The $\beta$'s that come out of GLS are estimates of the predictor effects that contribute to the mean response. So, to predict a new data point, yes, you would just plug in the predictor values into the linear predictor. The observed prediction error will usually take a form like

$$ \sum_{i} (Y_{i} - \hat{Y}_{i} )^2 $$

which potentially is the mean of a sum of non-independent random variables if the test data points are heteroskedastic/autocorrelated in the same way the training data was. Fortunately, the linearity of expectation,

$$ E \big( \sum_{i} X_{i} \big) = \sum_{i} E(X_{i}) $$

is true regardless of whether or not the $X_{i}$ are independent, so your prediction error will not be biased.

The $\beta$'s that come out of GLS are estimates of the predictor effects that contribute to the mean response and the assumption of GLS is that the mean of $Y|X$ is the same for each data point - only non-constant variance and serial correlation are allowed for. So, to predict a new data point, yes, you would just plug in the predictor values into the linear predictor. The observed prediction error will usually take a form like

$$ \sum_{i} (Y_{i} - \hat{Y}_{i} )^2 $$

which potentially is the mean of a sum of non-independent random variables if the test data points are heteroskedastic/autocorrelated in the same way the training data was. Fortunately, the linearity of expectation,

$$ E \big( \sum_{i} X_{i} \big) = \sum_{i} E(X_{i}) $$

is true regardless of whether or not the $X_{i}$ are independent so your prediction error will not be biased.

The $\beta$'s that come out of GLS are estimates of the predictor effects that contribute to the mean response. So, to predict a new data point, yes, you would just plug in the predictor values into the linear predictor. The observed predictor error will usually take a form like

$$ \sum_{i} (Y_{i} - \hat{Y}_{i} )^2 $$

is the mean of a sum of non-independent random variables if the test data points are not independent. Fortunately, the linearity of expectation,

$$ E \big( \sum_{i} X_{i} \big) = \sum_{i} E(X_{i}) $$

is true regardless of whether or not the $X_{i}$ are independent, so your prediction error will not be biased.

The $\beta$'s that come out of GLS are estimates of the predictor effects that contribute to the mean response. So, to predict a new data point, yes, you would just plug in the predictor values into the linear predictor. The observed prediction error will usually take a form like

$$ \sum_{i} (Y_{i} - \hat{Y}_{i} )^2 $$

which potentially is the mean of a sum of non-independent random variables if the test data points are heteroskedastic/autocorrelated in the same way the training data was. Fortunately, the linearity of expectation,

$$ E \big( \sum_{i} X_{i} \big) = \sum_{i} E(X_{i}) $$

is true regardless of whether or not the $X_{i}$ are independent, so your prediction error will not be biased.