Can regression forecasts of univariate time series be independent (of one another) Suppose I have short-term forecasts from two univarite regression models of the same time series.  I am choosing the models to be as different as possible in structure and assumptions. For instance, one may be an autoregression and the other fits a set of Fourier terms. Both models use only lags or transformed lags of the dependent variable as independent variables.
Is it likely that the two forecasts will be independent? Is it possible?
I am trying to understand why the average of several forecasts is often better than the individual forecasts, and whether there is anything I can do to make that more likely to be true, or true to a greater extent. It seems like the average should be better than the component forecasts if the components have expected value equal to the future value, variances are comparable in magnitude,  and the forecasts are independent.
 A: If two regression models of the same time series are trained on the same set of data, then the forecasts generated by these models are dependent on one another as both models are likely to have learned similar patterns from the data.
To try to make these two short-term regression models statistically independent in your scenario is to fit your models on different training sets of the same sample dataset, for instance, to use different periods of the same underlying time series or to split your time series into two sequence parts each of which is to be fed to different model, then the forecasts generated by these models would be possibly independent.
If each model is accurate and independent then averaging them together will likely result in improved overall forecast accurate with reduced variance, in the same fashion as random forest ensemble method. On the other hand, if the individual models are themselves inaccurate, then averaging them may not result in a significant improvement in accuracy.
