Difference between fixed effects and individual dummies

Dana Baade

Join Date: Jul 2022

Posts: 23
#1

Difference between fixed effects and individual dummies

22 Jul 2022, 03:34

Hi everyone,

I am running a regression with fixed effects and using xtreg. I was wondering what the difference is between

Code:

xtreg y x1 x2 i.time, fe cluster (id) predict u if e(sample), u list u

And running

Code:

reg y x1 x2 i.time i.id, cluster(id)

And looking at the individual dummy values for id. I am trying to have a look at the individual fixed effects and see whether the influence on y differs across id’s.
Tags: fixed effects

Carlo Lazzaro

Join Date: Apr 2014
Posts: 17852

22 Jul 2022, 05:29

Dana:
the relationship you're looking for is easy to get, but, since your second code will take forever, you'd be better off by reducing you sample to test it:

Code:

use "https://www.stata-press.com/data/r17/nlswork.dta"
. xtreg ln_wage wks_ue ttl_exp i.year if idcode<=3, fe vce(cluster idcode )

Fixed-effects (within) regression               Number of obs     =         32
Group variable: idcode                          Number of groups  =          3

R-squared:                                      Obs per group:
     Within  = 0.8596                                         min =          9
     Between = 0.5755                                         avg =       10.7
     Overall = 0.6766                                         max =         12

                                                F(2,2)            =          .
corr(u_i, Xb) = -0.6640                         Prob > F          =          .

                                 (Std. err. adjusted for 3 clusters in idcode)
------------------------------------------------------------------------------
             |               Robust
     ln_wage | Coefficient  std. err.      t    P>|t|     [95% conf. interval]
-------------+----------------------------------------------------------------
      wks_ue |  -.0059005   .0010199    -5.79   0.029    -.0102886   -.0015123
     ttl_exp |  -.1802504   .0103821   -17.36   0.003     -.224921   -.1355797
             |
        year |
         69  |   .3129576   .0059897    52.25   0.000      .287186    .3387291
         70  |   .3131149   .0403495     7.76   0.016     .1395049    .4867249
         71  |   .3734201   .2173143     1.72   0.228    -.5616077    1.308448
         72  |   .6047066   .2053181     2.95   0.099    -.2787059    1.488119
         73  |   .9161217   .0569372    16.09   0.004     .6711409    1.161103
         75  |   1.071495    .092861    11.54   0.007     .6719459    1.471043
         77  |   1.200775   .0786513    15.27   0.004     .8623657    1.539184
         78  |   1.549068   .2457137     6.30   0.024     .4918478    2.606289
         80  |   2.210631   .0980471    22.55   0.002     1.788768    2.632493
         83  |   2.314151   .0527242    43.89   0.001     2.087297    2.541005
         85  |   2.587935   .1320649    19.60   0.003     2.019705    3.156164
         87  |   2.877827   .1719838    16.73   0.004     2.137841    3.617814
         88  |   3.144313   .1948402    16.14   0.004     2.305984    3.982643
             |
       _cons |   1.371913   .0186366    73.61   0.000     1.291726    1.452101
-------------+----------------------------------------------------------------
     sigma_u |  .29256637
     sigma_e |  .18669493
         rho |  .71062703   (fraction of variance due to u_i)
------------------------------------------------------------------------------

. 
. predict fitted_fe, xb

. predict u_i, u


. reg ln_wage wks_ue ttl_exp i.idcode i.year if idcode<=3, vce(cluster idcode )

Linear regression                               Number of obs     =         32
                                                F(1, 2)           =          .
                                                Prob > F          =          .
                                                R-squared         =     0.8952
                                                Root MSE          =     .18669

                                 (Std. err. adjusted for 3 clusters in idcode)
------------------------------------------------------------------------------
             |               Robust
     ln_wage | Coefficient  std. err.      t    P>|t|     [95% conf. interval]
-------------+----------------------------------------------------------------
      wks_ue |  -.0059005   .0010903    -5.41   0.032    -.0105916   -.0012094
     ttl_exp |  -.1802504    .011099   -16.24   0.004    -.2280053   -.1324954
             |
      idcode |
          2  |  -.2142075   .0199583   -10.73   0.009    -.3000812   -.1283337
          3  |   .3644592   .0565652     6.44   0.023     .1210789    .6078395
             |
        year |
         69  |   .3129576   .0064032    48.87   0.000     .2854066    .3405085
         70  |   .3131149   .0431355     7.26   0.018      .127518    .4987118
         71  |   .3734201   .2323187     1.61   0.249    -.6261667    1.373007
         72  |   .6047066   .2194943     2.75   0.110    -.3397011    1.549114
         73  |   .9161217   .0608684    15.05   0.004     .6542262    1.178017
         75  |   1.071495   .0992726    10.79   0.008     .6443591     1.49863
         77  |   1.200775   .0840817    14.28   0.005     .8390003    1.562549
         78  |   1.549068    .262679     5.90   0.028      .418852    2.679285
         80  |   2.210631   .1048167    21.09   0.002     1.759641    2.661621
         83  |   2.314151   .0563646    41.06   0.001     2.071634    2.556669
         85  |   2.587935   .1411833    18.33   0.003     1.980472    3.195398
         87  |   2.877827   .1838584    15.65   0.004     2.086749    3.668906
         88  |   3.144313   .2082929    15.10   0.004     2.248101    4.040526
             |
       _cons |   1.295487   .0464494    27.89   0.001     1.095632    1.495343
------------------------------------------------------------------------------

