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RE: st: RE: Interpretation of quadratic terms
From
"Nick Cox" <[email protected]>
To
<[email protected]>
Subject
RE: st: RE: Interpretation of quadratic terms
Date
Tue, 9 Mar 2010 21:02:52 -0000
I think you're both right. In olden days, pre-emptive centring, as we
say in English, was a good idea in order to avoid numerical problems
with mediocre programs that did not handle near multicollinearity well.
Nowadays, decent programs including Stata take care that you get bitten
as little as possible by such problems. If course, if you really do have
multicollinearity, nothing much can help, except that Stata drops
predictors and flags the issue.
Nick
[email protected]
Rodolphe Desbordes
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'.
Rosie Chen
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.
Rodolphe Desbordes <[email protected]>
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.
. 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
------------------------------------------------------------------------
------
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