*Help* Interaction term categorical variable with dummy variable

[Lucas Venn]

Dear all,

Currently I am doing research on family firms performance relative to non-family firms.
One of the research questions is: “What is the relation between the type of CEO (founder, descendant, or professional outsider) and family firm performance? ”

For this, I made the following hypothesis:

H₀: There is no significant difference in performance between family firms with different CEO types.
H₁: Family firms with a descendant CEO underperform family firms with a professional CEO, which underperform family firms with a founder CEO.

After that, I designed the following model:

Return on assets = alpha + B₁(Family Firm) + B₂(CEO_founder) + B₃(CEO_descendant) + B₄(FamilyFirm * CEO_Founder) + B₅(FamilyFirm * CEO_descendant) + control variables

Important to know is that the CEO variable consists of:
CEO == 0 --> Founder CEO's
CEO == 1 --> Professional CEO's
CEO == 2 --> Descendant CEO's

I divided this variable into three dummies using the following code (and purposly left out one of the dummies from the equation since it is the base group, as usually with categorical variable equations). (SEE CODE BELOW)

CEO type is only defined for family firms, so if family firm == 1

Family firm is a dummy variable that equals 1 if the firm is a family firm, 0 otherwise.

The model however, excludes the two interaction terms due to collinearity. Am I specifying my model correct? If not, what would be a better option?

gen CEO_founder = 1 if (FamilyFirm==1 & ceo==0)
replace CEO_founder = 0 if (CEO_founder ==.)

gen CEO_professional = 1 if (FamilyFirm==1 & ceo==1)
replace CEO_professional = 0 if (CEO_professional ==.)

gen CEO_descendant = 1 if (FamilyFirm==1 & ceo==2)
replace CEO_descendant = 0 if (CEO_descendant ==.)

gen FamilyFirm_FounderCEO = FamilyFirm * CEO_founder
gen FamilyFirm_DescendantCEO = FamilyFirm * CEO_descendant

reg roa FamilyFirm CEO_founder CEO_descendant FamilyFirm_DescendantCEO FamilyFirm_FounderCEO  log_emp salesgrowth capitalstructure log_firmsize log_firmage risk investments Blockholders EquityBased i.sic2digits i.state1 i.fyear, robust

 reg roa  FamilyFirm_DescendantCEO FamilyFirm_FounderCEO  log_emp salesgrowth capitalstructure log_fi
> rmsize log_firmage risk investments Blockholders EquityBased i.sic2digits i.state1 i.fyear, robust

Linear regression                               Number of obs     =      3,777
                                                F(104, 3672)      =      29.90
                                                Prob > F          =     0.0000
                                                R-squared         =     0.3471
                                                Root MSE          =     .06258

