Problem Regarding Ommited Variable in Fixed Effects Model

[Aulia Irfan Mufti]

Hello everyone, I am currently in my thesis about Sustainability Reporting Quality. Basically i want to know that if specific characteristics of the board are really have a significant impact on SRQ. In my research, board charateristics represented by Board Age, BODMeeting, BOCMeeting, Board Ethnicity, Independent Comissionaire, ExGovoff, CurGovOff, Board Size, Board Female Director, and Firm Age. This research is basically a panel data research with time period is 2014-2018. So after coding the data, i try to run the regression, OLS, RE, and Fixed Effects. After I run Fixed Effect, I found that Board Ethnicity is ommited because of collinearity. My guess is that happened because Board Ethnicity showed that Board Ethnicity is invariant time variable in my observation because the coding show that the company here have heterogenistic ethnic board (arround 98%). My question is, should I drop this variable before doing the hausman test or just let them be ? Thank you for your attention.

[Carlo Lazzaro]

Aulia:
welcome to this forum.
As you surmise, the -fe- estimator wipes out all the time-invariant predictors.
That said, there's no scope in omitting the time-invariant regressor, because -hausman- runs on the coefficients estimated by both -fe- and -re- specifications, as you can see from the following toy-example:

. use "https://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 age, fe
note: 2.race omitted because of collinearity
note: 3.race omitted because of collinearity

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]
-------------+----------------------------------------------------------------
        race |
      black  |          0  (omitted)
      other  |          0  (omitted)
             |
         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

. estimates store fe

. xtreg ln_wage i.race age, re

Random-effects GLS 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 = 0.1032                                         avg =        6.1
     overall = 0.0945                                         max =         15

                                                Wald chi2(3)      =    3242.34
corr(u_i, X)   = 0 (assumed)                    Prob > chi2       =     0.0000

------------------------------------------------------------------------------
     ln_wage |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
        race |
      black  |  -.1209428   .0129079    -9.37   0.000    -.1462418   -.0956439
      other  |   .0981941   .0538424     1.82   0.068    -.0073351    .2037233
             |
         age |    .018534    .000331    55.99   0.000     .0178852    .0191828
       _cons |    1.15423   .0118069    97.76   0.000     1.131089    1.177371
-------------+----------------------------------------------------------------
     sigma_u |  .36581626
     sigma_e |  .30349389
         rho |  .59231394   (fraction of variance due to u_i)
------------------------------------------------------------------------------

. estimates store re

. hausman fe re

                 ---- Coefficients ----
             |      (b)          (B)            (b-B)     sqrt(diag(V_b-V_B))
             |       fe           re         Difference          S.E.
-------------+----------------------------------------------------------------
         age |    .0181349      .018534       -.0003991        .0001064
------------------------------------------------------------------------------
                           b = consistent under Ho and Ha; obtained from xtreg
            B = inconsistent under Ha, efficient under Ho; obtained from xtreg

    Test:  Ho:  difference in coefficients not systematic

                  chi2(1) = (b-B)'[(V_b-V_B)^(-1)](b-B)
                          =       14.08
                Prob>chi2 =      0.0002

.

[Aulia Irfan Mufti]

Thank you very much for your explanation, Mr Carlo. So, from your example we know that the hausmann test result is we choose Fixed Effect Models as our mode. My question is if we're going to use FEM as our models, how do we interprete the results for the omitted variables?. I am sorry if this question sounds silly, it is because I don't have any experience in research before. Once again, thanks for your input. 🙏

[Aulia Irfan Mufti]

Aulia:
welcome to this forum.
As you surmise, the -fe- estimator wipes out all the time-invariant predictors.
That said, there's no scope in omitting the time-invariant regressor, because -hausman- runs on the coefficients estimated by both -fe- and -re- specifications, as you can see from the following toy-example:

. use "https://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 age, fe
note: 2.race omitted because of collinearity
note: 3.race omitted because of collinearity

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]
-------------+----------------------------------------------------------------
race |
black | 0 (omitted)
other | 0 (omitted)
|
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

