# Forecasting with a VAR estimated by GLS versus OLS

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.

## 2 Answers

There is such thing as best linear unbiased prediction (BLUP). It is similar and actualy closely related to Gauss-Markov theorem about best linear unbiased estimator. If your forecasts are BLUP, then the MSE is smallest for the underlying model. If the disturbances are not correlated and homoscedastic, then the lowest MSE comes from OLS, if disturbances are correlated, then the lowest MSE comes from GLS.

Actually I have also been spending a lot of time being concentrated on this issue.

MSE, specifically in sample, is by definition least in OLS, which many times I simulated and got the results that MSE from OLS is less than MSE from GLS, however the differences between two were not large. Meanwhile, I could also check that S.E of GLS is a bit less than S.E of OLS as in textbook.

In a view of population a coefficient of GLS is unbiased as OLS and most efficient, which would give us least MSE in a population. So it shows greater MSE than MSE from OLS only in a current sample data.

p.s : Do you get an another insight on this issue?

• Thanks. I think we have similar views on this issue. I did not get another insight since the time I posted the question. – Richard Hardy May 7 '15 at 5:32