------------------------------------------------------------------------------------------
                         |               Robust
                     roa | Coefficient  std. err.      t    P>|t|     [95% conf. interval]
-------------------------+----------------------------------------------------------------
FamilyFirm_DescendantCEO |   .0105863   .0094473     1.12   0.263    -.0079361    .0291087
   FamilyFirm_FounderCEO |  -.0050199   .0065614    -0.77   0.444    -.0178843    .0078444
                 log_emp |   .0059451    .001816     3.27   0.001     .0023845    .0095056
             salesgrowth |   .0004484   .0000997     4.50   0.000     .0002529    .0006439
        capitalstructure |  -.0071573   .0096734    -0.74   0.459     -.026123    .0118084
            log_firmsize |  -.0138522   .0020049    -6.91   0.000     -.017783   -.0099214
             log_firmage |   .0033036   .0012909     2.56   0.011     .0007727    .0058345
                    risk |  -.0063298     .00048   -13.19   0.000     -.007271   -.0053886
             investments |   .0605475    .016407     3.69   0.000     .0283799    .0927152
            Blockholders |   .0052603   .0236589     0.22   0.824    -.0411255    .0516461
             EquityBased |  -.0201391   .0054034    -3.73   0.000    -.0307331    -.009545
                         |
              sic2digits |
                     13  |  -.0391027   .0185658    -2.11   0.035    -.0755031   -.0027023
                     14  |   .0208269   .0167784     1.24   0.215     -.012069    .0537227
                     15  |   .0322023   .0189297     1.70   0.089    -.0049114    .0693161
                     17  |  -.0288372   .0177959    -1.62   0.105    -.0637279    .0060536
                     20  |   .0152617   .0152625     1.00   0.317     -.014662    .0451855
                     21  |   .1120894   .0224992     4.98   0.000     .0679773    .1562015
                     22  |  -.0036684   .0173595    -0.21   0.833    -.0377036    .0303668
                     23  |   .0247952   .0174594     1.42   0.156    -.0094359    .0590263
                     24  |   .0031122   .0172642     0.18   0.857    -.0307361    .0369605
                     25  |  -.0051528   .0182351    -0.28   0.778    -.0409048    .0305991
                     26  |   .0086347   .0164204     0.53   0.599    -.0235592    .0408287
                     27  |   -.064818   .0238632    -2.72   0.007    -.1116043   -.0180316
                     28  |   .0259653    .015737     1.65   0.099    -.0048887    .0568194
                     29  |   .0262743   .0166107     1.58   0.114    -.0062929    .0588415
                     30  |   .0206258    .018899     1.09   0.275    -.0164278    .0576793
                     31  |   .0709935   .0364127     1.95   0.051    -.0003977    .1423847
                     33  |   .0359817   .0276374     1.30   0.193    -.0182045    .0901679
                     34  |   .0227177   .0167503     1.36   0.175    -.0101231    .0555585
                     35  |   .0037313   .0160719     0.23   0.816    -.0277794     .035242
                     36  |   .0335453   .0161451     2.08   0.038     .0018912    .0651995
                     37  |  -.0067882   .0158973    -0.43   0.669    -.0379566    .0243802
                     38  |   .0009343   .0157829     0.06   0.953    -.0300099    .0318784
                     39  |  -.0366462   .0279071    -1.31   0.189    -.0913611    .0180687
                     40  |   .0350545   .0165663     2.12   0.034     .0025744    .0675346
                     42  |   .0435524   .0168983     2.58   0.010     .0104213    .0766835
                     44  |  -.0013812   .0184086    -0.08   0.940    -.0374732    .0347108
                     45  |   .0113539   .0172787     0.66   0.511    -.0225229    .0452307
                     47  |   .0066101   .0220136     0.30   0.764      -.03655    .0497701
                     48  |  -.0059456   .0157025    -0.38   0.705    -.0367321     .024841
                     49  |   -.026402   .0153695    -1.72   0.086    -.0565357    .0037317
                     50  |   .0309275    .016992     1.82   0.069    -.0023872    .0642421
                     51  |  -.0336195   .0172331    -1.95   0.051    -.0674068    .0001679
                     52  |   .0852731   .0189178     4.51   0.000     .0481827    .1223636
                     53  |   .0089047   .0170008     0.52   0.600    -.0244272    .0422367
                     54  |  -.0067078   .0180693    -0.37   0.710    -.0421346     .028719
                     55  |   .0319856   .0167529     1.91   0.056    -.0008603    .0648315
                     56  |    .044288   .0194883     2.27   0.023     .0060791     .082497
                     57  |   .0126308   .0232134     0.54   0.586    -.0328816    .0581431
                     58  |   .0835831   .0198849     4.20   0.000     .0445965    .1225698
                     59  |   .0106676   .0175598     0.61   0.544    -.0237602    .0450955
                     70  |  -.0139763   .0189177    -0.74   0.460    -.0510664    .0231139
                     72  |   .0265694   .0198259     1.34   0.180    -.0123016    .0654403
                     73  |   .0079797   .0160309     0.50   0.619    -.0234507      .03941
                     78  |  -.0010061   .0195517    -0.05   0.959    -.0393393    .0373271
                     79  |  -.0962977   .0174154    -5.53   0.000    -.1304425   -.0621528
                     80  |   .0084036   .0158193     0.53   0.595     -.022612    .0394191
                     87  |   .0044515   .0181779     0.24   0.807    -.0311882    .0400912
                     99  |  -.0023553   .0184226    -0.13   0.898    -.0384749    .0337643
                         |
                  state1 |
                     AR  |   .0319428   .0105936     3.02   0.003     .0111728    .0527128
                     AZ  |     .05515   .0135813     4.06   0.000     .0285223    .0817777
                     CA  |   .0657816   .0097247     6.76   0.000     .0467152     .084848
                     CO  |   .0221536   .0134975     1.64   0.101    -.0043097    .0486169
                     CT  |    .035811   .0105448     3.40   0.001     .0151367    .0564853
                     DC  |   .0420347   .0108868     3.86   0.000       .02069    .0633795
                     DE  |  -.0240273   .0231936    -1.04   0.300    -.0695009    .0214462
                     FL  |   .0257033   .0103495     2.48   0.013      .005412    .0459947
                     GA  |   .0401811   .0101631     3.95   0.000     .0202552     .060107
                     ID  |   .0790621   .0222317     3.56   0.000     .0354745    .1226497
                     IL  |    .047402   .0096359     4.92   0.000     .0285097    .0662942
                     IN  |    .043314   .0108005     4.01   0.000     .0221383    .0644896
                     KS  |   .0960166    .012974     7.40   0.000     .0705797    .1214535
                     KY  |   .1073979   .0180018     5.97   0.000     .0721033    .1426925
                     LA  |   .0447665   .0120286     3.72   0.000      .021183    .0683499
                     MA  |    .052936   .0110284     4.80   0.000     .0313137    .0745584
                     MD  |   .0420008   .0124553     3.37   0.001     .0175807    .0664208
                     ME  |   .1264252   .0197528     6.40   0.000     .0876977    .1651528
                     MI  |   .0482097   .0108425     4.45   0.000     .0269517    .0694676
                     MN  |   .0541534   .0100041     5.41   0.000     .0345393    .0737675
                     MO  |   .0518473    .009738     5.32   0.000     .0327549    .0709396
                     NC  |   .0210059   .0078269     2.68   0.007     .0056604    .0363515
                     NE  |   .0536138    .012932     4.15   0.000     .0282592    .0789684
                     NJ  |   .0497018    .010249     4.85   0.000     .0296076     .069796
                     NV  |   .1471773   .0156233     9.42   0.000     .1165461    .1778084
                     NY  |    .044313   .0096834     4.58   0.000     .0253276    .0632983
                     OH  |   .0476847   .0101479     4.70   0.000     .0277886    .0675808
                     OK  |   .0568621   .0152607     3.73   0.000     .0269418    .0867823
                     OR  |   .0753758   .0136582     5.52   0.000     .0485974    .1021542
                     PA  |   .0387563   .0096929     4.00   0.000     .0197522    .0577603
                     RI  |   .0495705   .0129957     3.81   0.000      .024091      .07505
                     TN  |   .0442944   .0098257     4.51   0.000     .0250301    .0635588
                     TX  |   .0538165   .0106717     5.04   0.000     .0328936    .0747395
                     VA  |   .0609739   .0122018     5.00   0.000     .0370509    .0848969
                     WA  |   .0475422   .0106825     4.45   0.000      .026598    .0684864
                     WI  |   .0389844   .0095671     4.07   0.000     .0202271    .0577418
                         |
                   fyear |
                   2013  |   .0010796   .0048403     0.22   0.824    -.0084104    .0105696
                   2014  |   .0006056   .0048276     0.13   0.900    -.0088594    .0100707
                   2015  |   .0012205   .0051037     0.24   0.811    -.0087859    .0112269
                   2016  |   .0030816   .0049035     0.63   0.530    -.0065322    .0126954
                   2017  |  -.0023672   .0046176    -0.51   0.608    -.0114206    .0066861
                   2018  |   .0238586   .0050898     4.69   0.000     .0138794    .0338377
                   2019  |   .0219956   .0050068     4.39   0.000     .0121793    .0318119
                   2020  |   .0327799   .0057572     5.69   0.000     .0214923    .0440675
                   2021  |   .0306567   .0053962     5.68   0.000     .0200768    .0412366
                         |
                   _cons |    .146908   .0253466     5.80   0.000     .0972131    .1966028
------------------------------------------------------------------------------------------

