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RE: st: RE: Interpretation of quadratic terms
From
Rodolphe Desbordes <[email protected]>
To
"'[email protected]'" <[email protected]>
Subject
RE: st: RE: Interpretation of quadratic terms
Date
Tue, 9 Mar 2010 20:39:30 +0000
Dear Rosie,
My point is that centering does not reduce multicollinearity. As you can see in my example, the standard errors of the estimated marginal effects at the mean of `mpg' are the same using uncentered or centered values of `mpg'.
Rodolphe
-----Original Message-----
From: [email protected] [mailto:[email protected]] On Behalf Of Rosie Chen
Sent: mardi 9 mars 2010 20:30
To: [email protected]
Subject: Re: st: RE: Interpretation of quadratic terms
Thanks, Rodolphe, for this helpful demonstration. Agree that the major purpose of centering seems to be that we make the interpretation of X meaningful. I guess reducing multicollinearity is a bi-product of the benefit.
Rosie
----- Original Message ----
From: Rodolphe Desbordes <[email protected]>
To: "[email protected]" <[email protected]>
Sent: Mon, March 8, 2010 7:15:12 PM
Subject: RE: st: RE: Interpretation of quadratic terms
Dear Rosie,
Centering will not affect your estimates and their uncertainty. However, centering allows you to directly obtain the estimated effect of X on Y for a meaningful value of X, i.e. the mean of X.
Rodolphe
. sysuse auto.dta,clear
(1978 Automobile Data)
. gen double mpg2=mpg^2
. reg price mpg mpg2
Source | SS df MS Number of obs = 74
-------------+------------------------------ F( 2, 71) = 18.28
Model | 215835615 2 107917807 Prob > F = 0.0000
Residual | 419229781 71 5904644.81 R-squared = 0.3399
-------------+------------------------------ Adj R-squared = 0.3213
Total | 635065396 73 8699525.97 Root MSE = 2429.9
------------------------------------------------------------------------------
price | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
mpg | -1265.194 289.5443 -4.37 0.000 -1842.529 -687.8593
mpg2 | 21.36069 5.938885 3.60 0.001 9.518891 33.20249
_cons | 22716.48 3366.577 6.75 0.000 16003.71 29429.24
------------------------------------------------------------------------------
. sum mpg
Variable | Obs Mean Std. Dev. Min Max
-------------+--------------------------------------------------------
mpg | 74 21.2973 5.785503 12 41
. local m=r(mean)
. lincom _b[mpg]+2*_b[mpg2]*`m'
( 1) mpg + 42.59459 mpg2 = 0
------------------------------------------------------------------------------
price | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
(1) | -355.3442 58.86205 -6.04 0.000 -472.7118 -237.9766
------------------------------------------------------------------------------
. gen double mpgm=mpg-`m'
. gen double mpgm2=mpgm^2
. reg price mpgm mpgm2
Source | SS df MS Number of obs = 74
-------------+------------------------------ F( 2, 71) = 18.28
Model | 215835615 2 107917807 Prob > F = 0.0000
Residual | 419229781 71 5904644.81 R-squared = 0.3399
-------------+------------------------------ Adj R-squared = 0.3213
Total | 635065396 73 8699525.97 Root MSE = 2429.9
------------------------------------------------------------------------------
price | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
mpgm | -355.3442 58.86205 -6.04 0.000 -472.7118 -237.9766
mpgm2 | 21.36069 5.938885 3.60 0.001 9.518891 33.20249
_cons | 5459.933 343.8718 15.88 0.000 4774.272 6145.594
------------------------------------------------------------------------------
.
end of do-file
________________________________________
From: [email protected] [[email protected]] On Behalf Of Rosie Chen [[email protected]]
Sent: 08 March 2010 21:15
To: [email protected]
Subject: Re: st: RE: Interpretation of quadratic terms
Thanks a lot for the advice, Rodolphe. I found several resources that suggest centering before creating quadratic terms. Below is one example.
http://www.ats.ucla.edu/stat/mult_pkg/faq/general/curves.htm
Rosie
----- Original Message ----
From: Rodolphe Desbordes <[email protected]>
To: "[email protected]" <[email protected]>
Sent: Mon, March 8, 2010 1:42:31 PM
Subject: st: RE: Interpretation of quadratic terms
Dear Rosie,
If the coefficient on X is positive and the coefficient on X^2 is negative, that suggests that X has a positive effect on Y until a turning point is reached, e.g. 1.3/(2*0.2)=3.25. Beyond that value, X has a negative impact on Y.
Rodolphe
PS: I am not sure that `centering' reduces multicollinearity.
-----Original Message-----
From: [email protected] [mailto:[email protected]] On Behalf Of Rosie Chen
Sent: lundi 8 mars 2010 18:28
To: [email protected]
Subject: st: Interpretation of quadratic terms
Dear all,
I have a question regarding how to interpret quadratic terms in regression, and would appreciate your help very much.
Because the non-linear nature of the relationship between X and Y; I need to include quadratic terms in the model. To avoid multicollinearity problem with the original variable and its quadratic term, I centered the variable first (X) and then created the square term (Xsq). The model with the quadratic term (Xsq) was proved to be significantly better. Suppose the output is like the following (both coefficients are significant), how to interpret the results? The two signs are opposite. Could anyone provide some insight? Thank you very much in advance! --Rosie
y= a + 1.3*X - 0.2*Xsq + e
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