Stata FAQ: What is seemingly unrelated regression and how can I perform it in Stata?



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Stata FAQ
What is seemingly unrelated regression and how can I perform it in Stata?

A model may contain a number of linear equations. It would be unrealistic to expect that the equation errors would be uncorrelated. A set of equations that has contemporaneous cross-equation error correlation is called a seemingly unrelated regression (SUR) system. At first look, the equations seem unrelated, but the equations are related through the correlation in the errors. The Stata command to do seemingly unrelated regression is sureg.

We will illustrate sureg using the file hsb2.dta which contains 200 observations from the High School and Beyond study. hsb2.dta can be accessed directly over the Internet from the ATS website with the use command below.

use http://www.ats.ucla.edu/stat/stata/notes/hsb2, clear

describe 

Contains data from http://www.ats.ucla.edu/stat/stata/notes/hsb2.dta
  obs:           200                          highschool and beyond (200
                                                cases)
 vars:            11                          8 May 1999 14:55
 size:         9,600 (98.9% of memory free)
-------------------------------------------------------------------------------
   1. id        float  %9.0g                  
   2. female    float  %9.0g       gl         
   3. race      float  %12.0g      rl         
   4. ses       float  %9.0g       sl         
   5. schtyp    float  %9.0g       scl        type of school
   6. prog      float  %9.0g       sel        type of program
   7. read      float  %9.0g                  reading score
   8. write     float  %9.0g                  writing score
   9. math      float  %9.0g                  math score
  10. science   float  %9.0g                  science score
  11. socst     float  %9.0g                  social studies score
-------------------------------------------------------------------------------
Sorted by:

We will use two equations, one for read and one for math and run the sureg command.

sureg (read female ses socst)(math female ses science)


Seemingly unrelated regression
------------------------------------------------------------------
Equation      Obs  Parms        RMSE    "R-sq"       Chi2        P
------------------------------------------------------------------
read          200      3    7.940579    0.3972   117.2329   0.0000
math          200      3    7.200735    0.4063   116.6664   0.0000
------------------------------------------------------------------------------
         |      Coef.   Std. Err.       z     P>|z|       [95% Conf. Interval]
---------+--------------------------------------------------------------------
read     |
  female |  -1.399691   1.139324     -1.229   0.219      -3.632726    .8333432
     ses |   1.495314   .8298821      1.802   0.072       -.131225    3.121853
   socst |   .5155857   .0548183      9.405   0.000       .4081438    .6230277
   _cons |   24.30038   3.343654      7.268   0.000       17.74694    30.85382
---------+--------------------------------------------------------------------
math     |
  female |   1.031629   1.031014      1.001   0.317      -.9891222     3.05238
     ses |   1.657043   .7340503      2.257   0.024       .2183303    3.095755
 science |   .5058777   .0529013      9.563   0.000        .402193    .6095624
   _cons |   21.41615   3.416379      6.269   0.000       14.72017    28.11213
------------------------------------------------------------------------------

Let's contrast the results of the sureg command with two separate regressions using the regress command.

regress read  female ses socst

  Source |       SS       df       MS                  Number of obs =     200
---------+------------------------------               F(  3,   196) =   43.56
   Model |  8368.53693     3  2789.51231               Prob > F      =  0.0000
Residual |  12550.8831   196  64.0351177               R-squared     =  0.4000
---------+------------------------------               Adj R-squared =  0.3909
   Total |    20919.42   199  105.122714               Root MSE      =  8.0022

------------------------------------------------------------------------------
    read |      Coef.   Std. Err.       t     P>|t|       [95% Conf. Interval]
---------+--------------------------------------------------------------------
  female |  -1.511128   1.151079     -1.313   0.191      -3.781219    .7589629
     ses |   1.218366   .8399004      1.451   0.148      -.4380365    2.874768
   socst |   .5699327   .0562967     10.124   0.000       .4589077    .6809578
   _cons |   22.19363   3.400423      6.527   0.000       15.48751    28.89974
------------------------------------------------------------------------------

regress math  female ses science


  Source |       SS       df       MS                  Number of obs =     200
---------+------------------------------               F(  3,   196) =   45.62
   Model |  7181.43086     3  2393.81029               Prob > F      =  0.0000
Residual |  10284.3641   196  52.4712456               R-squared     =  0.4112
---------+------------------------------               Adj R-squared =  0.4022
   Total |   17465.795   199  87.7678141               Root MSE      =  7.2437

------------------------------------------------------------------------------
    math |      Coef.   Std. Err.       t     P>|t|       [95% Conf. Interval]
---------+--------------------------------------------------------------------
  female |   1.160903   1.041641      1.114   0.266      -.8933606    3.215167
     ses |   1.399639   .7423902      1.885   0.061      -.0644595    2.863737
 science |   .5753302    .054328     10.590   0.000       .4681876    .6824727
   _cons |   18.14428   3.481754      5.211   0.000       11.27777    25.01079
------------------------------------------------------------------------------

Note that the regression coefficients, standard errors, R²'s, etc. are different in sureg from those in the standard regressions. This is due to correlated errors in the two equations.

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Stata FAQ What is seemingly unrelated regression and how can I perform it in Stata?

Stata FAQ
What is seemingly unrelated regression and how can I perform it in Stata?