Date: Dec 2, 2012 5:48 PM
Author: Ray Koopman
Subject: Re: Interpretation of coefficients in multiple regressions which<br> model linear dependence on an IV

On Nov 21, 8:13 pm, djh <> wrote:
> In a different thread, Ray Koopman explained that if one
> suspects these regressions to be dependent on the IV ?u?:
> c on u
> c on e
> c on (e,u)
> then under the usual initial assumption that the dependence
> is linear, these three regressions should be modified to:
> c on (u, u^2) instead of c on u
> c on (e, u, u*e) instead of c on e
> c on (e, u, u*e, u^2) instead of c on (e,u)

In this post I want to talk about only your third case:

y = a0 + a1*x1 + a2*x2 + a3*x1*x2 + a4*x2^2.

(As before, I use the usual generic variable names so as not to
get caught up in any peculiarities of your particular variables.)

One interpretation of the model is that y is a linear function of x1
and x2: y = A0 + A1*x1 + A2*x2, with A0, A1, and A2 all being linear
functions of x2. (I'll let you do the algebra; it parallels that in
the first two models.) Note that this model does not treat x1 and x2
symmetrically: it matters which x is 1 and which is 2.

There are two average slopes. Call them Av1 and Av2:

dy/dx1 = a1 + a3*x2
dy/dx2 = a2 + a3*x1 + 2*a4*x2

Let m1 = mean_x1 and m2 = mean_x2.

Av1 = a1 + a3*m2
Av2 = a2 + a3*m1 + 2*a4*m2

var[Av1] = var[a1] + var[a3]*m2^2 + 2*cov[a1,a3]*m2

var[Av2] = var[a2] + var[a3]*m1^2 + 4*var[a4]*m2^2 +
2*cov[a2,a3]*m1 + 4*cov[a2,a4]*m2 + 4*cov[a3,a4]*m1*m2

cov[Av1,Av2] = cov[a1,a2] + var[a3]*m1*m2 +
cov[a1,a3]*m1 + cov[a2,a3]*m2 +
2*cov[a1,a4]*m2 + 2*cov[a3,a4]*m2^2

df = n-5.