Re: st: gradient and the inverse of the information matrix

[Maarten Buis <maartenlbuis@gmail.com>]

On Wed, May 1, 2013 at 11:40 PM, Jun Xu wrote:
> I am working on a single-equation categorical dependent variable model. I estimated the model with constraints imposed. Then if I understand the score test correctly, it would be a simple matrix operation of
>
> gradient * inv(information matrix) * gradient'
>
> where gradient = e(gradient) and inv(information matrix) = e(V)
>
> But the results do not match those from SAS, at least not to the point of having rounding errors. Either e(gradient) does not mean what it's named or the inverse of information matrix != e(V), I couldn't think of other possibilities.

The gradient returned in e(gradient) is what it is supposed to be, but
not what you want it to be. What e(gradient) gives you is the gradient
of the constrained model at the estimated parameters of the
constrained model. What you want is the gradient of the
_unconstrained_ model at the parameter values of the constrained
model. Many (but not all) Stata commands allow you to get the gradient
you are after by first estimating the constrained model, store the
parameters, use those parameters at starting values for an
unconstrained model using the -from()- option, and specify that you
want to get restults after 0 iterations (i.e. just the starting
values) using the -iter()- option. Here is an example using -logit-.

*------------------ begin example ------------------
// prepare some data
sysuse auto, clear
sum price
gen z_price = ( price - r(mean) ) / r(sd)
sum weight
gen z_weight = ( weight - r(mean) ) / r(sd)

// a logit with a silly constraint
constraint 1 z_price = -z_weight
logit foreign z_price z_weight, constraint(1)
est store c

// store the coefficients
tempname b0
matrix `b0' = e(b)

// repeat that model but get the gradient for the
// unconstrained model at the parameter values of
// the constrained model
logit foreign z_price z_weight, from(`b0') iter(0)

// compute the score statistic
matrix chi = e(gradient)*e(V)*e(gradient)'
matlist chi

// compare score statistic with Wald and likelihood ratio
// not surprisingly 74 observations is not quite asymptotia
logit foreign z_price z_weight
lrtest . c

test z_price = -z_weight
*------------------- end example -------------------
(For more on examples I sent to the Statalist see:
http://www.maartenbuis.nl/example_faq )

Hope this helps,
Maarten

---------------------------------
Maarten L. Buis
WZB
Reichpietschufer 50
10785 Berlin
Germany

http://www.maartenbuis.nl
---------------------------------
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