. predict fitted_OLS, xb


. list idcode fitted_fe u_i fitted_OLS in 1

     +------------------------------------------+
     | idcode     fitted_fe         u_i   fitted~S |
     |------------------------------------------|
  1. |      1   1.477956   -.0764263    1.40153 |
     +------------------------------------------+

. di 1.477956   -.0764263
1.4015297

.

Kind regards,
Carlo
(Stata 19.0)

Comment

Dana Baade

Join Date: Jul 2022
Posts: 23

22 Jul 2022, 06:12

Thank you Carlo,

I meant it a bit differently however. I was wondering is there was a way to see how the individual fixed effects influence the outcome y.

I though that the dummies and predict then should then give more or less the same result. Howver if I code:

Code:

list idcode fitted_fe u_i fitted_OLS if idcode==2

       +------------------------------------------+
       | idcode   fitted~e         u_i   fitted~S |
       |------------------------------------------|
   13. |      2    1.50497   -.2906338   1.214336 |
   14. |      2   1.736703   -.2906338   1.446069 |
   15. |      2   1.818076   -.2906338   1.527442 |
   16. |      2   2.044777   -.2906338   1.754144 |
   17. |      2   1.993807   -.2906338   1.703173 |
       |------------------------------------------|
   18. |      2    2.16185   -.2906338   1.871217 |
   19. |      2          .           .          . |
   20. |      2          .           .          . |
   21. |      2    2.06811   -.2906338   1.777477 |
   22. |      2    2.10139   -.2906338   1.810756 |
       |------------------------------------------|
   23. |      2          .           .          . |
   24. |      2   2.060893   -.2906338   1.770259 |
       +------------------------------------------

u_i is -0.29 and the dummy gives -0.21.

Code:

 reg ln_wage wks_ue ttl_exp i.idcode i.year if idcode<=3, vce(cluster idcode )

Linear regression                               Number of obs     =         32
                                                F(1, 2)           =          .
                                                Prob > F          =          .
                                                R-squared         =     0.8952
                                                Root MSE          =     .18669

                                 (Std. Err. adjusted for 3 clusters in idcode)
------------------------------------------------------------------------------
             |               Robust
     ln_wage |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
      wks_ue |  -.0059005   .0010903    -5.41   0.032    -.0105916   -.0012094
     ttl_exp |  -.1802504    .011099   -16.24   0.004    -.2280053   -.1324954
             |
      idcode |
          2  |  -.2142075   .0199583   -10.73   0.009    -.3000812   -.1283337
          3  |   .3644592   .0565652     6.44   0.023     .1210789    .6078395
             |
        year |
         69  |   .3129576   .0064032    48.87   0.000     .2854066    .3405085
         70  |   .3131149   .0431355     7.26   0.018      .127518    .4987118
         71  |   .3734201   .2323187     1.61   0.249    -.6261667    1.373007
         72  |   .6047066   .2194943     2.75   0.110    -.3397011    1.549114
         73  |   .9161217   .0608684    15.05   0.004     .6542262    1.178017
         75  |   1.071495   .0992726    10.79   0.008     .6443591     1.49863
         77  |   1.200775   .0840817    14.28   0.005     .8390003    1.562549
         78  |   1.549068    .262679     5.90   0.028      .418852    2.679285
         80  |   2.210631   .1048167    21.09   0.002     1.759641    2.661621
         83  |   2.314151   .0563646    41.06   0.001     2.071634    2.556669
         85  |   2.587935   .1411833    18.33   0.003     1.980472    3.195398
         87  |   2.877827   .1838584    15.65   0.004     2.086749    3.668906
         88  |   3.144313   .2082929    15.10   0.004     2.248101    4.040526
             |
       _cons |   1.295487   .0464494    27.89   0.001     1.095632    1.495343
------------------------------------------------------------------------------

it seems like if I use

Code:

display -.0764263-.2142075
-.2906338

I get u_i. Would this be a correct way to approach this? I then ofcourse do not get a p value.

Comment

Carlo Lazzaro

Join Date: Apr 2014

Posts: 17852
#4

22 Jul 2022, 13:49

Dana:
your last calculation does not sound correct.
That said, the -fe- estimator assumes that the -u- term is correlated with the vector of the regressors, whereas OLS does not follow this approach.
In addition, OLS does not imply demeaning.

