Use of time dummies in fixed effects estimation

[Emil Holm]

Dear all,

I am writing my Master's thesis, where I am doing an FE estimation on the probability of default of a panel data set from 2008-2018. I have been made aware that it could be a good idea to include time dummies to control for any aggregate time effect (i.e. changes in the economic environment).

The variable I am interested in is the first one "WOMEN", as this is the fraction of female directors in the firms.

When I run my regression without time dummies, I get significant results, but when I include them, the results turn insignificant. (See code)


First my estimation with time dummies

. xtreg $ylist $xlist i.Time, fe vce(robust)

Fixed-effects (within) regression               Number of obs     =     76,710
Group variable: ID                              Number of groups  =     12,895

R-sq:                                           Obs per group:
     within  = 0.2433                                         min =          1
     between = 0.2360                                         avg =        5.9
     overall = 0.2757                                         max =         46

                                                F(36,12894)       =      42.19
corr(u_i, Xb)  = 0.0226                         Prob > F          =     0.0000

                                    (Std. Err. adjusted for 12,895 clusters in ID)
----------------------------------------------------------------------------------
                 |               Robust
              PD |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-----------------+----------------------------------------------------------------
           Women |  -.0011513    .001198    -0.96   0.337    -.0034996     .001197
             ROA |   -.102301   .0060319   -16.96   0.000    -.1141244   -.0904776
            MTBt |   .0001097   .0000167     6.57   0.000      .000077    .0001424
          LIQUID |  -.0002312   .0000897    -2.58   0.010    -.0004069   -.0000554
            SOLV |    .007016   .0006786    10.34   0.000     .0056859    .0083461
           RISK1 |   .0176122    .000926    19.02   0.000     .0157971    .0194274
          MktCap |  -5.00e-09   6.51e-10    -7.69   0.000    -6.28e-09   -3.73e-09
            Inde |   .0022668    .000858     2.64   0.008      .000585    .0039486
     OtherBoards |   .0001514   .0001023     1.48   0.139    -.0000491    .0003519
           Bsize |  -.0004402   .0000529    -8.33   0.000    -.0005439   -.0003366
            dGov |   .0011423   .0004785     2.39   0.017     .0002044    .0020802
          dAudit |   .0000593   .0002138     0.28   0.782    -.0003599    .0004784
           dComp |    .000157   .0001896     0.83   0.408    -.0002146    .0005286
            dNom |  -.0012808    .000506    -2.53   0.011    -.0022726   -.0002889
         InstOwn |  -.0212428   .0022082    -9.62   0.000    -.0255713   -.0169144
             FCF |  -3.85e-08   9.35e-09    -4.12   0.000    -5.68e-08   -2.02e-08
         dEnergy |  -.0017655   .0007583    -2.33   0.020    -.0032519   -.0002792
      dMaterials |   -.000624   .0005299    -1.18   0.239    -.0016627    .0004146
   dConsumerDisc |   .0023597   .0005535     4.26   0.000     .0012748    .0034447
    dConsumerSta |   .0001703   .0007431     0.23   0.819    -.0012863    .0016268
     dHealthCare |   .0027769   .0007045     3.94   0.000      .001396    .0041578
     dFinancials |  -.0004171   .0037388    -0.11   0.911    -.0077456    .0069114
dInformationTech |   .0030258   .0007783     3.89   0.000     .0015002    .0045514
          dComms |   .0002314   .0009373     0.25   0.805    -.0016059    .0020686
      dUtilities |  -.0050889   .0007094    -7.17   0.000    -.0064794   -.0036985
     dRealEstate |  -.0060983   .0020713    -2.94   0.003    -.0101584   -.0020381
                 |
            Time |
              2  |  -.0041194   .0003411   -12.08   0.000     -.004788   -.0034509
              3  |  -.0039076    .000284   -13.76   0.000    -.0044642    -.003351
              4  |  -.0017163   .0001891    -9.08   0.000     -.002087   -.0013457
              5  |  -.0023712   .0002102   -11.28   0.000    -.0027832   -.0019593
              6  |  -.0010996   .0001915    -5.74   0.000     -.001475   -.0007242
              7  |  -.0013153   .0002051    -6.41   0.000    -.0017172   -.0009133
              8  |   -.001092   .0002049    -5.33   0.000    -.0014935   -.0006904
              9  |  -.0015867    .000247    -6.42   0.000    -.0020708   -.0011026
             10  |  -.0032568   .0002524   -12.90   0.000    -.0037515    -.002762
             11  |   -.001924   .0002559    -7.52   0.000    -.0024257   -.0014223
                 |
           _cons |   .0269272   .0023039    11.69   0.000     .0224112    .0314433
-----------------+----------------------------------------------------------------
         sigma_u |  .01838454
         sigma_e |  .00901617
             rho |  .80611808   (fraction of variance due to u_i)
----------------------------------------------------------------------------------

