Time-demeaning time-varying dummy variables

[Chizuru Hishiro]

Hi! I'm running a fixed effects regression (within group method) in which two of my explanatory variables are time-varying dummy variables. For example, employment status=1 if the individual is employed and 0 otherwise (the periods in my data are not consecutive years so there is variation across time for some individuals in terms of employment). I understand that using the within group transformation, we time-demean the explanatory variables. I understand this in the case of continuous variables where we get the average of the data for individual i across time then subtract it from the value for that individual i at time t. But I'm not sure if that works for time-varying dummy variables (and if that makes sense) as I don't get the intuition behind it. I can't get an answer anywhere as all of the resources/video tutorials I've consulted all use continuous time-varying explanatory variables as examples.

[Carlo Lazzaro]

Chizuru:
welcome to this forum.
Perhaps the following toy-example helps:

. use "https://www.stata-press.com/data/r18/nlswork.dta"
(National Longitudinal Survey of Young Women, 14-24 years old in 1968)

. xtreg ln_wage i.nev_mar i.year if idcode<=3, fe

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

R-squared:                                      Obs per group:
     Within  = 0.5763                                         min =         12
     Between = 0.3526                                         avg =       13.0
     Overall = 0.4751                                         max =         15

                                                F(15, 21)         =       1.90
corr(u_i, Xb) = -0.4328                         Prob > F          =     0.0857

------------------------------------------------------------------------------
     ln_wage | Coefficient  Std. err.      t    P>|t|     [95% conf. interval]
-------------+----------------------------------------------------------------
   1.nev_mar |  -.5062391   .4037468    -1.25   0.224    -1.345877    .3333984
             |
        year |
         69  |    .208967   .3868942     0.54   0.595    -.5956235    1.013557
         70  |  -.0423474   .3868942    -0.11   0.914    -.8469379     .762243
         71  |  -.3816407   .3225583    -1.18   0.250    -1.052437    .2891561
         72  |  -.2259469   .3225583    -0.70   0.491    -.8967436    .4448499
         73  |  -.0512957   .3225583    -0.16   0.875    -.7220924     .619501
         75  |   .0213776   .3225583     0.07   0.948    -.6494192    .6921743
         77  |   .0156442   .3225583     0.05   0.962    -.6551526    .6864409
         78  |   .2230704   .3225583     0.69   0.497    -.4477264    .8938671
         80  |   .2523643   .3225583     0.78   0.443    -.4184324    .9231611
         82  |   .1650993   .3396819     0.49   0.632    -.5413079    .8715065
         83  |   .2721994   .3225583     0.84   0.408    -.3985974    .9429961
         85  |   .3510093   .3225583     1.09   0.289    -.3197874    1.021806
         87  |   .2757865   .3225583     0.85   0.402    -.3950102    .9465833
         88  |   .2836143   .3225583     0.88   0.389    -.3871824    .9544111
             |
       _cons |   1.873077   .3274309     5.72   0.000     1.192147    2.554006
-------------+----------------------------------------------------------------
     sigma_u |  .23778325
     sigma_e |  .27357548
         rho |  .43034687   (fraction of variance due to u_i)
------------------------------------------------------------------------------
F test that all u_i=0: F(2, 21) = 6.37                       Prob > F = 0.0069


. bysort idcode: egen wanted=mean( nev_mar)

. bysort idcode: gen dem_nev_mar= nev_mar-wanted


. regress ln_wage i.idcode dem_nev_mar i.year if idcode<=3

      Source |       SS           df       MS      Number of obs   =        39
-------------+----------------------------------   F(17, 21)       =      2.83
       Model |  3.60402454        17  .212001443   Prob > F        =    0.0127
    Residual |  1.57171442        21  .074843544   R-squared       =    0.6963
-------------+----------------------------------   Adj R-squared   =    0.4505
       Total |  5.17573896        38  .136203657   Root MSE        =    .27358

------------------------------------------------------------------------------
     ln_wage | Coefficient  Std. err.      t    P>|t|     [95% conf. interval]
-------------+----------------------------------------------------------------
      idcode |
          2  |  -.3696661   .1152181    -3.21   0.004    -.6092751    -.130057
          3  |  -.4640525   .1106151    -4.20   0.000    -.6940893   -.2340158
             |
 dem_nev_mar |  -.5062391   .4037468    -1.25   0.224    -1.345877    .3333984
             |
        year |
         69  |    .208967   .3868942     0.54   0.595    -.5956235    1.013557
         70  |  -.0423474   .3868942    -0.11   0.914    -.8469379     .762243
         71  |  -.3816407   .3225583    -1.18   0.250    -1.052437    .2891561
         72  |  -.2259469   .3225583    -0.70   0.491    -.8967436    .4448499
         73  |  -.0512957   .3225583    -0.16   0.875    -.7220924     .619501
         75  |   .0213776   .3225583     0.07   0.948    -.6494192    .6921743
         77  |   .0156442   .3225583     0.05   0.962    -.6551526    .6864409
         78  |   .2230704   .3225583     0.69   0.497    -.4477264    .8938671
         80  |   .2523643   .3225583     0.78   0.443    -.4184324    .9231611
         82  |   .1650993   .3396819     0.49   0.632    -.5413079    .8715065
         83  |   .2721994   .3225583     0.84   0.408    -.3985974    .9429961
         85  |   .3510093   .3225583     1.09   0.289    -.3197874    1.021806
         87  |   .2757865   .3225583     0.85   0.402    -.3950102    .9465833
         88  |   .2836143   .3225583     0.88   0.389    -.3871824    .9544111
             |
       _cons |   1.957614   .2950919     6.63   0.000     1.343937    2.571291
------------------------------------------------------------------------------

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