Model averaging in prediction — “Wisdom of the Crowd”

Question

Suppose I'm trying to predict $Y$ (a real number) and I have $n$ experts who with guesses $Y_1,...Y_n$. Each prediction is a reasonable guess as to the value of Y in itself (hence the name "expert"), but we should be able to come up with a better prediction for $Y$ by combining the experts guesses. We could take our guess $Y_o$ to be

$$Y_o=\sum_{i=1}^n\frac{Y_i}{n}$$

but this isn't optimal if some experts are better than others.

I have a few questions related to this topic.

Question 1: Is this equal weights method used much? Are there any other simple methods? Can you refer me to (preferably relatively basic) information on the topic?

Question 2: I've seen model averaging discussed in this as a possible alternative to other methods of model selection for linear regression which may fail, for example, if there is multicollinearity. That's led me to wonder whether the following solution might be justifiable:

We want (perhaps out of naivete) to regress $Y$ against predictors $X1,X2$ and $W$. Suppose, in fact, that $X1$ and $X2$ themselves are predictors of $Y$. Using multilinear regression for $Y$ with all three variables is problematic because, being predictors of $Y$, $X1$ and $X2$ exhibit multicollinearity, but what about creating two multilinear regression models

$$Y=\alpha_o + \alpha_1X1 + \alpha_2W$$

and

$$Y = \beta_o + \beta_1X2 + \beta2W$$

and then using some average-like combination of these two regression-based predictors to predict $Y$? What weights might we use? Again, is there any literature on this topic?

Thank you.

EDIT: Since I posted this, I've read about stacked regression, as described here. Does the situation I've described sound suitable for applying stacked regression?

Answer 1

Just to help keep perspective, it is useful to remember that model averaging is a complex and hard to interpret process that does not surpass a global penalized model in terms of predictive accuracy. That is, if all the models are part of the same family so that a global all-inclusive model can be formed. So if you fit a more saturated model (e.g., include main effects of all potential predictors and expand them with regression splines so as to not assume linearity) but you use penalized maximum likelihood estimation or Bayesian estimation so as to avoid overfitting, you will get a very competitive result that is easier to interpret, as you can estimate the partial effect of any one variable by simply looking at its partial coefficients. Model simplification (also called pre-conditioning) can also be of help here.