# st: RE: [Non Stata] Estimation strategy for a belief learning model.

 From austin nichols To statalist@hsphsun2.harvard.edu Subject st: RE: [Non Stata] Estimation strategy for a belief learning model. Date Mon, 8 Aug 2005 12:45:10 -0400

```I'm not going to read Cheung and Friedman (1997) to figure
out what the real model would be, I'm just going to assume
you want to see the theoretical prediction "explains" each
individual person's beliefs at each point in time, based on
some initial starting point, using some test of whether the
coefficients on the theoretical predictions are non-zero.

It's not clear whether you are inferring beliefs from play,
or just asking players to report beliefs. This distinction
could have a huge impact on the error structure.

Nevertheless, here is one way to go:

Express beliefs as odds ratios: p(A)/p(not A) so the
left-hand-side variable ranges from zero to infinity.  Then
take the logs of the odds ratios. Now the LHS variable
ranges from negative infinity to positive infinity.  Do the
same for your theoretical predictions. If probabilities of
zero or one are possible, you might have problems, both
practically and theoretically, since this would be an
absorbing state for any future updating of beliefs.

Now you've got three real-valued LHS variables, call them A
B C, in three equations, to regress on predictions from
your model. They no longer sum to one, but the deviations
are clearly correlated across the equations. Estimate each
regression and then use -suest- to adjust for correlations
of errors across all 3 simultaneously.  You probably want
to cluster by individual.

I don't think you really want a learning curve model (aka
latent growth curve model, see HLM software for these
models), but you do want to allow for round-specific fixed
effects, probably.  Just -tab round, gen(rd)- and then
include rd* as regressors in each model, like so...
g C=ln(pc/(1-pc))
reg C theoreticalC rd* , score(sC)
suest A B C, cluster(id)

Maybe you want indiv fixed effects of some kind, too...

If you want to see a graph of the log-odds-ratio versus
probability, try:
clear
range p 0 1 100
g lor=ln(p/(1-p))
line lor p

-----Original Message-----
From: Antoine Terracol [mailto:Antoine.Terracol@univ-paris1.fr]
To: statalist@hsphsun2.harvard.edu

In the context of an economic experiment, I have data on
the elicited beliefs of individuals on the probability that
their opponent plays strategy A, B or C in the next round.

I want to fit a learning model (Cheung and Friedman (1997)
"Individual Learning in Normal Form Games: Some Laboratory
Results," Games and Economic Behavior, 19, 46-76.) on the
data to see wether it describes reasonably well the
learning process of individuals.

For a given strategy, say strategy A, I can estimate the
parameters of
Actual Belief = Theoretical Belief + epsilon where
epsilon is a disturbance random variable However, because I
only use the belief for a given strategy, I throw away
valuable information (beliefs on strategy B) I obviously
(??) can't simply multiply the likelihood contributions for
both beliefs since they are almost mechanically negatively
correlated; and I'm not sure whether using a bivariate
distribution for the error terms would be appropriate. I'd
appreciate any hint or advice on the subject...

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