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st: standard error obtained from ttest and linear regression


From   "Aijing Shang" <shang@ispm.unibe.ch>
To   <statalist@hsphsun2.harvard.edu>
Subject   st: standard error obtained from ttest and linear regression
Date   Thu, 19 Feb 2004 17:02:24 +0100

By chance, I found the standard error of mean values obtained from ttest is
different from linear regression. Please look at the following example.
The data is auto.dta in STATA.

. ttest price, by(foreign)

Two-sample t test with equal variances

----------------------------------------------------------------------------
--
   Group |     Obs        Mean    Std. Err.   Std. Dev.   [95% Conf.
Interval]
---------+------------------------------------------------------------------
--
Domestic |      52    6072.423    429.4911    3097.104    5210.184
6934.662
 Foreign |      22    6384.682    558.9942    2621.915     5222.19
7547.174
---------+------------------------------------------------------------------
--
combined |      74    6165.257    342.8719    2949.496    5481.914
6848.6
---------+------------------------------------------------------------------
--
    diff |           -312.2587    754.4488               -1816.225
1191.708
----------------------------------------------------------------------------
--
Degrees of freedom: 72

                Ho: mean(Domestic) - mean(Foreign) = diff = 0

     Ha: diff < 0               Ha: diff != 0              Ha: diff > 0
       t =  -0.4139                t =  -0.4139              t =  -0.4139
   P < t =   0.3401          P > |t| =   0.6802          P > t =   0.6599


. reg price foreign

      Source |       SS       df       MS              Number of obs =
74
-------------+------------------------------           F(  1,    72) =
0.17
       Model |  1507382.66     1  1507382.66           Prob > F      =
0.6802
    Residual |   633558013    72  8799416.85           R-squared     =
0.0024
-------------+------------------------------           Adj R-squared
= -0.0115
       Total |   635065396    73  8699525.97           Root MSE      =
2966.4

----------------------------------------------------------------------------
--
       price |      Coef.   Std. Err.      t    P>|t|     [95% Conf.
Interval]
-------------+--------------------------------------------------------------
--
     foreign |   312.2587   754.4488     0.41   0.680    -1191.708
1816.225
       _cons |   6072.423    411.363    14.76   0.000     5252.386
6892.46
----------------------------------------------------------------------------
--

ttest and linear regression provide exactly the same estimates of mean
difference of price between domestic and foreign autos, which is 312.2587 in
this example. As well as its standard error, which is 754.4488. However,
look at _cons in the linear regression. In this example, the coefficient of
_cons should be the mean price of domestic auto, since varible "foreign" is
coded as 0 if the autos were from domestic, 1 if foreign. The coefficient of
_cons is exactly the same as the mean value in ttest, which is 6072.423. But
its standard error in ttest is 429.4911, whereas 411.363 in linear
regression. Can anybody tell me how this can happen? Thank you very much in
advance.

Aijing Shang

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