[Carlo Lazzaro]

Jon:
good to know. Fluent indeed (most foreign Italian speakers I know would not have used the word "anzi", that may sound something like "even better"), bravo!
That said, authors seems to have used a model that is half-way between a pooled OLS (that implies clustering) and a -fe- regression (that implies creating a categorical variable for -panelid-, but without clustering).
As econometricians should avoid reinventing the wheel and, at the same time, staying away from doing analyses following the "business as usual " approach, I would strongly recmmend you to check whether the methodology described by authors is in line with the current methodological standard of your reserach field.
I do share your opionion about asking the authors directly instead of banging my head against the wall trying to find an explanation to something that I do not/cannot know.

[Carlo Lazzaro]

Jon:
as a second thought, you may want to consider Sergio Correia 's -reghdfe- (-search reghdfe-).
As an aside, the authors you quoted reported they did a multivariate regression analysis: however, the technical (but often misused) meaning of multivariate is: having more than one regressand in the left-hand side of the regressione equation (see -help mvreg-).

[Jon Hoefer]

Thanks for your further suggestion,
I have just run the reghdfe code:

. reghdfe ROA Family_Firm_Identifier Founder_Identity FirmAge FirmSize Indebtedness , absorb(Fiscal_Year SIC)
(dropped 7 singleton observations)
(MWFE estimator converged in 7 iterations)

HDFE Linear regression                            Number of obs   =      1,934
Absorbing 2 HDFE groups                           F(   5,   1797) =      24.01
                                                  Prob > F        =     0.0000
                                                  R-squared       =     0.4978
                                                  Adj R-squared   =     0.4598
                                                  Within R-sq.    =     0.0626
                                                  Root MSE        =     0.0817

----------------------------------------------------------------------------------------
                   ROA |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-----------------------+----------------------------------------------------------------
Family_Firm_Identifier |   .0146939    .009365     1.57   0.117    -.0036735    .0330613
      Founder_Identity |  -.0293939   .0102303    -2.87   0.004    -.0494585   -.0093294
               FirmAge |   .2215533   .2225699     1.00   0.320    -.2149697    .6580762
              FirmSize |  -.0013989   .0022335    -0.63   0.531    -.0057795    .0029816
          Indebtedness |  -.1601288   .0151588   -10.56   0.000    -.1898595   -.1303981
                 _cons |  -1.492851   1.687947    -0.88   0.377    -4.803397    1.817695
----------------------------------------------------------------------------------------

Absorbed degrees of freedom:
-----------------------------------------------------+
 Absorbed FE | Categories  - Redundant  = Num. Coefs |
-------------+---------------------------------------|
 Fiscal_Year |        23           0          23     |
         SIC |       110           1         109     |
-----------------------------------------------------+

Results seem quite promising, in particular the adj. R^2. Just so that I understand correctly what it does, is this a kind of "hybrid" model? And would I know need to apply the specification tests, such as VIF etc?

[Carlo Lazzaro]

Jon:
no, hybrid model are a different beast (see -mundlak-).
It is a good habit to apply postestimation routines to all the regression models.
However, I'm not aware of such a routine for -reghdfe-.

[Jeff Wooldridge]

Jon, some comments.

1. Your ability to keep a time-invariant variable while adding fixed effects "by hand" is an illusion. You should not be doing this. There is only one true fixed effects estimator. xtreg, fe does it properly, as Carlo emphasized. By putting in the dummies "by hand" you are deluding yourself. Stata is simply dropping variables until there is no collinearity left. From an identification perspective, you cannot estimate coefficients on the time-constant variables.

2. If you want to use fixed effects for all other coefficients but still include a time-constant variable, use the correlated random effects approach due to Mundlak. Some people on this site call them "hybrid" models. You include the time averages of all time-varying variables. No fixed effects. You can do this with xtreg, re or just use pooled OLS and cluster your standard errors by firm -- which you should be doing anyway.

3. You shouldn't be looking at VIFs at all, let alone on the firm dummies. Look at the coefficients and standard errors of the variables you care about. I can have a VIF of 300 on a variable I don't care about -- incuding the firm dummies -- and I wouldn't care one bit. In fact, I could have a very large VIF on a variable I do care about and it wouldn't directly matter. What matters is how large is the coefficient to its standard error.

4. SInce you should be clustering your standard errors, you should use a robust form of the Hausman test. Happily, the Mundlak formulation does that directly: a joint test, made cluster-robust, on the time averages. There have been several threads about that on this site.

But the most important thing is to use xtreg, which properly time demeans all variables and drops out time constant variables. If you must have the time constant variable in there, use the Mundlak approach.

