Model averaging when linear and quadratic effects are modeled in a global model I am trying to derived estimates of model-averaged parameter effects on a fairly complicated set of models using an information-theoretic approach. I have several models that investigate continuous and categorical variables, interactions of these variables, and I have a global model which includes all of these variables and interactions. My question is, when I am model averaging, I read that the beta estimates for the slope of the effect of a quadratic term are represented by x+x^2. But, I also have a linear model of this same variable (e.g., x=linear time and x+x^2 = quadratic time). 
My first question is: Do I use both beta estimates for the quadratic model (e.g., x and x^) as part of calculating my model-averaged estimate?
Second question is: In my global model, the way I've set up my models (which I assume must be incorrect) I only have a beta estimate for x and x^2 I assume these are to be interpreted as the quadratic term x+x^2). Should I somehow be able to have an extra parameter in the global model, so that I have two estimates for x? An x to represent the linear term by itself, and a different estimate for x that is part of the x+x^2? Thanks for any help you can give to explain this.
 A: If you're just averaging model fits/predictions, it won't matter, but when you say "the quadratic term" it implies you're interested in averaging parameter estimates, in which case you should be dealing with the actual term with an $x^2$ in it, rather than the combination of linear and quadratic. 
However, in that case, it would be best to work with orthogonal polynomials, because then the presence of higher order terms doesn't impact the lower order estimates; you can average over the parameter estimates without their meaning changing.
Compare what happens without and then with orthogonal polynomials:
First, fitting linear and then linear+quadratic:
> lm(dist~speed,cars)

Coefficients:
(Intercept)        speed  
    -17.579        3.932  

> lm(dist~speed+I(speed^2),cars)

Coefficients:
(Intercept)        speed   I(speed^2)  
    2.47014      0.91329      0.09996  

As you see, the coefficient of speed is substantively affected by the presence of the speed^2 term. By comparison, when we use orthogonal polynomials, the fit is the same in both cases, but the coefficient of the linear term is the same in both models:
> lm(dist~poly(speed,1),cars)

Coefficients:
   (Intercept)  poly(speed, 1)  
         42.98          145.55  

> lm(dist~poly(speed,2),cars)

Coefficients:
    (Intercept)  poly(speed, 2)1  poly(speed, 2)2  
          42.98           145.55            23.00  

In this situation, when a term is present in the model, its meaning is the same whether there are higher order terms or not; model averaging over parameters can be explicitly seen as shrinkage toward zero (since the averaged estimate will be a weighted average of 0, when it's absent, and the estimate, when it's in the model).
