I understand that heteroscedasticity leads to problems with coefficient estimates of a model, but I'd like to know how it relates to predictive accuracy. After creating my original linear model, I am able to model the magnitude of the residuals (R^2~.06) using the variables from my original model. I tried Weighted least squares, however can't get anything to improve the predictive accuracy of my original model. I am only concerned with predictive accuracy and so would like to know if there is anything else I could try and also if the existence of heteroskasdicity implies that predictions could definitely be improved.
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Heteroscedasticity does not distort your predictive accuarcy but only inference. Therefore, if your aim is just prediction and you do not want to make statements like "$\beta$ is significant" then you do not have to mind about the heteroscedasticity. The technical reason for this is, that you do not need homoscedasticity for consistency or unbiasdeness. Remember that $$\hat{\beta}= \beta +(X'X)^{-1}X'e$$ and therefore $$E[\hat{\beta}]=\beta$$ and $$plim[\hat{\beta}]=\beta$$ if $E[(X'X)^{-1}X'e]=0$ or $plim[(X'X)^{-1}X'e]=0$, respectively. But no assumptions about $V[e|X]$ are needed.
I would recommend you to go with the unbiased estimates and estimate additonally a heteroscedasticity robust variance-covariance matrix (white-huber standard errors) of the style $ (X'X)^{-1}X'ee'X(X'X)^{-1}$ but I would not recommend to use FGLS (which includes weighted least squares) for prediction as the FGLSE (feasible generalized least squares estimator) is biased.
