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## Why do estimation commands sometimes omit variables?

 Title Estimation commands and omitted variables Author James Hardin, StataCorp

When you run a regression (or other estimation command) and the estimation routine omits 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 omitted 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
note: weight omitted because of collinearity.

 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 Coefficient Std. err. t P>|t| [95% conf. interval] price -.0017329 .0006706 -2.58 0.012 -.0030711 -.0003947 weight 0 (omitted) 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 omitted one of the variables that was in the dependency that we created. Which variable it omits is somewhat arbitrary, but it will always omit one of the variables in the dependency. To find out what that dependency is, we can run the regression using the omitted 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) > 99999.00 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 Coefficient Std. err. t P>|t| [95% conf. interval] price -.4166667 2.11e-08 -2.0e+07 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 2.3e+07 0.000 .4166667 .4166667 displacement .4999999 2.91e-07 1.7e+06 0.000 .4999993 .5000005 _cons -.0002082 .0001657 -1.26 0.213 -.0005388 .0001224

The regression that we ran where the omitted 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.