[Clyde Schechter]

As I do not fully understand your research question (more on this below) I am not sure if you are specifying your model correctly or not. But I am quite sure that your Stata code and this data set cannot estimate the model you have chosen.

The key to this is that CEO type is only defined for family firms. So, CEO type has missing value for any non-family firm. Consequently, because only observations with non-missing values for all model variables are used in estimation, in your estimation sample family_firm is always equal to 1. Because it is always equal to 1, it, is colinear with the constant term, and any interaction term involving it is colinear with the other variable in the interaction. Consequently, family_firm itself and both interaction terms are omitted due to this colinearity.

The proper resolution of this problem depends on what your actual research question is. The actual hypotheses you stated refer only to the effect of CEO type within family firms and says nothing at all about contrasting family firms with non-family firms. So if the hypotheses you stated properly reflect your research question, the variable family_firm is irrelevant, and so are any observations in your data set with family_firm == 0. The solution in this case is to remove all references to family_firm from the model.

reg roa CEO_founder CEO_descendant  log_emp salesgrowth capitalstructure log_firmsize log_firmage risk investments Blockholders EquityBased i.sic2digits i.state1 i.fyear, robust

Actually, you would be better off eliminating your homebrew indicator ("dummy") variables and use factor-variable notation with the original 3-level ceo variable instead:

reg roa i.ceo  log_emp salesgrowth capitalstructure log_firmsize log_firmage risk investments Blockholders EquityBased i.sic2digits i.state1 i.fyear, robust

On the other hand, if your research question involves understanding (simultaneously) the effect of ceo type and a contrast between family-firms and non-family firms, then your code is fine (although here I would even more strongly urge you to drop the homebrew indicator and interaction terms and use factor variable notation) but your data set is incapable of supporting such an analysis. You would need instead to provide non-missing values for the ceo type variable for the non-family firms. I suppose that the most sensible way to do that would be to set ceo to professional in most of these cases, or to founder if the ceo was the firm's founder (even if it was never a family firm). But I don't work in finance or economics, and that is a substantive question, not a statistical one--so my answer is just what an educated layman might say about this based on experience and common sense. Really, you should think about how such a designation would work with respect to your hypotheses, and then, if in doubt, consult a colleague in your discipline.

