears as an integral limit.

The 2nd moment of data can be:

E(St^2)=integrate( (x - c(y))^2* f(x, y), dxdy),

These moments involve the parameters we like to estimate.

It turns out that these theoretical moments can only be computed
numerically.

How can we do GMM for these cases?

Thanks a lot!
Herman Rubin
science forum Guru

Joined: 25 Mar 2005
Posts: 730

Posted: Thu Jul 20, 2006 2:58 pm    Post subject: Re: possible to use Generalized Method of Moments for this problem?

Lucy <comtech.usa@gmail.com> wrote:
 Quote: Hi all, In this problem, I try to use Generalized Method of Moments(GMM) to obtain estimate of some parameters. For a time series of data, I will be able to find its sample moments, 1st, 2nd, 3rd, 4th order, etc.

Can you find averages of other functions? Third and
fourth moments start getting very imprecise.

 Quote: However, the theoretical moments are very complicated: The 1st moment of the data: E(St)=integrate( (x - c(y))* f(x, y), dxdy), where f(x, y) is a very complicated joint density function of x and y. c(y) is a function which involves integral of some other complicated form from 0 to y. So in c(y), y appears as an integral limit. The 2nd moment of data can be: E(St^2)=integrate( (x - c(y))^2* f(x, y), dxdy), These moments involve the parameters we like to estimate. It turns out that these theoretical moments can only be computed numerically.

First, I suggest you consult with a good numerical
analyst, preferably one who also understands statistics.
This problem looks somewhat difficult for email, but
might be possible. A good numerical analyst might be
able to find a way to greatly simplify the calculations.

Even then, you might have to use simulation. I suggest
you find a way to use the SAME random variables for
all values of the parameters you try, and also, if
possible, to compute enough derivatives to help in
the resulting numerical work. The method of synthetic
variables may be useful; it is not used anywhere near
as much as it should be.

A good mathematical statistician may be able to look
at the likelihood function and suggest generalized
moments to use.

 Quote: How can we do GMM for these cases? Any advice is appreciated! Thanks a lot!

--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558

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