Then my model without time dummies

 . xtreg $ylist $xlist , fe vce(robust)

Fixed-effects (within) regression               Number of obs     =     76,710
Group variable: ID                              Number of groups  =     12,895

R-sq:                                           Obs per group:
     within  = 0.2338                                         min =          1
     between = 0.2244                                         avg =        5.9
     overall = 0.2638                                         max =         46

                                                F(26,12894)       =      52.71
corr(u_i, Xb)  = 0.0224                         Prob > F          =     0.0000

                                    (Std. Err. adjusted for 12,895 clusters in ID)
----------------------------------------------------------------------------------
                 |               Robust
              PD |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-----------------+----------------------------------------------------------------
           Women |  -.0021433   .0010889    -1.97   0.049    -.0042777   -8.82e-06
             ROA |  -.1033834   .0059445   -17.39   0.000    -.1150355   -.0917312
            MTBt |   .0001104   .0000167     6.63   0.000     .0000777    .0001431
          LIQUID |  -.0002607   .0000901    -2.89   0.004    -.0004373   -.0000842
            SOLV |   .0068866   .0006663    10.34   0.000     .0055805    .0081926
           RISK1 |    .012754   .0005728    22.27   0.000     .0116312    .0138768
          MktCap |  -5.18e-09   6.75e-10    -7.68   0.000    -6.51e-09   -3.86e-09
            Inde |   .0017683   .0008672     2.04   0.041     .0000684    .0034682
     OtherBoards |   .0001932   .0001038     1.86   0.063    -.0000103    .0003966
           Bsize |  -.0004775    .000053    -9.01   0.000    -.0005814   -.0003736
            dGov |   .0012372   .0005022     2.46   0.014     .0002527    .0022217
          dAudit |  -1.80e-06   .0002154    -0.01   0.993    -.0004241    .0004205
           dComp |   .0000934   .0001928     0.48   0.628    -.0002845    .0004712
            dNom |  -.0014574   .0005306    -2.75   0.006    -.0024975   -.0004174
         InstOwn |  -.0213736   .0022335    -9.57   0.000    -.0257516   -.0169956
             FCF |  -4.17e-08   9.68e-09    -4.31   0.000    -6.06e-08   -2.27e-08
         dEnergy |    -.00168   .0007594    -2.21   0.027    -.0031685   -.0001915
      dMaterials |  -.0005221   .0005387    -0.97   0.332     -.001578    .0005337
   dConsumerDisc |   .0025209    .000564     4.47   0.000     .0014154    .0036265
    dConsumerSta |  -.0001026   .0007424    -0.14   0.890    -.0015579    .0013527
     dHealthCare |   .0026983   .0007112     3.79   0.000     .0013042    .0040924
     dFinancials |  -.0000391    .003853    -0.01   0.992    -.0075915    .0075133
dInformationTech |   .0030545   .0007851     3.89   0.000     .0015156    .0045933
          dComms |   .0003357   .0009567     0.35   0.726    -.0015396     .002211
      dUtilities |  -.0057208   .0007085    -8.07   0.000    -.0071095    -.004332
     dRealEstate |  -.0062761   .0021122    -2.97   0.003    -.0104163    -.002136
           _cons |   .0278123   .0023418    11.88   0.000      .023222    .0324026
-----------------+----------------------------------------------------------------
         sigma_u |  .01851273
         sigma_e |  .00907152
             rho |  .80637699   (fraction of variance due to u_i)
----------------------------------------------------------------------------------

I have also tested using timeparm if the time dummies are significantly different:

  testparm i.Time

 ( 1)  2.Time = 0
 ( 2)  3.Time = 0
 ( 3)  4.Time = 0
 ( 4)  5.Time = 0
 ( 5)  6.Time = 0
 ( 6)  7.Time = 0
 ( 7)  8.Time = 0
 ( 8)  9.Time = 0
 ( 9)  10.Time = 0
 (10)  11.Time = 0

       F( 10, 12894) =   38.22
            Prob > F =    0.0000

What are your experiences with time dummies in panel data? Should they be included or not?

