*
* Example 18.7 from pp 530-531
*
open data tablefd-1[1].txt
data(format=prn,org=columns) 1 20 id y e
*
set ysq  = y**2
set logy = log(y)
set recy = 1/y
*
nonlin p lambda
*
frml yfrml    = y-p/lambda
frml ysqfrml  = y**2-p*(p+1)/lambda**2
frml recyfrml = 1/y-lambda/(p-1)
frml logyfrml = log(y)-%digamma(p)+log(lambda)
*
compute p=2,lambda=1.0
*
* Maximum likelihood
*
frml loglike = p*log(lambda)-%lngamma(p)-lambda*y+(p-1)*log(y)
maximize(method=bhhh) loglike
*
instruments constant
*
* The RATS NLSYSTEM instruction handles the weight matrix a bit differently than
* the description on page 539: it computes a new one after each iteration of the
* non-linear optimization. The following two operation will produce the same
* results as the process described in the text:
*
*   Step 1: Computes the minimum distance estimator using the identity matrix
*    as the sigma matrix of the residuals.
*   Step 2: Use NLSYSTEM with the "SECONDSTEP" option. This computes a single
*    weight matrix at the initial parameter values.
*
* Note that the standard errors computed in the text aren't correct. You can't do
* the simple calculation using a different weight matrix than was used in
* estimating the coefficients.
*
nlsystem(instruments,nosigma,cv=%identity(4)) / yfrml ysqfrml logyfrml recyfrml
nlsystem(instruments,nosigma,zudep,secondstep) / yfrml ysqfrml logyfrml recyfrml