Difference in the estimate between subsample and interaction model

[Jeffery Xu]

Hi all

This question might be stupid but I am really curious.

Suppose I want to run the following two models. The first is the interaction model with full dataset. The second is the subsample when foreign = 0.

.
sysuse auto
reg weight  c.price##i.foreign turn length

      Source |       SS           df       MS      Number of obs   =        74
-------------+----------------------------------   F(5, 68)        =    208.98
       Model |  41400010.4         5  8280002.08   Prob > F        =    0.0000
    Residual |  2694167.97        68  39620.1172   R-squared       =    0.9389
-------------+----------------------------------   Adj R-squared   =    0.9344
       Total |  44094178.4        73  604029.841   Root MSE        =    199.05

---------------------------------------------------------------------------------
         weight | Coefficient  Std. err.      t    P>|t|     [95% conf. interval]
----------------+----------------------------------------------------------------
          price |    .064213   .0103255     6.22   0.000     .0436088    .0848173
                |
        foreign |
       Foreign  |  -99.87397   140.4027    -0.71   0.479    -380.0432    180.2953
                |
foreign#c.price |
       Foreign  |     -.0246   .0193774    -1.27   0.209    -.0632669     .014067
                |
           turn |   20.14046   11.44118     1.76   0.083    -2.690055    42.97098
         length |   23.29932   2.314677    10.07   0.000     18.68046    27.91819
          _cons |  -2477.284   308.2764    -8.04   0.000     -3092.44   -1862.128
---------------------------------------------------------------------------------

. reg weight  c.price##i.foreign turn length if foreign == 0
note: 0.foreign omitted because of collinearity.
note: 0.foreign#c.price omitted because of collinearity.

      Source |       SS           df       MS      Number of obs   =        52
-------------+----------------------------------   F(3, 48)        =    163.81
       Model |  22465799.4         3  7488599.81   Prob > F        =    0.0000
    Residual |  2194267.87        48  45713.9139   R-squared       =    0.9110
-------------+----------------------------------   Adj R-squared   =    0.9055
       Total |  24660067.3        51  483530.732   Root MSE        =    213.81

---------------------------------------------------------------------------------
         weight | Coefficient  Std. err.      t    P>|t|     [95% conf. interval]
----------------+----------------------------------------------------------------
          price |   .0625345   .0111822     5.59   0.000     .0400512    .0850179
                |
        foreign |
      Domestic  |          0  (omitted)
                |
foreign#c.price |
      Domestic  |          0  (omitted)
                |
           turn |   17.54976   13.18924     1.33   0.190     -8.96897     44.0685
         length |   24.26179   2.716546     8.93   0.000     18.79981    29.72377
          _cons |  -2548.501   338.2455    -7.53   0.000    -3228.589   -1868.413
---------------------------------------------------------------------------------

Could anyone explain why the coefficient of "Price" is different between the interaction model and subsample analysis? Given the coefficient estimate of "Price" is equal to the slope when foreign = 0. Why this value is different from the one estimated using the subsample?

[Ken Chui]

That is only true if "turn" and "length" are associated with "weight" in ways that are completely independent by "foreign." Once a sub-sample is created, the confounding relationship among the variables are no longer the same. Perhaps looking at this pair of command may help. In order to recreate the stratified model in an interaction model, everything needs the same interaction term:

sysuse auto, clear

reg weight c.price##i.foreign c.turn##i.foreign c.length##i.foreign

reg weight price turn length if foreign == 0