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st: Beta coefficients are not equal to coefficients on standardized variables?


From   Roberto Liebscher <roberto.liebscher@ku-eichstaett.de>
To   statalist@hsphsun2.harvard.edu
Subject   st: Beta coefficients are not equal to coefficients on standardized variables?
Date   Fri, 15 Jun 2012 17:22:51 +0200

There is one thing that makes me puzzling about the - beta - option in regression commands. In a simple example using the lifeexp dataset I first used the built-in function - beta - :

sysuse auto

regress lexp gnppc popgrowth, beta


. regress lexp gnppc popgrowth, beta

Source | SS df MS Number of obs = 63 -------------+------------------------------ F( 2, 60) = 36.20 Model | 777.530873 2 388.765436 Prob > F = 0.0000 Residual | 644.405635 60 10.7400939 R-squared = 0.5468 -------------+------------------------------ Adj R-squared = 0.5317 Total | 1421.93651 62 22.9344598 Root MSE = 3.2772

------------------------------------------------------------------------------
        lexp |      Coef.   Std. Err.      t    P>|t|    Beta
-------------+----------------------------------------------------------------
       gnppc |    .000293   .0000419     6.99   0.000 .6506803
   popgrowth |  -.9833919    .485387    -2.03   0.047 -.1885781
       _cons |   70.67366   .8071596    87.56   0.000       .
------------------------------------------------------------------------------



Then I standardized the variables by hand and re-ran the regression with the new variables:

. egen popgrowth_std = std(popgrowth)

. egen lexp_std = std(lexp)

. egen gnppc_std = std(gnppc)
(5 missing values generated)

regress lexp_std gnppc_std popgrowth_std


Source | SS df MS Number of obs = 63 -------------+------------------------------ F( 2, 60) = 36.20 Model | 34.9700449 2 17.4850225 Prob > F = 0.0000 Residual | 28.9826364 60 .483043939 R-squared = 0.5468 -------------+------------------------------ Adj R-squared = 0.5317 Total | 63.9526813 62 1.03149486 Root MSE = .69501

------------------------------------------------------------------------------
lexp_std | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
gnppc_std | .6608475 .0945336 6.99 0.000 .4717521 .8499428 popgrowth_~d | -.1942026 .0958554 -2.03 0.047 -.3859419 -.0024633 _cons | -.0042032 .0875655 -0.05 0.962 -.1793602 .1709538
------------------------------------------------------------------------------


Now the coefficients are slightly different. For example the coefficient on gnppc_std is 0.6608475 whereas it has been 0.6506803 in the first calculation.

Is this caused by rounding errors? Or is there any other explanation for this?

Thanks in advance.

Roberto
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