Thanks in advance

[Carlo Lazzaro]

Emil:
in your second model the statistical significance of -female- is, at best, tenuous (p=0.049).
Besides, looking at the 95% CIs of the same variable in both models, they do not look so different to me.
Hence, I would stick with the regression code that includes -i.time- in the right-hand side of your regression equation.
As a second thought (and stating that my last experience with corporate finance dates back to the last millennium), in my opinion it makes pefectly sense that, other things being equal, the probability of default is influenced/predicted by financial indicators, such as Return on Assets (ROA) and the like and that bosses' gender has a negligible role.
THat said, I would be much more concerned about other topics that you do not seem to mention in your post:
- is your model correctly specified (ie, does it give a true and far view of the data generating process)?
- is -fe- the right specification for your regression model?

[Emil Holm]

Dear Carlo,

Thank you for your reply.

With regards to our questions:
1) I believe my specifications are in order. How would you normally test for under/over identification?

2) First I tested the pooled OLS vs RE with a Breusch Pagan test and secondly I tested RE vs FE using a Hausman test. Both were significant, why I ended up with FE.
I have also applied robust standard errors due to my Wald test showed signs of heteroskedasticity.

[Carlo Lazzaro]

Emil:
thanks for clarifying.
About how to test your model specification, you may want to take a look at this thread: https://www.statalist.org/forums/for...ng-stata/page2, #22.

[Emil Holm]

Carlo:

Running the test as you posted in the thread yields the following:

. xtreg $ylist $xlist i.Time, fe vce(robust)

Fixed-effects (within) regression               Number of obs     =     76,710
Group variable: ID                              Number of groups  =     12,895

R-sq:                                           Obs per group:
     within  = 0.2433                                         min =          1
     between = 0.2360                                         avg =        5.9
     overall = 0.2757                                         max =         46

                                                F(36,12894)       =      42.19
corr(u_i, Xb)  = 0.0226                         Prob > F          =     0.0000

