Re: [Gretl-users] Restriction on model coefficients
On Thu, 4 Feb 2016, Carlo Maria Petrangelo wrote:
Hello everyone,
I need to estimate an equation with tsls. This equation has a restriction on
parameters. The restriction has inside itself values coming from the
estimation parameters of another equation and it is the following:
b3=[a1/(1-a2)](1-b2)-b1, on which "a" indicate a parameter of the first
equation (already estimated) and "b" indicate a parameter of the second
equation (which has to be estimated with tsls) and the numbers indicate the
parameter associated to a regressor.
How can I estimated the second equation once I have estimated the first.
Premise: for this kind of things, your best bet is ALWAYS two estimate the
two equations jointly. Otherwise, you have to make a few adjustments to
you standard errors which are not only boring to compute, but also prone
to sizeable finite-sample bias.
That said, I see three alternatives:
* write the log-likelihood for the two equations under normality and use
mle
* re-express your problem as a suitable set of moment conditions and use
gmm
* use a lego-like approach:
- estimate 1st stage via ols and get a1 and a2, plus residuals u
- assuming your "main" equation looks like
y = x1 * b1 + x2 * b2 + x3 * b3
compute w = y - (a1/(1-a2))*x3
compute z1 = x1 - x3
compute z2 = x2 - (a1/(1-a2))*x3
- run ols on
w = z1 * b1 + z2 * b2 + gamma * u
- bootstrap the whole thing
this will give you consistent estimates of a1, a2, b1 and b2, from which
you can compute b3 via the delta method.
HTH,
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Riccardo (Jack) Lucchetti
Dipartimento di Scienze Economiche e Sociali (DiSES)
Università Politecnica delle Marche
(formerly known as Università di Ancona)
r.lucchetti(a)univpm.it
http://www2.econ.univpm.it/servizi/hpp/lucchetti
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