JW

[Jon Hoefer]

Dear Jeff,

thank you very much for your detailed explanations and what an honour to receive your feedback. Generally the willingness to help on this site is really mind blowing and exceeds by far what I expected when signing up a few days ago. Back to topic: I will look into the Mundlak approach. Due to me building on the work of previous research, in fact I try to replicate it using a different sample, would it still make sense to once apply the model as done by the aforementioned authors to ensure comparability and then use the model suggested by you (as indeed, my main variable of interest is time invariant and using a standard FE would hence be useless)? Also, in the second part of my analysis I want to examine a potential mediation effect, previously not considered (which in fact is supposed to be my contribution / extension of previous work). Will using the Mundlak approach make the later mediation analysis somehow more complex / trickier to execute? As I only have one single mediator and use "manually" bootstrapped CI's (following Hayes) to infer about the indirect effect's significance, it shouldn't, right?

[Jon Hoefer]

So I tried to read into Mundlak (including the chapter 14 of Jeff Wooldridge 's Introductory econometrics : a modern approach 5th edition) and do more or less understand what it does and how this is beneficial in my case. However, I am not sure which of the two xthybrid commands are delivering the CRE model, you were referring to before (the one with or without the cre option)? I will report both versions of it. I also clustered the s.e. as suggested on firm level (GVKEY) and included Fiscal Year dummies (the Dummy*Carlo Lazzaro As I cannot include the factor notation, I have to come back to my manually derived dummies that you weren't too happy about in your first response, right? You earlier said that they can sometimes be erroneous. Could you maybe be so kind as to elaborate what instances you had in mind, such that I can double check I am not running into one of the traps you thought of?).
One other question arising is why I cannot include my industry dummies anymore (it will give me: "no observations r(2000)")?
Ultimately, Jeff points out in his book that the interpretation of the time invariant coefficients have to be taken with a grain of salt. As again the variable Family Firm Status is exactly such a variable (and the one I care most about), does this approach make sense in my case after all?
And as we are discussing interpretations anyway, how do I interpret the coefficients? Do I have to add the coefficients for the same variable (say indebtedness) to derive the typical one margin increase leads to c.p. change in y?
And is the included Wald test the "sort of better Hausman" test, you referred to? In that case both models would again reject H0 that RE is efficient!
As you see, I believe that the Hybrid /CRE is probably pushing me to the limit of my econometrical knowledge and in fact none of the many economic PhD students I know can really help me out, which makes me wondering if I should really build my entire thesis on the model?

Below as promised the code with and w/o the cre option.

xthybrid ROA Family_Firm_Identifier Founder_Identity FirmAge , use(FirmSize Indebtedness  Dummy* ) clusterid(GVKEY) full se t p star vce(cluster GVKEY)


---------------------------------------------------------------------------------------------------------------------------------------------------------------------
Model model
---------------------------------------------------------------------------------------------------------------------------------------------------------------------

Mixed-effects GLM                               Number of obs     =      1,941
Family:                Gaussian
Link:                  identity
Group variable:           GVKEY                 Number of groups  =        198

                                                Obs per group:
                                                              min =          1
                                                              avg =        9.8
                                                              max =         23

