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st: Interpretation of margins in the presence of fixed effects


From   Dana Shills <[email protected]>
To   "[email protected]" <[email protected]>
Subject   st: Interpretation of margins in the presence of fixed effects
Date   Thu, 5 Sep 2013 22:02:27 -0400

I have read the manual on the margins command in detail and it is still not clear to me what role do fixed effects (or say dummy variables) play in the computation of predictive margins.
 
Suppose I want to look at the relation between firm size and age (specifically 9 age dummies) with and w/o industry dummies. The summary stats of these three variables are below

    Variable |       Obs        Mean    Std. Dev.       Min        Max
-------------+--------------------------------------------------------
        size |        97    123.0103    166.3225          2        806
      agedum |       100        3.78    2.213731          1          9
        inum |       100        13.6    8.952174          1         31
 
Case I: WITHOUT INDUSTRY DUMMIES

. reg size i.agedum
 
      Source |       SS       df       MS              Number of obs =      97
-------------+------------------------------           F(  8,    88) =    4.04
       Model |  713993.178     8  89249.1473           Prob> F      =  0.0004
    Residual |  1941671.81    88  22064.4524           R-squared     =  0.2689
-------------+------------------------------           Adj R-squared =  0.2024
       Total |  2655664.99    96   27663.177           Root MSE      =  148.54
 
------------------------------------------------------------------------------
        size |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
      agedum |
          2  |      117.4   52.62008     2.23   0.028     12.82866    221.9713
          3  |   30.73333    51.9304     0.59   0.555    -72.46742    133.9341
          4  |   28.45263   51.30546     0.55   0.581    -73.50619    130.4115
          5  |   108.5429   67.99285     1.60   0.114    -26.57865    243.6644
          6  |   103.5429   67.99285     1.52   0.131    -31.57865    238.6644
          7  |      -14.8   76.70628    -0.19   0.847    -167.2376    137.6376
          8  |      323.4   76.70628     4.22   0.000     170.9624    475.8376
          9  |     275.15   83.58873     3.29   0.001      109.035     441.265
             |
       _cons |       48.6   38.35314     1.27   0.208    -27.61881    124.8188
------------------------------------------------------------------------------
 
. margins agedum
 
Adjusted predictions                              Number of obs   =         97
Model VCE    : OLS
 
Expression   : Linear prediction, predict()
 
------------------------------------------------------------------------------
             |            Delta-method
             |     Margin   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
      agedum |
          1  |       48.6   38.35314     1.27   0.205    -26.57078    123.7708
          2  |        166    36.0265     4.61   0.000     95.38935    236.6106
          3  |   79.33333   35.01147     2.27   0.023     10.71212    147.9546
          4  |   77.05263   34.07766     2.26   0.024     10.26164    143.8436
          5  |   157.1429   56.14325     2.80   0.005     47.10411    267.1816
          6  |   152.1429   56.14325     2.71   0.007     42.10411    262.1816
          7  |       33.8   66.42959     0.51   0.611     -96.3996    163.9996
          8  |        372   66.42959     5.60   0.000     241.8004    502.1996
          9  |     323.75   74.27054     4.36   0.000     178.1824    469.3176
 
I understand that the margins command is giving the predicted employment of each age bin. So the average employment of firms in the second age group (5-10 years) is 166 employees if all firms in the dataset were treated to be between 5-10 years old. And it is easy to see that the predicted margins are just the regression coefficients adjusted for the constant.
 
Case II: WITH INDUSTRY DUMMIES
 
. reg size i.agedum i.inum
 
      Source |       SS       df       MS              Number of obs =      97
-------------+------------------------------           F( 38,    58) =    1.67
       Model |  1385935.82    38  36471.9953           Prob> F      =  0.0390
    Residual |  1269729.17    58  21891.8822           R-squared     =  0.5219
-------------+------------------------------           Adj R-squared =  0.2086
       Total |  2655664.99    96   27663.177           Root MSE      =  147.96
 
