Why do estimation commands sometimes drop variables?

When you run a regression (or other estimation command) and the estimation routine drops a variable, it does so because of a dependency among the independent variables in the proposed model. You can identify this dependency by running a regression where you specify the dropped variable as the dependent variable and the remaining variables as the independent variables. Below, we generate a dependency on purpose to illustrate:

. sysuse auto
(1978 Automobile Data)

. generate newvar = price + 2.4*weight - 1.2*displ

. regress trunk price weight mpg foreign newvar displ

      Source |       SS       df       MS              Number of obs =      74
-------------+------------------------------           F(  5,    68) =   12.03
       Model |  626.913967     5  125.382793           Prob > F      =  0.0000
    Residual |  708.707655    68  10.4221714           R-squared     =  0.4694
-------------+------------------------------           Adj R-squared =  0.4304
       Total |  1335.62162    73  18.2961866           Root MSE      =  3.2283

------------------------------------------------------------------------------
       trunk |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
       price |  -.0017329   .0006706    -2.58   0.012    -.0030711   -.0003947
      weight |  (dropped)
         mpg |  -.0709254   .1125374    -0.63   0.531    -.2954903    .1536395
     foreign |   1.374419   1.287406     1.07   0.289    -1.194561    3.943399
      newvar |   .0015145   .0005881     2.58   0.012     .0003411     .002688
displacement |    .007182   .0092692     0.77   0.441    -.0113143    .0256783
       _cons |   4.170958   5.277511     0.79   0.432    -6.360151    14.70207
------------------------------------------------------------------------------

The regression dropped one of the variables that was in the dependency that we created. Which variable it drops is somewhat random, but it will always drop one of the variables in the dependency. To find out what that dependency is, we can run the regression using the dropped variable as our dependent variable and the remaining independent variables from the original regression as the independent variables in this regression.

. regress weight price mpg foreign newvar displ

      Source |       SS       df       MS              Number of obs =      74
-------------+------------------------------           F(  5,    68) =       .
       Model |  44094178.4     5  8818835.68           Prob > F      =  0.0000
    Residual |  6.9847e-07    68  1.0272e-08           R-squared     =  1.0000
-------------+------------------------------           Adj R-squared =  1.0000
       Total |  44094178.4    73  604029.841           Root MSE      =   .0001

------------------------------------------------------------------------------
      weight |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
       price |  -.4166667   2.11e-08        .   0.000    -.4166667   -.4166667
         mpg |   4.40e-06   3.53e-06     1.25   0.217    -2.65e-06    .0000115
     foreign |    .000041   .0000404     1.02   0.314    -.0000396    .0001217
      newvar |   .4166667   1.85e-08        .   0.000     .4166667    .4166667
displacement |   .4999999   2.91e-07        .   0.000     .4999993    .5000005
       _cons |  -.0002082   .0001657    -1.26   0.213    -.0005388    .0001224
------------------------------------------------------------------------------

The regression that we ran where the dropped variable was the dependent variable has an R-squared value of 1.00 and the residual sum of squares is zero (well, nearly). Also, the coefficients of the regression show the relationship between the price, newvar, and displ variables. The output of this regression tells us that we have the dependency

weight = -.4166667*price + .4166667*newvar + .4999999*displacement 

which is equivalent to the dependency that we defined above.