Integration method: mvaghermite                 Integration pts.  =          7

                                                Wald chi2(51)     =     127.62
Log pseudolikelihood =  2123.9675               Prob > chi2       =     0.0000
                                             (Std. Err. adjusted for 198 clusters in GVKEY)
-------------------------------------------------------------------------------------------
                          |               Robust
                      ROA |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
--------------------------+----------------------------------------------------------------
      R__Founder_Identity |  -.0103487   .0107998    -0.96   0.338     -.031516    .0108187
R__Family_Firm_Identifier |   .0062879   .0080315     0.78   0.434    -.0094535    .0220294
               R__FirmAge |  -.3661062    .284958    -1.28   0.199    -.9246135    .1924012
              W__FirmSize |  -.0004327   .0198675    -0.02   0.983    -.0393723     .038507
          W__Indebtedness |  -.2011179   .0940852    -2.14   0.033    -.3855215   -.0167143
             W__Dummy1996 |   .0238792   .0263619     0.91   0.365    -.0277891    .0755476
             W__Dummy1997 |   .0278088   .0259882     1.07   0.285    -.0231271    .0787447
             W__Dummy1998 |   .0262129   .0230787     1.14   0.256    -.0190206    .0714464
             W__Dummy1999 |   .0258633   .0221711     1.17   0.243    -.0175912    .0693178
             W__Dummy2000 |   .0194638   .0195582     1.00   0.320    -.0188695    .0577972
             W__Dummy2001 |  -.0051138   .0196393    -0.26   0.795     -.043606    .0333785
             W__Dummy2002 |   .0026989   .0189194     0.14   0.887    -.0343823    .0397802
             W__Dummy2003 |   .0040163   .0189091     0.21   0.832    -.0330449    .0410776
             W__Dummy2004 |   .0021752   .0177665     0.12   0.903    -.0326465     .036997
             W__Dummy2005 |   .0019852   .0160678     0.12   0.902    -.0295072    .0334776
             W__Dummy2006 |   .0060733   .0152538     0.40   0.691    -.0238235    .0359702
             W__Dummy2007 |   .0004823   .0148036     0.03   0.974    -.0285323    .0294969
             W__Dummy2008 |  -.0055196   .0151505    -0.36   0.716     -.035214    .0241748
             W__Dummy2009 |  -.0153233   .0160633    -0.95   0.340    -.0468068    .0161602
             W__Dummy2010 |  -.0023362   .0155486    -0.15   0.881    -.0328109    .0281385
             W__Dummy2011 |  -.0008953   .0144983    -0.06   0.951    -.0293115    .0275209
             W__Dummy2012 |  -.0192225   .0216484    -0.89   0.375    -.0616525    .0232076
             W__Dummy2013 |  -.0018771   .0132719    -0.14   0.888    -.0278896    .0241354
             W__Dummy2014 |   .0014494   .0127034     0.11   0.909    -.0234487    .0263476
             W__Dummy2015 |  -.0075441   .0140946    -0.54   0.592     -.035169    .0200809
             W__Dummy2016 |  -.0075926   .0122263    -0.62   0.535    -.0315556    .0163705
             W__Dummy2017 |  -.0079407   .0102742    -0.77   0.440    -.0280777    .0121963
             W__Dummy2018 |          0  (omitted)
              B__FirmSize |  -.0057417   .0041781    -1.37   0.169    -.0139305    .0024471
          B__Indebtedness |  -.0070658     .05781    -0.12   0.903    -.1203714    .1062397
             B__Dummy1996 |   .0377186   .0901597     0.42   0.676    -.1389912    .2144285
             B__Dummy1997 |   .0089758   .1204319     0.07   0.941    -.2270663     .245018
             B__Dummy1998 |   .0170458   .1137641     0.15   0.881    -.2059278    .2400193
             B__Dummy1999 |   .1108761   .0977229     1.13   0.257    -.0806573    .3024095
             B__Dummy2000 |   .1088523   .1028957     1.06   0.290    -.0928195    .3105241
             B__Dummy2001 |  -.1445377   .1845951    -0.78   0.434    -.5063375    .2172621
             B__Dummy2002 |   .0901492   .2034074     0.44   0.658    -.3085219    .4888204
             B__Dummy2003 |   .0573947   .2110652     0.27   0.786    -.3562855    .4710749
             B__Dummy2004 |   .2313114   .2242215     1.03   0.302    -.2081547    .6707775
             B__Dummy2005 |  -.0291301   .1914772    -0.15   0.879    -.4044185    .3461583
             B__Dummy2006 |   .0910931   .1002774     0.91   0.364     -.105447    .2876331
             B__Dummy2007 |   .0408657    .091365     0.45   0.655    -.1382064    .2199379
             B__Dummy2008 |  -.2681968   .2088886    -1.28   0.199    -.6776109    .1412173
             B__Dummy2009 |   .2378223   .1988754     1.20   0.232    -.1519664    .6276109
             B__Dummy2010 |   .0666368   .1259459     0.53   0.597    -.1802126    .3134862
             B__Dummy2011 |   .1987823   .1405588     1.41   0.157     -.076708    .4742726
             B__Dummy2012 |   -.166531   .2602637    -0.64   0.522    -.6766385    .3435764
             B__Dummy2013 |   .2862411   .2627209     1.09   0.276    -.2286825    .8011647
             B__Dummy2014 |  -.1186477   .1093414    -1.09   0.278     -.332953    .0956576
             B__Dummy2015 |   .0503885   .0927551     0.54   0.587    -.1314082    .2321852
             B__Dummy2016 |    .145158   .0929272     1.56   0.118     -.036976    .3272919
             B__Dummy2017 |   .0704575    .127713     0.55   0.581    -.1798555    .3207704
             B__Dummy2018 |          0  (omitted)
                    _cons |   2.902238   2.166216     1.34   0.180    -1.343468    7.147944
--------------------------+----------------------------------------------------------------
GVKEY                     |
                var(_cons)|   .0052415   .0009267                      .0037065    .0074122
--------------------------+----------------------------------------------------------------
                var(e.ROA)|    .005297   .0013802                      .0031786    .0088272
-------------------------------------------------------------------------------------------

