Re: [Gretl-devel] Restricted estimation in bulti-in ML setups?
On Thu, 9 Feb 2017, Sven Schreiber wrote:
Hi,
I think it is not currently possible to do restricted ML estimation with the
built-in estimators (apart from the special case OLS), otherwise I would have
formulated this as a question and sent to gretl-users instead. Before someone
answers "do your own mle block", I'm aware of that possibility but
that's not
the point.
For example in the ordered probit model I have checked that one can test
restrictions on the thresholds (cut points) with a standard "restrict ... end
restrict" command block. This raised the question of why actually it isn't
possible (easily...) to impose that restriction in ML estimation.
Note that the probit is just an example, in principle I guess this would
apply to many (all?) estimators that are built on ML and which already allow
to test a restriction on an element of $coeff.
I could imagine that in some model contexts arbitrarily restricting some
parameter might induce ill-behaved likelihood function shapes. But in
principle it should be OK, no?
In principle, for linear restrictions a general solution to what you
propose could be achieved by resorting to the following apparatus: call g
the score vector and express the linear restrictions as Rb = d, where b
are the unrestricted parameters.
By working on the null space of R, you can turn the constraints from
implicit to explicit form as b = S \theta + s (like we do in the SVAR
package) so that the score vector for the unrestricted parameters becomes
g'S (and the Hessian S'HS if needed) by the chain rule. Then you run BFGS
or Newton on \theta and map back the solution to b via \hat{b} =
S\hat{\theta} + s plus all the delta-method adjustments to $vcv. Clearly,
as you say, there's no guarantee that the resulting problem should be well
behaved, but I guess that's unavoidable.
The problem is that the possibility of going through the above should be
contemplated in the C code, and at present isn't (and believe me, doing so
for all the ML estimators we have would be a major piece of work).
Moreover, in the special case of index models, the same effect can be
achieved more simply, as linear restrictions on coefficients can often be
re-expressed by a redefinition of the explanatory variables; see example
below:
<hansl>
set verbose off
set seed 110217
nulldata 1024
x1 = uniform()
x2 = uniform()
y = 1 + x1 + x2 + normal() > 0
probit y 0 x1 x2 --p-values
pnames = varname($xlist) # used later
L0 = $lnl # used later
# test b2 == b3 via Wald test
restrict
b2 - b3 = 0
end restrict
# same restriction via LR
z = x1 + x2
probit y 0 z --quiet
L1 = $lnl
LR = 2 * (L0 - L1)
S = {1,0; 0, 1; 0, 1}
btilde = S*$coeff
v = qform(S, $vcv)
cs = btilde ~ sqrt(diag(v))
modprint cs pnames # restricted model
printf "LR test = %g (pvalue = %g)\n", LR, pvalue(x, 1, LR)
</hansl>
As for nonlinear restrictions, things would be even more complex, and I
don't think there's anything that can be done automatically.
-------------------------------------------------------
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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