Problems with interpretations of FE model and hybrid model with Mundlak approach

[Dita Rach]

Hello all, this is my first time on the forum, so I apologize for any mistakes in the way I ask my questions.

I am running a panel data analysis with multiple firms of 5 countries for 9 years (2014-2022) to examine the interaction effect of Corporate Social Responsibility (L_CSR) and Conservatism-Optimism (CONV) on Earnings Management (ABS_DA). Following to the Hausman test, I run xtreg for fixed-effects (FE) model. I also test the interaction term of CONV (time-invariant variable) and L_CSR with variable CSR_CONV. Xtreg shows insignificant coefficient from the interaction term (shown in Code 1). As the CONV variable is being omitted because it is time-invariant, I run the hybrid model with the Mundlak approach, which accounts for time-invariant variables. The Mundlak approach yielded a significant negative coefficient for the interaction term (shown in Code 2), supporting my hypothesis that CONV moderates the relationship between L_CSR and ABS_DA.

My question is: When interpreting these results, should I rely on the findings from the Mundlak approach since it accounts for time-invariant variables (however my focus is on the interaction term between CONV and L_CSR)? or can I present the results of the FE model and using the hybrid model as additional analysis? Thank you very much for your help...

1) Here is the FE code & result:

xtreg ABS_DA L_CSR CONV CSR_CONV L_LEV L_SIZE L_GROWTH L_MB L_ROA L_BR L_TURN L_MC i.year, fe vce(cluster Firm_id)
note: CONV omitted because of collinearity.

Fixed-effects (within) regression               Number of obs     =      3,623
Group variable: Firm_id                         Number of groups  =        998

R-squared:                                      Obs per group:
     Within  = 0.1602                                         min =          1
     Between = 0.0020                                         avg =        3.6
     Overall = 0.0477                                         max =          9

                                                F(18, 997)        =      11.71
corr(u_i, Xb) = -0.4405                         Prob > F          =     0.0000

                              (Std. err. adjusted for 998 clusters in Firm_id)
------------------------------------------------------------------------------
             |               Robust
      ABS_DA | Coefficient  std. err.      t    P>|t|     [95% conf. interval]
-------------+----------------------------------------------------------------
       L_CSR |   -.001022    .000358    -2.85   0.004    -.0017245   -.0003194
         CONV|          0  (omitted)
     CSR_CONV|   .0006112   .0007518     0.81   0.416    -.0008641    .0020864
       L_LEV |  -.0128577   .0500331    -0.26   0.797    -.1110399    .0853246
      L_SIZE |    .005521   .0151557     0.36   0.716    -.0242197    .0352617
    L_GROWTH |   .0004559   .0098745     0.05   0.963    -.0189214    .0198332
        L_MB |  -.0043127   .0008162    -5.28   0.000    -.0059143    -.002711
       L_ROA |  -.0156184   .0837509    -0.19   0.852    -.1799667    .1487299
        L_BR |   .0091417   .0462473     0.20   0.843    -.0816115    .0998948
      L_TURN |   .0258815   .0246094     1.05   0.293    -.0224107    .0741737
        L_MC |  -.0006197   .0001711    -3.62   0.000    -.0009555    -.000284
             |
        year |
       2015  |   .0884555   .0163791     5.40   0.000      .056314     .120597
       2016  |   .0077182   .0054587     1.41   0.158    -.0029937      .01843
       2017  |   .0240472   .0070449     3.41   0.001     .0102227    .0378717
       2018  |   .0308461   .0097658     3.16   0.002     .0116822      .05001
       2019  |   .0285223   .0096993     2.94   0.003     .0094888    .0475557
       2020  |   .1324932   .0174992     7.57   0.000     .0981538    .1668327
       2021  |   .1655739   .0217027     7.63   0.000     .1229856    .2081621
       2022  |   .0804342   .0201485     3.99   0.000     .0408959    .1199725
             |
       _cons |  -.0405993    .335684    -0.12   0.904    -.6993276    .6181289
-------------+----------------------------------------------------------------
     sigma_u |  .09668809
     sigma_e |  .11243648
         rho |  .42511838   (fraction of variance due to u_i)
------------------------------------------------------------------------------

2) Here is the Hybrid code & result:

xtreg ABS_DA L_CSR mean_CSR CONV CSR_CONV L_LEV mean_LEV L_SIZE mean_SIZE L_GROWTH mean_GROWTH L_MB mean_MB L_ROA mean_ROA L_BR mean_BR L_TURN mean_TURN L_MC mean_MC i.year, re vce(cluster Firm_id)

Random-effects GLS regression                   Number of obs     =      3,623
Group variable: Firm_id                         Number of groups  =        998

