## Nonlinear estimation: cross-section data

Econometrics questions and discussions

### Nonlinear estimation: cross-section data

Hi all,

I have a problem that involves nonlinear estimation, period-by-period.

I have (say) 8 equations of the form
f(delta,theta(i)) = epsilon(i)

and (say) 28 equations of the form
g(delta,theta(i)) = epsilon(i) - epsilon(j).

I can successfully use FIND to estimate all the parameters (delta vector, plus all the theta(i)), if I minimize the squared epsilons AND the squares of (epsilon(i) - epsilon(j)). (This post has been edited, as I had an error.) I think that what I am doing is akin to GMM or indirect inference - and if so, as any covariance matrix yields consistent estimates, I can run this unweighted - but I am not sure about that.)

Since this is period-by-period, I don't think NLSYSTEM is appropriate.

I also plan to bootstrap the standard errors by simulating new data using the estimated parameters and sampling from the estimated epsilons.

Thanks everyone.

Randy
randal_verbrugge

Posts: 14
Joined: Mon Sep 23, 2013 10:43 am

### Re: Nonlinear estimation: cross-section data

I'm a bit confused about where the "J" is coming from in the "G" equations. Plus, how many elements are in each THETA?
TomDoan

Posts: 7240
Joined: Wed Nov 01, 2006 5:36 pm

### Re: Nonlinear estimation: cross-section data

For each i, there is a theta variable which refers to unobserved variance of the prior. For the system, the delta vector consists of:
a) mean of signal
b) upward bias of signal, for those who think it is upward biased
c) variance of signal, for those who think it is upward biased
d) downward bias of signal, for those who think it is downward biased
e) variance of signal, for those who think it is downward biased

The Bayesian updating equation for each person is one equation for each person. The objective function is minimizing the errors on these equations.

I have some cross-equation-restrictions that I also impose (respondent i versus respondent j), that allow me far more equations than parameters.

Nonetheless, while relative variances seem to be well-identified (e.g., theta(i) versus c)), absolute levels of variances (such as theta(i) itself and c) itself) are difficult to pin down. I am using a normalization that assumes that those who do not update have a very, very firm prior belief. It still requires some playing around to find starting values that lead to minimum errors on the Bayesian updating equations.

I am using FIND. Simplex seems to converge, although perhaps not to a global min. Genetic takes forever and often does not converge. Simulated annealing takes approximately 365 days to converge (actually I have no idea; one run did not complete overnight and I killed it.)
randal_verbrugge

Posts: 14
Joined: Mon Sep 23, 2013 10:43 am 