------------------------------------------------------------------------------
        size |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
      agedum |
          2  |   185.1145   63.99243     2.89   0.005     57.01978    313.2093
          3  |   105.8429   73.84605     1.43   0.157    -41.97599    253.6619
          4  |    74.4776   66.75151     1.12   0.269    -59.14007    208.0953
          5  |   171.1386   81.31018     2.10   0.040     8.378543    333.8986
          6  |   219.9226   87.65383     2.51   0.015     44.46437    395.3808
          7  |   23.76708   83.98693     0.28   0.778     -144.351    191.8852
          8  |   477.1547   96.73068     4.93   0.000     283.5272    670.7822
          9  |    595.833   129.2686     4.61   0.000     337.0737    854.5923
             |
        inum |
          2  |  -234.5949   168.2358    -1.39   0.169    -571.3555    102.1657
          3  |  -17.52719   160.0994    -0.11   0.913    -338.0009    302.9465
          4  |  -55.42242   169.1826    -0.33   0.744    -394.0781    283.2333
          5  |  -271.0341   187.8332    -1.44   0.154    -647.0231     104.955
          6  |  -118.9656   166.5016    -0.71   0.478    -452.2547    214.3236
          7  |  -95.84294   221.8941    -0.43   0.667    -540.0123    348.3264
          8  |  -206.4776    219.635    -0.94   0.351    -646.1248    233.1696
          9  |  -234.8429    195.681    -1.20   0.235    -626.5411    156.8552
         10  |   -83.4776    219.635    -0.38   0.705    -523.1248    356.1696
         11  |  -451.0993    188.756    -2.39   0.020    -828.9356   -73.26306
         12  |  -271.1145   218.8122    -1.24   0.220    -709.1148    166.8857
         13  |  -68.56328   168.2744    -0.41   0.685    -405.4011    268.2746
         14  |  -55.88331   171.9721    -0.32   0.746    -400.1229    288.3562
         15  |  -265.9787    191.091    -1.39   0.169     -648.489    116.5315
         16  |  -125.1974   175.2724    -0.71   0.478    -476.0432    225.6484
         17  |  -147.8429   221.8941    -0.67   0.508    -592.0123    296.3264
         18  |  -148.5633   168.2744    -0.88   0.381    -485.4011    188.2746
         19  |  -181.3566   181.0713    -1.00   0.321    -543.8101     181.097
         20  |  -297.1145   218.8122    -1.36   0.180    -735.1148    140.8857
         21  |  -13.84294   221.8941    -0.06   0.950    -458.0123    430.3264
         22  |  -193.3253   167.3785    -1.16   0.253    -528.3697    141.7191
         23  |  -160.5306   193.9322    -0.83   0.411    -548.7281    227.6669
         24  |  -145.7978   165.7959    -0.88   0.383    -477.6744    186.0788
         25  |     -109.5   181.2121    -0.60   0.548    -472.2354    253.2354
         26  |  -290.1386   224.4886    -1.29   0.201    -739.5012    159.2241
         27  |  -183.1603   191.4903    -0.96   0.343    -566.4698    200.1492
         28  |   -77.4776    219.635    -0.35   0.726    -517.1248    362.1696
         29  |  -120.0795   177.5111    -0.68   0.501    -475.4067    235.2476
         30  |  -346.9226   226.8633    -1.53   0.132    -801.0388    107.1937
         31  |  -204.4776    219.635    -0.93   0.356    -644.1248    235.1696
             |
       _cons |        134   147.9591     0.91   0.369    -162.1722    430.1722
------------------------------------------------------------------------------
 
. margins agedum
 
Predictive margins                                Number of obs   =         97
Model VCE    : OLS
 
Expression   : Linear prediction, predict()
 
------------------------------------------------------------------------------
             |            Delta-method
             |     Margin   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
      agedum |
          1  |  -22.27385   49.35852    -0.45   0.652    -119.0148    74.46706
          2  |   162.8407   40.73507     4.00   0.000     83.00143      242.68
          3  |   83.56909   47.28917     1.77   0.077    -9.115992    176.2542
          4  |   52.20375   41.96549     1.24   0.214     -30.0471    134.4546
          5  |   148.8647   66.87327     2.23   0.026      17.7955    279.9339
          6  |   197.6487   71.66732     2.76   0.006     57.18337    338.1141
          7  |   1.493234   70.09108     0.02   0.983    -135.8828    138.8692
          8  |   454.8808   75.76749     6.00   0.000     306.3793    603.3824
          9  |   573.5592   111.6518     5.14   0.000     354.7258    792.3926
 
When we include the age dummies, it is not clear to me how we arrive at the number -22.27385 for the first age bin. Also how can size be negative?? What is the exact interpretation of this number? 

(Btw I just created a random sample of 100 observations from a larger dataset for purposes of illustration)
 
Thank you for your help.
 
Dana 		 	   		  
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