Computation of the standard error of a FGLS coefficient estimate

Hi,

I use the MATLAB code below to calculate the standard errors of the coeffficients estimated by Weighted Least Squares (WLS), using some data at hand. I wanted to check if my calculation is correct. Therefore I used Stata, an econoemtric software, which computes the standard errors of WLS coefficients estimates. Stata documentation at http://www.stata.com/manuals13/rvwls.pdf gives the formula used by Stata. The coefficient estimates I obtain in MATLAB are exactly the same coefficient estimates I obtain in Stata, but the standard errors differ. In particular, the standard errors I obtain in MATLAB are always a constant fraction of those I obtain in Stata. That is, the former figures are always about half the size of the latter figures. I am surprised because the formula of the WLS coefficient estimates is (X'*inv(Omega)*X)\(X'*inv(Omega)*Y) and it is straightforward to obtain the variance-covariance matrix of the WLS coefficient estimates, which is just the inverse of (X'*inv(Omega)*X). I cannot seem to pinpoint the reason why the standard errors differ.

Tunga

• Matlab code

clear;

load 'data';

Y = unaid;

N = length(Y);

X = [ones(N,1) dur ncb rank year];

beta_hat = (X'*X)\(X'*Y);

yhat = X*beta_hat;

e = Y-yhat;

log_e2 = log(e.^2);

beta_hat2 = (X'*X)\(X'*log_e2);

yhat2 = X*beta_hat2;

O = diag(exp(yhat2));

OI = eye(N)/O;

beta_hat_fgls = (X'*OI*X)\(X'*OI*Y);

beta_hat_varcov_fgls = (X'*OI*X)\eye(5);

beta_hat_se_fgls = sqrt(diag(beta_hat_varcov_fgls));

Standard errors of five variables:

2.087

0.013

0.044

0.025

0.024

• Stata code

clear all

use "data", clear

reg unaid dur ncb rank year

predict e, resid

gen e2 = e^2

gen le = log(e2)

reg le dur ncb rank year

predict le_hat

gen w_hat = (exp(le_hat))

reg unaid dur ncb rank year [aweight=1/w_hat]

Standard errors of five variables:

3.883

0.025

0.081

0.048

0.044