Specification Errors

Linear Statistical Models: Regression

Specification Errors

Updates for Stata 11

Specification Errors

Recall that specification errors include:

Omission of relevant variables.
Inclusion of irrelevant variables.
Nonlinearity.
Nonadditivity (Interaction).
Both nonlinearity and nonadditivity can be thought of as special cases of omitted variables.

Omitting Relevant Variables

When relevant variables are omitted from the model and they are correlated with variables in the model -
estimation of the regression coefficient is biased.
Since omitting relevant variables when they are uncorrelated introduces no bias in estimation -
One may be temped to leave these variables out to be safe -
But this, in turn, leads to higher standard errors.

Including Irrelevant Variables

Estimation of regression coefficient not biased -
But there is an increase in the standard error of estimate.
Beware of the "play it safe" method of throwing variables into the model.
Less serious than omitting relevant variables.

Some Stata Commands

linktest - tests for specification error.
ovtest - tests for omitted variables.

use http://www.philender.com/courses/data/hsbdemo, clear

regress read write female i.ses

      Source |       SS       df       MS              Number of obs =     200
-------------+------------------------------           F(  4,   195) =   35.32
       Model |  8789.20628     4  2197.30157           Prob > F      =  0.0000
    Residual |  12130.2137   195  62.2062242           R-squared     =  0.4201
-------------+------------------------------           Adj R-squared =  0.4083
       Total |    20919.42   199  105.122714           Root MSE      =  7.8871

------------------------------------------------------------------------------
        read |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
       write |    .666923   .0631027    10.57   0.000     .5424716    .7913743
      female |  -3.997829   1.182598    -3.38   0.001    -6.330155   -1.665504
             |
         ses |
          2  |   1.727183   1.427627     1.21   0.228    -1.088388    4.542754
          3  |   3.967854   1.610743     2.46   0.015     .7911399    7.144568
             |
       _cons |   17.24087   3.271046     5.27   0.000      10.7897    23.69204
------------------------------------------------------------------------------

testparm i.ses

 ( 1)  2.ses = 0
 ( 2)  3.ses = 0

       F(  2,   195) =    3.11
            Prob > F =    0.0467

linktest

      Source |       SS       df       MS              Number of obs =     200
-------------+------------------------------           F(  2,   197) =   72.36
       Model |  8859.13636     2  4429.56818           Prob > F      =  0.0000
    Residual |  12060.2836   197  61.2197139           R-squared     =  0.4235
-------------+------------------------------           Adj R-squared =  0.4176
       Total |    20919.42   199  105.122714           Root MSE      =  7.8243

------------------------------------------------------------------------------
        read |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
        _hat |  -.2489419   1.171552    -0.21   0.832    -2.559335    2.061451
      _hatsq |   .0121192   .0113394     1.07   0.286    -.0102429    .0344814
       _cons |   31.63874   29.92719     1.06   0.292    -27.38005    90.65752
------------------------------------------------------------------------------

ovtest

Ramsey RESET test using powers of the fitted values of read
       Ho:  model has no omitted variables
                 F(3, 192) =      1.44
                  Prob > F =      0.2337
                  
use http://www.philender.com/courses/data/gpa, clear

regress gpa mat ar

  Source |       SS       df       MS                  Number of obs =      30
---------+------------------------------               F(  2,    27) =   13.09
   Model |  5.13721671     2  2.56860836               Prob > F      =  0.0001
Residual |  5.29744993    27  .196201849               R-squared     =  0.4923
---------+------------------------------               Adj R-squared =  0.4547
   Total |  10.4346666    29  .359816091               Root MSE      =  .44295

------------------------------------------------------------------------------
     gpa |      Coef.   Std. Err.       t     P>|t|       [95% Conf. Interval]
---------+--------------------------------------------------------------------
     mat |   .0249295   .0104491      2.386   0.024       .0034898    .0463692
      ar |   .2997867    .115247      2.601   0.015       .0633194    .5362541
   _cons |   .5738174   .6029815      0.952   0.350      -.6633984    1.811033
------------------------------------------------------------------------------

linktest

  Source |       SS       df       MS                  Number of obs =      30
---------+------------------------------               F(  2,    27) =   21.11
   Model |  6.36484445     2  3.18242222               Prob > F      =  0.0000
Residual |   4.0698222    27  .150734155               R-squared     =  0.6100
---------+------------------------------               Adj R-squared =  0.5811
   Total |  10.4346666    29  .359816091               Root MSE      =  .38824

------------------------------------------------------------------------------
     gpa |      Coef.   Std. Err.       t     P>|t|       [95% Conf. Interval]
---------+--------------------------------------------------------------------
    _hat |   9.044484   2.824042      3.203   0.003       3.250028    14.83894
  _hatsq |  -1.181952   .4141642     -2.854   0.008      -2.031747   -.3321575
   _cons |  -13.47598   4.756589     -2.833   0.009      -23.23569   -3.716264
------------------------------------------------------------------------------

ovtest

Ramsey RESET test using powers of the fitted values of gpa
       Ho:  model has no omitted variables
                  F(3, 24) =      9.83
                  Prob > F =      0.0002
                  
regress gpa mat ar greq grev

      Source |       SS       df       MS              Number of obs =      30
-------------+------------------------------           F(  4,    25) =   11.13
       Model |  6.68313355     4  1.67078339           Prob > F      =  0.0000
    Residual |   3.7515331    25  .150061324           R-squared     =  0.6405
-------------+------------------------------           Adj R-squared =  0.5829
       Total |  10.4346666    29  .359816091           Root MSE      =  .38738

------------------------------------------------------------------------------
         gpa |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         mat |   .0208961   .0095488     2.19   0.038     .0012299    .0405623
          ar |   .1442335   .1130013     1.28   0.214    -.0884969     .376964
        greq |   .0039983   .0018307     2.18   0.039      .000228    .0077686
        grev |   .0015236   .0010502     1.45   0.159    -.0006392    .0036865
       _cons |  -1.738107   .9507399    -1.83   0.079    -3.696192    .2199789
------------------------------------------------------------------------------

linktest

      Source |       SS       df       MS              Number of obs =      30
-------------+------------------------------           F(  2,    27) =   26.99
       Model |   6.9555881     2  3.47779405           Prob > F      =  0.0000
    Residual |  3.47907855    27  .128854761           R-squared     =  0.6666
-------------+------------------------------           Adj R-squared =  0.6419
       Total |  10.4346666    29  .359816091           Root MSE      =  .35896

------------------------------------------------------------------------------
         gpa |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
        _hat |   4.488587    2.40314     1.87   0.073    -.4422485    9.419423
      _hatsq |  -.5179375   .3561891    -1.45   0.157    -1.248777    .2129022
       _cons |  -5.757461   3.986623    -1.44   0.160    -13.93734    2.422414
------------------------------------------------------------------------------

ovtest

Ramsey RESET test using powers of the fitted values of gpa
       Ho:  model has no omitted variables
                  F(3, 22) =      1.71
                  Prob > F =      0.1932

Linear Statistical Models Course

Phil Ender, 17sep10, 13dec00