. xthybrid ROA Family_Firm_Identifier Founder_Identity FirmAge , use(FirmSize Indebtedness  Dummy* ) clusterid(GVKEY) cre full se t p star vce(cluster GVKEY)


---------------------------------------------------------------------------------------------------------------------------------------------------------------------
Model model
---------------------------------------------------------------------------------------------------------------------------------------------------------------------

Mixed-effects GLM                               Number of obs     =      1,954
Family:                Gaussian
Link:                  identity
Group variable:           GVKEY                 Number of groups  =        198

                                                Obs per group:
                                                              min =          1
                                                              avg =        9.9
                                                              max =         23

Integration method: mvaghermite                 Integration pts.  =          7

                                                Wald chi2(27)     =      57.07
Log pseudolikelihood =   2029.357               Prob > chi2       =     0.0006
                                             (Std. Err. adjusted for 198 clusters in GVKEY)
-------------------------------------------------------------------------------------------
                          |               Robust
                      ROA |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
--------------------------+----------------------------------------------------------------
      R__Founder_Identity |   -.006073     .01073    -0.57   0.571    -.0271033    .0149574
R__Family_Firm_Identifier |   .0180115   .0076213     2.36   0.018     .0030741    .0329489
               R__FirmAge |  -.4793204   .2979393    -1.61   0.108    -1.063271    .1046299
              D__FirmSize |  -.0046321   .0041958    -1.10   0.270    -.0128557    .0035915
          D__Indebtedness |   .0007006   .0579116     0.01   0.990    -.1128039    .1142052
             D__Dummy1996 |   .0258907   .0934996     0.28   0.782    -.1573652    .2091466
             D__Dummy1997 |   .0008085   .1248419     0.01   0.995    -.2438771    .2454941
             D__Dummy1998 |   .0052591   .1173628     0.04   0.964    -.2247678    .2352861
             D__Dummy1999 |    .104818   .0995797     1.05   0.293    -.0903546    .2999907
             D__Dummy2000 |   .0926501   .1059297     0.87   0.382    -.1149683    .3002684
             D__Dummy2001 |  -.1729816   .1859386    -0.93   0.352    -.5374145    .1914513
             D__Dummy2002 |   .1047877   .2064867     0.51   0.612    -.2999188    .5094942
             D__Dummy2003 |   .0231679   .2073122     0.11   0.911    -.3831566    .4294924
             D__Dummy2004 |   .2353138   .2225295     1.06   0.290    -.2008359    .6714635
             D__Dummy2005 |  -.0404735    .197668    -0.20   0.838    -.4278957    .3469488
             D__Dummy2006 |   .0819201   .1021544     0.80   0.423    -.1182989    .2821391
             D__Dummy2007 |   .0269258   .0944368     0.29   0.776    -.1581669    .2120186
             D__Dummy2008 |  -.1913619   .2230692    -0.86   0.391    -.6285696    .2458458
             D__Dummy2009 |   .1529891   .2201896     0.69   0.487    -.2785746    .5845527
             D__Dummy2010 |   .0492606   .1293585     0.38   0.703    -.2042773    .3027985
             D__Dummy2011 |   .2144549   .1406649     1.52   0.127    -.0612432    .4901529
             D__Dummy2012 |  -.2163049   .2674437    -0.81   0.419    -.7404849    .3078752
             D__Dummy2013 |   .2722216   .2743383     0.99   0.321    -.2654715    .8099148
             D__Dummy2014 |  -.1189433   .1110566    -1.07   0.284    -.3366102    .0987237
             D__Dummy2015 |   .0369637   .0962076     0.38   0.701    -.1515997    .2255272
             D__Dummy2016 |   .1206928   .0935769     1.29   0.197    -.0627146    .3041002
             D__Dummy2017 |   .0628153   .1354148     0.46   0.643    -.2025927    .3282234
             D__Dummy2018 |          0  (omitted)
                    _cons |    3.75446   2.265285     1.66   0.097    -.6854178    8.194337
--------------------------+----------------------------------------------------------------
GVKEY                     |
                var(_cons)|    .005286   .0009453                      .0037231    .0075051
--------------------------+----------------------------------------------------------------
                var(e.ROA)|    .005981   .0017219                      .0034018    .0105157
-------------------------------------------------------------------------------------------

[Carlo Lazzaro]

Jon:
the risk of being error prone with creating categorical variables by hand rests on the fact that, for a given observation/year combination, you may inavertently omit the correct 1 or 0 value.