                                    (Std. Err. adjusted for 12,895 clusters in ID)
----------------------------------------------------------------------------------
                 |               Robust
              PD |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-----------------+----------------------------------------------------------------
           Women |  -.0011513    .001198    -0.96   0.337    -.0034996     .001197
             ROA |   -.102301   .0060319   -16.96   0.000    -.1141244   -.0904776
            MTBt |   .0001097   .0000167     6.57   0.000      .000077    .0001424
          LIQUID |  -.0002312   .0000897    -2.58   0.010    -.0004069   -.0000554
            SOLV |    .007016   .0006786    10.34   0.000     .0056859    .0083461
           RISK1 |   .0176122    .000926    19.02   0.000     .0157971    .0194274
          MktCap |  -5.00e-09   6.51e-10    -7.69   0.000    -6.28e-09   -3.73e-09
            Inde |   .0022668    .000858     2.64   0.008      .000585    .0039486
     OtherBoards |   .0001514   .0001023     1.48   0.139    -.0000491    .0003519
           Bsize |  -.0004402   .0000529    -8.33   0.000    -.0005439   -.0003366
            dGov |   .0011423   .0004785     2.39   0.017     .0002044    .0020802
          dAudit |   .0000593   .0002138     0.28   0.782    -.0003599    .0004784
           dComp |    .000157   .0001896     0.83   0.408    -.0002146    .0005286
            dNom |  -.0012808    .000506    -2.53   0.011    -.0022726   -.0002889
         InstOwn |  -.0212428   .0022082    -9.62   0.000    -.0255713   -.0169144
             FCF |  -3.85e-08   9.35e-09    -4.12   0.000    -5.68e-08   -2.02e-08
         dEnergy |  -.0017655   .0007583    -2.33   0.020    -.0032519   -.0002792
      dMaterials |   -.000624   .0005299    -1.18   0.239    -.0016627    .0004146
   dConsumerDisc |   .0023597   .0005535     4.26   0.000     .0012748    .0034447
    dConsumerSta |   .0001703   .0007431     0.23   0.819    -.0012863    .0016268
     dHealthCare |   .0027769   .0007045     3.94   0.000      .001396    .0041578
     dFinancials |  -.0004171   .0037388    -0.11   0.911    -.0077456    .0069114
dInformationTech |   .0030258   .0007783     3.89   0.000     .0015002    .0045514
          dComms |   .0002314   .0009373     0.25   0.805    -.0016059    .0020686
      dUtilities |  -.0050889   .0007094    -7.17   0.000    -.0064794   -.0036985
     dRealEstate |  -.0060983   .0020713    -2.94   0.003    -.0101584   -.0020381
                 |
            Time |
              2  |  -.0041194   .0003411   -12.08   0.000     -.004788   -.0034509
              3  |  -.0039076    .000284   -13.76   0.000    -.0044642    -.003351
              4  |  -.0017163   .0001891    -9.08   0.000     -.002087   -.0013457
              5  |  -.0023712   .0002102   -11.28   0.000    -.0027832   -.0019593
              6  |  -.0010996   .0001915    -5.74   0.000     -.001475   -.0007242
              7  |  -.0013153   .0002051    -6.41   0.000    -.0017172   -.0009133
              8  |   -.001092   .0002049    -5.33   0.000    -.0014935   -.0006904
              9  |  -.0015867    .000247    -6.42   0.000    -.0020708   -.0011026
             10  |  -.0032568   .0002524   -12.90   0.000    -.0037515    -.002762
             11  |   -.001924   .0002559    -7.52   0.000    -.0024257   -.0014223
                 |
           _cons |   .0269272   .0023039    11.69   0.000     .0224112    .0314433
-----------------+----------------------------------------------------------------
         sigma_u |  .01838454
         sigma_e |  .00901617
             rho |  .80611808   (fraction of variance due to u_i)
----------------------------------------------------------------------------------

. *testparm i.Time
. predict fitted, xb
(58,638 missing values generated)

. g sq_fitted=fitted^2
(58,638 missing values generated)

. xtreg $ylist fitted sq_fitted , fe vce(robust)

Fixed-effects (within) regression               Number of obs     =     76,710
Group variable: ID                              Number of groups  =     12,895

R-sq:                                           Obs per group:
     within  = 0.2652                                         min =          1
     between = 0.2504                                         avg =        5.9
     overall = 0.3035                                         max =         46

                                                F(2,12894)        =     537.56
corr(u_i, Xb)  = 0.0572                         Prob > F          =     0.0000

                                (Std. Err. adjusted for 12,895 clusters in ID)
------------------------------------------------------------------------------
             |               Robust
          PD |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
      fitted |   .7021101   .0813152     8.63   0.000     .5427202       .8615
   sq_fitted |   10.03006   4.111815     2.44   0.015     1.970291    18.08982
       _cons |    .001003   .0002139     4.69   0.000     .0005837    .0014224
-------------+----------------------------------------------------------------
     sigma_u |  .01822713
     sigma_e |  .00888251
         rho |  .80809114   (fraction of variance due to u_i)
------------------------------------------------------------------------------

. test sq_fitted=0

 ( 1)  sq_fitted = 0

       F(  1, 12894) =    5.95
            Prob > F =    0.0147

Does this mean that my model is misspecified? If yes, how can I correct it?

[Carlo Lazzaro]

Emil:
yes it is.
Usually, misspecification is due to missing squared terms and/or interactions or forgotten predictors.
In other cases it may be a sign of endogeneity.
Check if any of the above affects your regression model.

[Emil Holm]

Carlo:

Ok, thank you.
Frankly, I am unsure which of my variables should be a squared term (or the likes).
With regards to endogeneity, I have a suspicion that there is a problem of reverse causality in the model.

Do you have any "magical econometrical tricks" that I can use?

[Carlo Lazzaro]

Emil:
sorry, no.
However, my first attempt would consider starting with a more parsimonious model and see what happens, both in terms of results and postestimation.
That said, I would take a look at the literature in yiour research field and see what others did in the past when presented with the same research topic.