Differences in correlation coefficients produced by corr and reg

[Jack Peters]

For certain variables, corr and reg produce different correlation coefficients that cannot be explained by rounding error. For example:

webuse lifeexp, clear

* Correlation coefficient from corr (1)
corr lexp gnppc

* To get the correlation coefficient from reg, first standardize both variables to have mean 0 and standard deviation 1
egen std_lexp = std(lexp)
egen std_gnppc = std(gnppc)

* Correlation coefficient from reg (2)
reg std_lexp std_gnppc

On Stata 17, (1) is 0.718 and (2) is 0.729.

summarize std_lexp std_gnppc

and

corr std_lexp std_gnppc

show that this behavior is not explained by failure of standardization. Both corr and reg use the same observations. What can explain this disparity? It must be something to do with these variables in particular, as for most variables (e.g. those in auto, nlsw88, or this dataset), the correlations produced by corr and reg are almost identical up to rounding error.

[Andrew Musau]

You are using a different sample to standardize the variables and this little detail matters.

webuse lifeexp, clear 

* Correlation coefficient from corr (1)
corr lexp gnppc

* To get the correlation coefficient from reg, first standardize both variables to have mean 0 and standard deviation 1

qui reg lexp gnppc
foreach var in lexp gnppc{
    qui sum `var' if e(sample)
    g std_`var'= (`var'- r(mean))/r(sd) 
}
reg std_lexp std_gnppc, nocons

Res.:

. corr lexp gnppc
(obs=63)

             |     lexp    gnppc
-------------+------------------
        lexp |   1.0000
       gnppc |   0.7182   1.0000


. reg std_lexp std_gnppc, nocons

      Source |       SS           df       MS      Number of obs   =        63
-------------+----------------------------------   F(1, 62)        =     66.05
       Model |  31.9801064         1  31.9801064   Prob > F        =    0.0000
    Residual |  30.0198953        62   .48419186   R-squared       =    0.5158
-------------+----------------------------------   Adj R-squared   =    0.5080
       Total |  62.0000017        63   .98412701   Root MSE        =    .69584

------------------------------------------------------------------------------
    std_lexp |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
   std_gnppc |   .7181979   .0883716     8.13   0.000     .5415455    .8948503
------------------------------------------------------------------------------

Anyway, a quicker way to test your theory is to invoke the -beta- option of regress:

webuse lifeexp, clear 

*Regress -beta-
regress lexp gnppc, beta

Res.:

. *Regress -beta-

. 
. regress lexp gnppc, beta

      Source |       SS           df       MS      Number of obs   =        63
-------------+----------------------------------   F(1, 61)        =     64.98
       Model |  733.446453         1  733.446453   Prob > F        =    0.0000
    Residual |  688.490055        61  11.2867222   R-squared       =    0.5158
-------------+----------------------------------   Adj R-squared   =    0.5079
       Total |  1421.93651        62  22.9344598   Root MSE        =    3.3596

------------------------------------------------------------------------------
        lexp |      Coef.   Std. Err.      t    P>|t|                     Beta
-------------+----------------------------------------------------------------
       gnppc |   .0003234   .0000401     8.06   0.000                 .7181979
       _cons |   69.44836   .5479821   126.73   0.000                        .
------------------------------------------------------------------------------

[Jack Peters]

Thanks so much!