Dear Listers,
The
following code creates 5 observations on x, which has a density
theta*x^(theta-1). I then estimate theta using Stata's ml routine,
first with method lf and then with d0. In each case theta is estimated
with and without a constraint, the constraint being that theta = 3.
The point of this exercise is to get Stata to calculate the gradient
and the variance using formulas for the unconstrained likelihood but at
the constrained value of theta, as one would do in a lagrange
multiplier test. The formula for the variance of theta is theta^2/n,
which for the constrained model is 1.8. This is exactly what is
produced by method d0 (and d1, though that's not shown), but quite far
from what is produced by method lf. Does this not imply something is
wrong with method lf?
version 11.0
clear all
capture log close
set more off
set seed 875411
local obs 5
set obs `obs'
local theta = 2
gen u = runiform()
gen x = exp(ln(u)/`theta')
gen lnx = ln(x)
sum lnx
sca meanlnx = r(mean)
sca theta1 = -1/meanlnx
sca vartheta1 = theta1^2/`obs'
sca list theta1 vartheta1
cap prog drop mlexamp1
prog define mlexamp1
args lnf theta
qui replace `lnf' = ln(`theta') + (`theta'-1)*ln($ML_y1)
end
cap prog drop mlexamp2
prog define mlexamp2
args todo b lnf g
tempvar theta
mleval `theta' = `b'
quietly {
mlsum `lnf' = ln(`theta') + (`theta'-1)*ln($ML_y1)
}
end
local null = 3
/* lf version */
ml model lf mlexamp1 (x =)
ml maximize
ml model lf mlexamp1 (x =)
ml init `null', copy
ml maximize, iter(0)
mat list e(V)
/* d0 version */
ml model d0 mlexamp2 (x =)
ml maximize
ml model d0 mlexamp2 (x =)
ml init `null', copy
ml maximize, iter(0)
mat list e(V)
Thanks,
Greg
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