Kind regards,
Carlo
(Stata 19.0)
Comment
Dana Baade

Join Date: Jul 2022

Posts: 23
#5

23 Jul 2022, 09:10

Thank you Carlo.

since it is not correct, how come u_i is equal to the dummy+u_i of id(1)? Furthermore what would you suggest if I want to see for which panel the fixed effects have the largest impact on y?
Comment
Carlo Lazzaro

Join Date: Apr 2014

Posts: 17852
#6

23 Jul 2022, 11:32

Dana:
sorry, my mistake.
Your calculation was indeed correct.
Admittedly; i've never challenged myself with this issue, as the fixed effect is a panel-specific constant and, as such, has no within-panel variance

Kind regards,
Carlo
(Stata 19.0)
1 like
Comment
Dana Baade

Join Date: Jul 2022

Posts: 23
#7

23 Jul 2022, 12:37

Thank you Carlo. I agree and I will see
Comment

Andrew Musau

Join Date: Oct 2014
Posts: 10481

23 Jul 2022, 12:37

Originally posted by Dana Baade View Post

u_i is equal to the dummy+u_i of id(1)?

Correct. The reason is because id=1 is the base (omitted) category in the regression with dummy variables.

I want to see for which panel the fixed effects have the largest impact on y?

Unless you have a large \(T\) dimension, you should not compare the fixed effects as they are not estimated consistently. If and only if the \(T\) dimension is large, then you can use margins to predict the panel-specific intercepts and then use test for a comparison across any pair. The dummy coefficients from regress compare each panel against the base.

Code:

webuse grunfeld, clear
regress invest mvalue i.company
margins company, post
di _b[3.company]- _b[1.company]
test 1.company=3.company

Res.:

Code:

. regress invest mvalue i.company

      Source |       SS           df       MS      Number of obs   =       200
-------------+----------------------------------   F(10, 189)      =    106.36
       Model |   7947627.4        10   794762.74   Prob > F        =    0.0000
    Residual |  1412316.51       189  7472.57415   R-squared       =    0.8491
-------------+----------------------------------   Adj R-squared   =    0.8411
       Total |  9359943.92       199  47034.8941   Root MSE        =    86.444

------------------------------------------------------------------------------
      invest |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
      mvalue |   .1898776   .0179944    10.55   0.000     .1543819    .2253733
             |
     company |
          2  |   250.9496   50.53491     4.97   0.000     151.2647    350.6345
          3  |  -51.44415   50.99737    -1.01   0.314    -152.0413    49.15302
          4  |   169.3784   70.98565     2.39   0.018     29.35244    309.4043
          5  |   232.7314   78.71866     2.96   0.004      77.4514    388.0115
          6  |    190.568   75.54874     2.52   0.012     41.54088     339.595
          7  |   234.0336    80.0986     2.92   0.004     76.03153    392.0357
          8  |   130.3806   71.35614     1.83   0.069    -10.37613    271.1374
          9  |   193.4162   76.99706     2.51   0.013     41.53223    345.3003
         10  |   204.4981   81.43403     2.51   0.013     43.86171    365.1345
             |
       _cons |  -214.8799   80.34482    -2.67   0.008    -373.3677   -56.39209
------------------------------------------------------------------------------

.
. margins company, post

Predictive margins                              Number of obs     =        200
Model VCE    : OLS

Expression   : Linear prediction, predict()

------------------------------------------------------------------------------
             |            Delta-method
             |     Margin   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
     company |
          1  |  -9.492932   61.63044    -0.15   0.878    -131.0648     112.079
          2  |   241.4567   25.10364     9.62   0.000     191.9373     290.976
          3  |  -60.93708   24.75706    -2.46   0.015    -109.7727   -12.10142
          4  |   159.8854   20.55464     7.78   0.000     119.3395    200.4314
          5  |   223.2385   24.65136     9.06   0.000     174.6113    271.8657
          6  |    181.075   22.70359     7.98   0.000       136.29      225.86
          7  |   224.5407    25.5895     8.77   0.000      174.063    275.0184
          8  |   120.8877   20.69455     5.84   0.000     80.06575    161.7097
          9  |   183.9233   23.55442     7.81   0.000       137.46    230.3866
         10  |   195.0052   26.54116     7.35   0.000     142.6502    247.3601
------------------------------------------------------------------------------

.
. di _b[3.company]- _b[1.company]
-51.444149

.
. test 1.company=3.company

 ( 1)  1bn.company - 3.company = 0

       F(  1,   189) =    1.02
            Prob > F =    0.3144

.

Last edited by Andrew Musau; 23 Jul 2022, 13:03.

Comment

Dana Baade

Join Date: Jul 2022

Posts: 23
#9

23 Jul 2022, 12:41

Thank you Andrew
Comment

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