Regarding time-fixed effect in panel regression

[Adnan Habib]

I have a basic econometrics concept query. I have a model where the dependent variable is both time and firm varying, but the indepdent variable is only time varying. In this case if I use time-fixed effect will it acccurately capture the effect of my time-varying independent variable?

[Yao Zhao]

Let's say your dependent variable is Yit. i means firm and t means time.
Your independent variable is only time varying, thus it's Xt.
By using time-fixed effect, all variables that are time varying would be absorbed by time-fixed effect.

Thus, you cannot use time-fixed effect in this case. The model is wrong.

But you can use cross-sectional fixed effect, in your case, it's firm fixed effects.

[Carlo Lazzaro]

I'm probably missing out on something here, but if the independent variable is time-varying, the -fe- estimator will give back its coefficient:

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

. xtreg ln_wage age, fe

Fixed-effects (within) regression               Number of obs     =     28,510
Group variable: idcode                          Number of groups  =      4,710

R-sq:                                           Obs per group:
     within  = 0.1026                                         min =          1
     between = 0.0877                                         avg =        6.1
     overall = 0.0774                                         max =         15

                                                F(1,23799)        =    2720.20
corr(u_i, Xb)  = 0.0314                         Prob > F          =     0.0000

------------------------------------------------------------------------------
     ln_wage |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         age |   .0181349   .0003477    52.16   0.000     .0174534    .0188164
       _cons |   1.148214   .0102579   111.93   0.000     1.128107     1.16832
-------------+----------------------------------------------------------------
     sigma_u |  .40635023
     sigma_e |  .30349389
         rho |  .64192015   (fraction of variance due to u_i)
------------------------------------------------------------------------------
F test that all u_i=0: F(4709, 23799) = 8.81                 Prob > F = 0.0000

.

Conversely, if the predictor is time-invariant, -fe- cannot estimate it, as subtracting from a costant its (panel) mean, returns zero:

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

. xtreg ln_wage i.race, fe
note: 2.race omitted because of collinearity
note: 3.race omitted because of collinearity

Fixed-effects (within) regression               Number of obs     =     28,534
Group variable: idcode                          Number of groups  =      4,711

R-sq:                                           Obs per group:
     within  = 0.0000                                         min =          1
     between = 0.0050                                         avg =        6.1
     overall =      .                                         max =         15

                                                F(0,23823)        =       0.00
corr(u_i, Xb)  =      .                         Prob > F          =          .

------------------------------------------------------------------------------
     ln_wage |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
        race |
      black  |          0  (omitted)
      other  |          0  (omitted)
             |
       _cons |   1.674907   .0018961   883.35   0.000     1.671191    1.678624
-------------+----------------------------------------------------------------
     sigma_u |  .42456905
     sigma_e |  .32028665
         rho |  .63731204   (fraction of variance due to u_i)
------------------------------------------------------------------------------
F test that all u_i=0: F(4710, 23823) = 8.44                 Prob > F = 0.0000

.

[Adnan Habib]

Thanks to both Yao and Carlo. Yao's explanation actulally makes sense.
Carlo, to clear your confusion. The independent variable is time-varying and yes fixed effect does chunk out a coefficient. I am just concerned whether that coefficient truly reflects the causal relation between the independent and dependent variable.

[Carlo Lazzaro]

Adnan:
thanks for clarifying.
However, your question is more substantive than statistical: probably the answer you're looking for can be provided by the literature of your research field.