st: RE: statalist-digest V4 #4698

[Philip Bromiley <bromiley@uci.edu>]

This response brings up a couple of issues.

Here's the model discussed:
  (1)   y = b1*x1 + b2*(x1)^2 + b3*x2 + constant

First, if you have a linear and a squared term, the parameter on the linear term is not irrelevant.  Obviously, the derivative of y with respect to x depends on b1 and b2.  You can even have a situation where both b1 and b2 are statistically insignificant individually (due to the correlation of x1 and x1^2) but jointly are significant.

However, there is a broader issue that has bothered me - does centering really fix a colinearity problem with a squared variable?  

I thought Bresley et al's Regression Diagnostics claimed the improvements were an illusion.  You're getting a different coefficient on x1 when you center it to create x1 before you do the squares because you've changed the origin.  If this were a standard interaction problem (y = b1*x1 + b2*(x2) + b3*x1*x2), b1 is the influence of x1 when x2=0 so adding or subtracting a constant from b2 will naturally change b1.  When I simulate the problem in equation 1, centering x1 before calculating x1*x1 does not change the parameter (or the standard error) for b2, i.e., the parameter on (x1)^2.  However, it does seem to change the estimated value of b1.

Would anyone like to comment on this issue?

Phil

 
Philip Bromiley
Dean's Professor in Strategic Management
Merage School of Business
University of California, Irvine
Irvine, CA 92697-3125
Bus: (949) 824-6657
Cell: (949) 943-6489
Date: Sat, 10 Nov 2012 12:16:30 -0500
From: "JVerkuilen (Gmail)" <jvverkuilen@gmail.com>
Subject: Re: st: Can multicollinearity problems be resolved by using residuals from another regression?

On Thu, Nov 8, 2012 at 9:36 PM, A. Shaul <3c5171@gmail.com> wrote:
> Dear Statalist,
>
> I expect a non-linear effect of an exogenous variable, x1, on a 
> dependent variable, y. The variable x1 is affected by another 
> exogenous variable, x2. The variable x2 affects x1 directly and also y 
> directly. The variable x1 does not affect x2. I am only interested in 
> the partial effect of x1 on y while controlling for x2 --- or at least 
> while controlling for the part of the variation in x2 that affects y 
> directly.
>
> I have the following regression equation:
>
>    (1)   y = b1*x1 + b2*(x1)^2 + b3*x2 + constant

I'm not 100% sure what you're doing but when you have polynomial terms like this collinearity is inevitable. Before doing anything odd, center x1 and then compute x1^2, and regress on the centered variables. (You may want to rescale x1 as well but centering does the
work.) This will give you a statistically equivalent model that breaks the collinearity between x1 and x1^2.

Usually though you're not interpreting x1 terms directly anyhow, so whether x1 or x1^2 is statistically significant individually is irrelevant. Certainly the linear term for x1 is irrelevant if the term for x1^2 is significant. You can test for x1 effects as a block using
- -testparm-.

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