Suppose I have a VAR model with different regressors in different equations
(this could be due to restricting some coefficients of a full VAR($p$) model to zero or having some different exogenous regressors in different equations).
Suppose also that the model is correctly specified.
Such a model can be estimated by GLS or by equation-by-equation OLS.
Both GLS and OLS estimators will be unbiased.
GLS estimator will be more efficient than equation-by-equation OLS estimator.
(Suppose for simplicity that the sample size is large enough so that we can rely on asymptotics where GLS is more efficient than OLS.)
Question 1: When I do forecasting, will the model estimated by GLS yield smaller mean squared error (MSE) than the one estimated by OLS?
I guess the answer is "YES". GLS will yield more precise estimates of model coefficients, which in turn will yield more accurate forecasts. But I am a bit confused: OLS will by definition give the smallest MSE in sample, for each equation in the system. Thus it will look better than GLS in terms of MSE in sample.
Question 2: Does that mean that the in-sample MSE from OLS will be overly optimistic (with reference to forecasting), especially when compared with the in-sample MSE from GLS?
Here is a remotely related question comparing OLS and GLS estimation of a VAR model.