Fixed effects coefficients interpretation

[Ethan OLeary]

Dear all,

I have searched for a while and found what I believe to be the answer however I wanted to double check my interpretation.

I have a fixed effects regression which I have conducted using -xtreg, fe-. This has fitted a model, I've done the relevant tests but I am now trying to interpret my coefficient. Several documents and my own knowledge affirm that the output stata produces shows the coefficients to the variables as if they were computed using LSDV analysis.

Thus if my stata output of -xtreg y x, fe- shows a coefficient of 2, I should interpret this as a unit change in x increases y by 2 and NOT that a unit change in the deviation of x from the individual mean of x increases the deviation of y from the individual mean of y by 2?

Many thanks for your help.

Ethan

[Andrew Musau]

There are several ways to estimate a fixed effects model and these do not have a bearing on the interpretation of the model. Pick up any standard Econometrics textbook and look for a discussion of the Frisch-Waugh theorem that establishes the equivalence of dummy variable regression and the within transformation.

[Carlo Lazzaro]

Ethan:
as an aside to Andrew's helpful reply, in my opinion a good advice is to -predict- the fitted values after -regress- or -xtreg- and try to re-calculate by hand their values based on the resulting coefficients (personally, I've learnt and I'm still learning a lot from this appraoch):

. use "https://www.stata-press.com/data/r16/nlswork.dta"
(National Longitudinal Survey.  Young Women 14-26 years of age in 1968)

. reg ln_wage i.idcode age if idcode<=3, vce(cluster idcode)

Linear regression                               Number of obs     =         39
                                                F(0, 2)           =          .
                                                Prob > F          =          .
                                                R-squared         =     0.5552
                                                Root MSE          =     .25646

                                 (Std. Err. adjusted for 3 clusters in idcode)
------------------------------------------------------------------------------
             |               Robust
     ln_wage |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
      idcode |
          2  |  -.3831397   .0218939   -17.50   0.003    -.4773415   -.2889378
          3  |  -.6053814   .1065503    -5.68   0.030     -1.06383   -.1469324
             |
         age |   .0307608   .0218939     1.40   0.295     -.063441    .1249627
       _cons |   1.230398   .5765394     2.13   0.166     -1.25025    3.711047
------------------------------------------------------------------------------

. predict fitted_OLS, xb
(24 missing values generated)

. list ln_wage age fitted_OLS if idcode==1 & _n==1

       +---------------------------+
       |  ln_wage   age   fitted~S |
       |---------------------------|
    1. | 1.451214    18   1.784093 |
       +---------------------------+

. di  1.230398 + 18*.0307608
1.7840924

. xtreg ln_wage age if idcode<=3, fe vce(cluster idcode)

Fixed-effects (within) regression               Number of obs     =         39
Group variable: idcode                          Number of groups  =          3

R-sq:                                           Obs per group:
     within  = 0.3795                                         min =         12
     between = 0.6496                                         avg =       13.0
     overall = 0.1214                                         max =         15

                                                F(1,2)            =       2.09
corr(u_i, Xb)  = -0.2960                        Prob > F          =     0.2854

                                 (Std. Err. adjusted for 3 clusters in idcode)
------------------------------------------------------------------------------
             |               Robust
     ln_wage |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         age |   .0307608    .021294     1.44   0.285    -.0608597    .1223813
       _cons |   .8796702   .6071507     1.45   0.284    -1.732688    3.492029
-------------+----------------------------------------------------------------
     sigma_u |  .30623358
     sigma_e |  .25645513
         rho |  .58777837   (fraction of variance due to u_i)
------------------------------------------------------------------------------

. predict fitted_fe, xb


. list ln_wage age fitted_fe if idcode==1 & _n==1

       +---------------------------+
       |  ln_wage   age   fitted~e |
       |---------------------------|
    1. | 1.451214    18   1.433365 |
       +---------------------------+

. . di 18*.0307608+.8796702
1.4333646

.
. predict fitted_u, u

. list ln_wage age fitted_fe fitted_u if idcode==1 & _n==1

       +--------------------------------------+
       |  ln_wage   age   fitted~e   fitted_u |
       |--------------------------------------|
    1. | 1.451214    18   1.433365   .3507281 |
       +--------------------------------------+
  
    . di 18*.0307608+.8796702+.3507281
1.7840927

That said, you can interpret your -xtreg- coefficients in the same way you would do for their OLS counterparts (provided that you include -i.panelid- in the right-hand side of your -regress- equation), being aware that, other things being equal, -xtreg- and -regress- predicted values differ for the value of -u-.