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Can the general linear model be solved by generalized least squares?

The model that generalized least squares (GLS) can solve doesn't assume anything on the covariance matrix of the errors, which I think is the same as a general linear model?

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  1. Yes. "General linear model" is a tag synonym for "multiple regression" at this site. GLS can of course solve multiple regression. According to the first Wikipedia page, "the errors are usually assumed to follow a multivariate normal distribution. If the errors do not follow a multivariate normal distribution, generalized linear models may be used to relax assumptions about errors". So general linear model can also be solved by other typical estimation methods, like ordinary least squares and maximum likelihood.
  2. I think there is no explicit connection between generalized least squares and general linear model. The key point is how to understand the term "general linear model". If we regard it as a synonym for "multiple regression", then we can also have assumptions about the errors. Wikipedia also addressed "the general linear model is a generalization of multiple linear regression model to the case of more than one dependent variable." In this way, it is another problem about multivariate analysis.
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  • $\begingroup$ Thanks! Isn't ordinary least square assuming the covaraince matrix is diagonal and the diagonal entries are the same? $\endgroup$ – Tim Jul 28 '13 at 15:40
  • $\begingroup$ Yes, the errors are assumed to be homoscedastic and serially uncorrelated to make the OLS estimator optimal in the class of linear unbiased estimators. GLS has almost no connection with general linear model, as I edited above. $\endgroup$ – Randel Jul 28 '13 at 16:04

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