R-squared:                                      Obs per group:
     Within  = 0.1562                                         min =          1
     Between = 0.0754                                         avg =        3.6
     Overall = 0.1368                                         max =          9

                                                Wald chi2(28)     =     238.51
corr(u_i, X) = 0 (assumed)                      Prob > chi2       =     0.0000

                              (Std. err. adjusted for 998 clusters in Firm_id)
------------------------------------------------------------------------------
             |               Robust
      ABS_DA | Coefficient  std. err.      z    P>|z|     [95% conf. interval]
-------------+----------------------------------------------------------------
       L_CSR |  -.0004348   .0002774    -1.57   0.117    -.0009784    .0001089
    mean_CSR |   .0007772   .0002963     2.62   0.009     .0001965    .0013578
         CONV|   .0499429   .0206724     2.42   0.016     .0094258    .0904601
     CSR_CONV|  -.0006156   .0003614    -1.70   0.088    -.0013239    .0000927
       L_LEV |  -.0257943   .0502612    -0.51   0.608    -.1243044    .0727158
    mean_LEV |   .2086541   .0610436     3.42   0.001     .0890108    .3282974
      L_SIZE |   .0206234   .0140414     1.47   0.142    -.0068973    .0481441
   mean_SIZE |  -.0217453   .0146853    -1.48   0.139     -.050528    .0070373
    L_GROWTH |  -.0004204   .0099079    -0.04   0.966    -.0198396    .0189988
 mean_GROWTH |  -.0057372   .0180673    -0.32   0.751    -.0411484     .029674
        L_MB |  -.0037132    .000759    -4.89   0.000    -.0052008   -.0022257
     mean_MB |   .0016251   .0008872     1.83   0.067    -.0001138     .003364
       L_ROA |  -.0689865     .08122    -0.85   0.396    -.2281747    .0902017
    mean_ROA |   .2126758   .1113415     1.91   0.056    -.0055496    .4309012
        L_BR |   .0240608   .0458199     0.53   0.600    -.0657445     .113866
     mean_BR |   .0592471   .0853255     0.69   0.487    -.1079878     .226482
      L_TURN |   .0242805   .0243608     1.00   0.319    -.0234658    .0720267
   mean_TURN |   -.034911   .0251526    -1.39   0.165    -.0842092    .0143871
        L_MC |  -.0002032   .0001271    -1.60   0.110    -.0004524     .000046
     mean_MC |   .0000132    .000144     0.09   0.927    -.0002691    .0002955
             |
        year |
       2015  |   .0810946   .0167123     4.85   0.000     .0483391    .1138501
       2016  |   .0027252   .0045284     0.60   0.547    -.0061502    .0116007
       2017  |   .0135633   .0056442     2.40   0.016     .0025009    .0246258
       2018  |   .0116815   .0060492     1.93   0.053    -.0001747    .0235377
       2019  |    .012064   .0064146     1.88   0.060    -.0005084    .0246365
       2020  |   .0988781   .0109463     9.03   0.000     .0774237    .1203324
       2021  |   .1154714   .0128408     8.99   0.000     .0903039    .1406388
       2022  |   .0296696   .0095399     3.11   0.002     .0109718    .0483674
             |
       _cons |   .0218019   .0605764     0.36   0.719    -.0969257    .1405295
-------------+----------------------------------------------------------------
     sigma_u |  .03200595
     sigma_e |  .11243648
         rho |  .07495647   (fraction of variance due to u_i)
------------------------------------------------------------------------------

[Clyde Schechter]

Your hybrid model is not correctly implemented. Your interaction term, CSR_CONV also needs a CSR_CONV_mean term to accompany it here. CSR_CONV is not time-invariant, and it will not be, and should not be, omitted. Done correctly, by the way, you will find that the estimates you get for the non-mean variables are the same as you got from -xtreg, fe-.

There is also the question of why you are doing this. The omission of CONV in the fixed-effects model arises because CONV is time invariant. Consequently, a within-firm effect of CONV is, in principle, impossible to estimate from this kind of data and analysis. The estimate of a CONV effect you will get from the hybrid model is the between-firm effect of CONV on your outcome. Normally, that between-firms effect is not of interest. On top of that, because CONV participates in an interaction with CSR, the result you will get is the between-firms effect of CONV conditional on CSR = 0. This is usually of even less interest than an unconditional between-firms effect. So unless there is something specific in your research goals that calls for estimating this between-firm effect of CONV conditional on CSR = 0, the hybrid model is just wasted effort here.

Finally, I should point out that if you really do need to use the hybrid model, and if you are running Stata version 18.5, you can get it much more simply, without having to create your own mean variables, by using -xtreg, cre-.

[Dita Rach]

Dear Mr Schechter,
Thank you very much for your comprehensive answer. After some consideration, the hybrid model may not be necessary for my research goals.