. estimates store fe

. xtreg ln_wage i.race age, re

Random-effects GLS 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 = 0.1032 avg = 6.1
overall = 0.0945 max = 15

Wald chi2(3) = 3242.34
corr(u_i, X) = 0 (assumed) Prob > chi2 = 0.0000

------------------------------------------------------------------------------
ln_wage | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
race |
black | -.1209428 .0129079 -9.37 0.000 -.1462418 -.0956439
other | .0981941 .0538424 1.82 0.068 -.0073351 .2037233
|
age | .018534 .000331 55.99 0.000 .0178852 .0191828
_cons | 1.15423 .0118069 97.76 0.000 1.131089 1.177371
-------------+----------------------------------------------------------------
sigma_u | .36581626
sigma_e | .30349389
rho | .59231394 (fraction of variance due to u_i)
------------------------------------------------------------------------------

. estimates store re

. hausman fe re

---- Coefficients ----
| (b) (B) (b-B) sqrt(diag(V_b-V_B))
| fe re Difference S.E.
-------------+----------------------------------------------------------------
age | .0181349 .018534 -.0003991 .0001064
------------------------------------------------------------------------------
b = consistent under Ho and Ha; obtained from xtreg
B = inconsistent under Ha, efficient under Ho; obtained from xtreg

Test: Ho: difference in coefficients not systematic

chi2(1) = (b-B)'[(V_b-V_B)^(-1)](b-B)
= 14.08
Prob>chi2 = 0.0002

.

Thank you very much for your explanation, Mr Carlo. So, from your example we know that the hausmann test result is we choose Fixed Effect Models as our mode. My question is if we're going to use FEM as our models, how do we interprete the results for the omitted variables?. I am sorry if this question sounds silly, it is because I don't have any experience in research before. Once again, thanks for your input. 🙏

[Carlo Lazzaro]

Aulia:
there are no silly questions, but issues that should be addressed given of the original poster's current knowledge of the topic she/he is interested in.
As we know that -fe- omits time-invariant predictors altogether, there's no way to obtain their coefficients: hence, there's nothing to waste your time about.
If you're intrerested in delving into this methodologically demanding area, reading the last available version of Jeff Wooldridge 's handbook on Econometric analysis of cross section and panel data is surely a must; Stata users dealing with microeconometrics also like the valuable https://www.stata.com/bookstore/micr...metrics-stata/.
As an aside, please call me Carlo, jus like all on (and many more) this list do. Thanks.

[Chris Boudreaux]

If you are interested in obtaining estimates for time-invariant variables within a fixed effects model, you might consider the correlated random effects model (or hybrid model). Schunck (2013) has an excellent article on this topic with Stata examples.

Schunck, R. (2013). Within and between estimates in random-effects models: Advantages and drawbacks of correlated random effects and hybrid models. The Stata Journal, 13(1), 65-76.

[Aulia Irfan Mufti]

Aulia:
there are no silly questions, but issues that should be addressed given of the original poster's current knowledge of the topic she/he is interested in.
As we know that -fe- omits time-invariant predictors altogether, there's no way to obtain their coefficients: hence, there's nothing to waste your time about.
If you're intrerested in delving into this methodologically demanding area, reading the last available version of Jeff Wooldridge 's handbook on Econometric analysis of cross section and panel data is surely a must; Stata users dealing with microeconometrics also like the valuable https://www.stata.com/bookstore/micr...metrics-stata/.
As an aside, please call me Carlo, jus like all on (and many more) this list do. Thanks.

thanks again, Carlo. Now I know what I have to do about this.

[Aulia Irfan Mufti]

If you are interested in obtaining estimates for time-invariant variables within a fixed effects model, you might consider the correlated random effects model (or hybrid model). Schunck (2013) has an excellent article on this topic with Stata examples.

Schunck, R. (2013). Within and between estimates in random-effects models: Advantages and drawbacks of correlated random effects and hybrid models. The Stata Journal, 13(1), 65-76.

this is quite helpful. I do really appreciate it. Thank you, Mr Chris. 🙏