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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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