[R] Fitting mixed-effects models with lme with fixed error term variances

[Virgilio Gómez-Rubio]

Dear R users,

I am writing to you because I have a few question on how to fix
the error term variances in lme in the hope that you could help me. To
my knowledge, the closest possibility is to fix the var-cov structure,
but not the whole var-cov matrix. I found an old thread (a few years
ago) about this, and it seems that the only alternative is to write the
likelihood down and use optim or a similar function to get the MLE of
the parameters of the model. 

I was also trying to compute Small Area Estimators using area level
models, so that I have one observation per area. I tried to fit the
following model:

Yhat_i = X beta + u_i + e_i

where Yhat is a direct estimator of the target variable, X, the area-
level  covariates, u_i the random effects independent and distributed as
N(0, sigma2_u) and e_i the 'error' terms, which are distributed as N(0,
sigma2_e/n_i), where n_i is the sample size in area i. 

This model should be unidentifiable because we have one observation per
group and we have to estimate two area level terms: u_i and e_i. But, to
my surprise, I have been able to fit that model using lme and get
sensible results. 

Am I missing something here? I have checked Pinheiro and Bates' book but
I have not been able to find an answer to my questions.

Many thanks for your help. Best regards,

Virgilio