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nlme::gls potential bug

Andy Beet 
Hi there,


I have been using the nlme::gls package created in R to fit a pretty 
simple model (linear with AR error)

y(t) = beta*x(t) + e(t)? ??? ??? ??? where e(t) ~ rho*e(t-1) + Z(t) ?? 
 ??? and Z(t)~ N(0,sig^2)

I call the R routine

glsObj <- nlme::gls(y ~ x -1, data=data, correlation = 
nlme::corAR1(form= ~x), method="ML")

All seems fine.


In addition, I have also coded the likelihood myself and maximized it 
for beta, rho and sigma.

I get the exact same estimates of beta and rho, (as nlme::gls) but the 
estimate of sigma is not the same and i can not figure out why.

The maximum likelihood estimator for sigma under this model is

sig^2 = (( 1-rho^2)u(1)^2 + sum((u(t)- rho*u(t-1))^2)/n

where the sum is t=2,...,n and

u(t) = y(t) - X(t)*beta


I have read the mixed-effects models in S and S-Plus book (nlme::gls 
code is based directly on this) and this problem is specified on page 
204 eq (5.5). I have also calculated sigma based on (5.7) -after the 
transformation documented (5.2) -and i do not get the same value as 
either the package or my implementation.

Any advice would be most welcomed. Is there a bug in the estimation of 
sigma in this package?

Thanks

Andy