Hausman Test for FE and RE

[Kathy Kim]

Hi, I have a question, if my Hausman test is like below, which model should I use? Is it FE or RE?

. hausman fe re

                 ---- Coefficients ----
             |      (b)          (B)            (b-B)     sqrt(diag(V_b-V_B))
             |       fe           re         Difference       Std. err.
-------------+----------------------------------------------------------------
   div_yield |    .0171481     .0126373        .0045108         .051287
   firm_size |   -.1940015    -.3004789        .1064773        .3209853
------------------------------------------------------------------------------
                          b = Consistent under H0 and Ha; obtained from xtreg.
           B = Inconsistent under Ha, efficient under H0; obtained from xtreg.

Test of H0: Difference in coefficients not systematic

    chi2(2) = (b-B)'[(V_b-V_B)^(-1)](b-B)
            =   0.12
Prob > chi2 = 0.9432

I also did this command to rerun RE, then test for presence of u_i:

. quietly xtreg caar div_yield firm_size, re

. xttest0

Breusch and Pagan Lagrangian multiplier test for random effects

        caar[companyid,t] = Xb + u[companyid] + e[companyid,t]

        Estimated results:
                         |       Var     SD = sqrt(Var)
                ---------+-----------------------------
                    caar |   112.5604       10.60945
                       e |   107.0638       10.34717
                       u |   6.004459         2.4504

        Test: Var(u) = 0
                             chibar2(01) =     2.89
                          Prob > chibar2 =   0.0444

[Carlo Lazzaro]

Kathy:
you should go -xtreg,re- because:
1) -hausman- does not eject the null (its H0 is: -re- is the way to go;
2) -xttest0- outcome tells you that your dataset actually shows ecidence of a panel-wise effect.

As an aside, with two predictors only your model is at high risk of misspecification.

[Kathy Kim]

Hi Mr Carlo, thank you for responding.

Will use the RE model.


As an aside, with two predictors only your model is at high risk of misspecification - Does this mean that I have to add more variables?

[Carlo Lazzaro]

Kathy:
first off, please call me Carlo. Thanks.
Two precitors are very unlikely to give back any informative result (being it statistically significant or not).
That said, there's a pretty easy to implement procedure to check the correct specification form of your regressand (in general terms, this test tells you whether your regression is correctly specified or not; see alo -linktest-, that unfrtunately you cannot call after -xtreg-):

. use "https://www.stata-press.com/data/r17/nlswork.dta"
(National Longitudinal Survey of Young Women, 14-24 years old in 1968)

. xtreg ln_wage c.age##c.age

Random-effects GLS regression                   Number of obs     =     28,510
Group variable: idcode                          Number of groups  =      4,710

R-squared:                                      Obs per group:
     Within  = 0.1087                                         min =          1
     Between = 0.1015                                         avg =        6.1
     Overall = 0.0870                                         max =         15

                                                Wald chi2(2)      =    3388.51
corr(u_i, X) = 0 (assumed)                      Prob > chi2       =     0.0000

------------------------------------------------------------------------------
     ln_wage | Coefficient  Std. err.      z    P>|z|     [95% conf. interval]
-------------+----------------------------------------------------------------
         age |   .0590339   .0027172    21.73   0.000     .0537083    .0643596
             |
 c.age#c.age |  -.0006758   .0000451   -15.00   0.000    -.0007641   -.0005876
             |
       _cons |   .5479714   .0397476    13.79   0.000     .4700675    .6258752
-------------+----------------------------------------------------------------
     sigma_u |   .3654049
     sigma_e |  .30245467
         rho |  .59342665   (fraction of variance due to u_i)
------------------------------------------------------------------------------

. predict fitted, xb
(24 missing values generated)

. g sq_fitted=fitted^2
(24 missing values generated)

. xtreg ln_wage fitted sq_fitted

Random-effects GLS regression                   Number of obs     =     28,510
Group variable: idcode                          Number of groups  =      4,710

R-squared:                                      Obs per group:
     Within  = 0.1088                                         min =          1
     Between = 0.1045                                         avg =        6.1
     Overall = 0.0887                                         max =         15

                                                Wald chi2(2)      =    3407.81
corr(u_i, X) = 0 (assumed)                      Prob > chi2       =     0.0000

------------------------------------------------------------------------------
     ln_wage | Coefficient  Std. err.      z    P>|z|     [95% conf. interval]
-------------+----------------------------------------------------------------
      fitted |   2.805959   .4327827     6.48   0.000      1.95772    3.654197
   sq_fitted |  -.5516341   .1320951    -4.18   0.000    -.8105358   -.2927324
       _cons |  -1.468083   .3527217    -4.16   0.000    -2.159405   -.7767613
-------------+----------------------------------------------------------------
     sigma_u |  .36481589
     sigma_e |  .30242516
         rho |  .59269507   (fraction of variance due to u_i)
------------------------------------------------------------------------------

. test sq_fitted

 ( 1)  sq_fitted = 0

           chi2(  1) =   17.44
         Prob > chi2 =    0.0000

.

As the -test- outcome reaches statistical significance, the regression is (as expected) misspecified. More predictors and/or interactions are called for to give a fair and true view of the data generating process.

[Kathy Kim]

Thank you very much for the detailed explanation, Carlo. I really appreciate it.