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Re: st: Intepretation of interaction terms


From   Maarten Buis <maartenlbuis@gmail.com>
To   statalist@hsphsun2.harvard.edu
Subject   Re: st: Intepretation of interaction terms
Date   Mon, 23 May 2011 14:07:09 +0200

On Mon, May 23, 2011 at 1:45 PM, lreine ycenna wrote:
> I have linear equations, all varables are numeric and I'm using
> xtreg,rreg and pooled:
>
> (1) y = a + b + axb
>
> (2) y = a + b + c + axb + axc
>
>
> In (1), the coefficient of a becomes the effect of a when b is zero;
> coef of b becomes the effect of b when a is zero.
> In (2), since not all variables are interacted, we cannot inteprete it
> as we do in (2). Therefore, the coef of b is influenced by c, and the
> coef of c is influenced by b.

In (2) the coefficient of a (b1) is the effect of a (x1) when b (x2)
and c (x3) are both 0. You can see this by rewriting (2) as (2a). Here
I called a b and c as x1 x2 and x3 and explicitly added the
coefficients (b1 till b5).

(2a) y = b0 + b1 x1 + b2 x2 + b3 x3 + b4 x1 x2 + b5 x1 x3
      y =  b0 + (b1 +  b4 x2 + b5 x3) x1  + b2 x2 + b3 x3

So the effect of x1 is (b1 +  b4 x2 + b5 x3), which will become b1
when both x2 and x3 are 0.

The coefficient of b (b2) is the effect of b (x2) when a (x1) is 0.
You can see this by rewriting (2) as (2b).

(2b) y = b0 + b1 x1 + b2 x2 + b3 x3 + b4 x1 x2 + b5 x1 x3
      y =  b0 + b1 x1 + (b2 +  b4 x1) x2 + b3 x3 +b5 x1 x3

So the effect of x2 is (b2 +  b4 x1) , which will become b2 when x1 is 0.

The coefficient of c (b3) is the effect of c (x3) when a (x1) is 0.
You can see this by rewriting (2) as (2c)

(2c) y = b0 + b1 x1 + b2 x2 + b3 x3 + b4 x1 x2 + b5 x1 x3
      y =   b0 + b1 x1 + b2 x2 +( b3 +  b5 x1) x3 + b4 x1 x2

So the effect of x3 is (b3 +  b5 x1) , which will become b3 when x1 is 0.

Hope this helps,
Maarten

--------------------------
Maarten L. Buis
Institut fuer Soziologie
Universitaet Tuebingen
Wilhelmstrasse 36
72074 Tuebingen
Germany


http://www.maartenbuis.nl
--------------------------
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