[Lucas Venn]

Dear dr. Schechter,


Very clear answer to me, which is highly appreciated; thanks a lot!
Indeed, the latter is what I want to do. Therefore, the mistake was indeed the dataset. Intuitively it is correct to put the CEO's of non-family firms to professionals so I did now.
In that case, if I understand correctly, you would suggest to use such a model instead of my current model:

reg roa familyfirm i.ceo log_emp salesgrowth other control variables, robust

Whereas my professor wants me to use the interaction term. Why do you think without the interaction term is better? (Just for improving my understandings)


Best regards,
Lucas van de Ven

[Clyde Schechter]

No, that's not what I meant. I said to remove the interaction terms from the model because they included family firm, and this was in the context of a data set where there were no values for ceo among the non-family firms.

But you are no longer in that context. You have changed the ceo variable so that it applies to both family and non-family firms. So now it is quite possible to put family firm and the interactions back into the model. So most likely you need is:

reg roa i.familyfirm##i.ceo log_emp salesgrowth other control variables, robust

This is going back to your original model--but with the data set fixed it is now possible to estimate it. So you can go ahead and do this.

The only reservation I have now is that your stated hypotheses from #1 do not make any use of the results from the non-family firms. So you need to expand your hypotheses to include some comparisons between family and non-family firms, and, ideally, specifically some contrast of the effect of different CEO types in the family vs non-family firm contexts. Once you do that, it appears that I am in agreement with your professor.

[Lucas Venn]

Dear dr. Schechter,

I tested the model and adjusted my hypothesis. The only thing I conceptually do not yet understand is what is the difference of also including the family firm and ceo variable seperately or not?
So:
reg roa familyfirm ceo i.familyfirm##i.ceo
versus
reg roa i.familyfirm##i.ceo

[Carlo Lazzaro]

Lucas:
your first cide is redundant, as it includes the conditional main effects for -familyfirm- and -ceo- twice:

. use "C:\Program Files\Stata17\ado\base\a\auto.dta"
(1978 automobile data)

. reg price i.foreign i.rep78 i.foreign##i.rep78
note: 1.foreign#1b.rep78 identifies no observations in the sample.
note: 1.foreign#2.rep78 identifies no observations in the sample.
note: 1.foreign#5.rep78 omitted because of collinearity.

      Source |       SS           df       MS      Number of obs   =        69
-------------+----------------------------------   F(7, 61)        =      0.39
       Model |    24684607         7  3526372.43   Prob > F        =    0.9049
    Residual |   552112352        61  9051022.16   R-squared       =    0.0428
-------------+----------------------------------   Adj R-squared   =   -0.0670
       Total |   576796959        68  8482308.22   Root MSE        =    3008.5

-------------------------------------------------------------------------------
        price | Coefficient  Std. err.      t    P>|t|     [95% conf. interval]
--------------+----------------------------------------------------------------
      foreign |
     Foreign  |   2088.167   2351.846     0.89   0.378     -2614.64    6790.974
              |
        rep78 |
           2  |   1403.125   2378.422     0.59   0.557    -3352.823    6159.073
           3  |   2042.574   2204.707     0.93   0.358    -2366.011    6451.159
           4  |   1317.056   2351.846     0.56   0.578    -3385.751    6019.863
           5  |       -360   3008.492    -0.12   0.905    -6375.851    5655.851
              |
foreign#rep78 |
   Foreign#1  |          0  (empty)
   Foreign#2  |          0  (empty)
   Foreign#3  |  -3866.574   2980.505    -1.30   0.199    -9826.462    2093.314
   Foreign#4  |  -1708.278   2746.365    -0.62   0.536    -7199.973    3783.418
   Foreign#5  |          0  (omitted)
              |
        _cons |     4564.5   2127.325     2.15   0.036      310.651    8818.349
-------------------------------------------------------------------------------

. reg price  i.foreign##i.rep78
note: 1.foreign#1b.rep78 identifies no observations in the sample.
note: 1.foreign#2.rep78 identifies no observations in the sample.
note: 1.foreign#5.rep78 omitted because of collinearity.

      Source |       SS           df       MS      Number of obs   =        69
-------------+----------------------------------   F(7, 61)        =      0.39
       Model |    24684607         7  3526372.43   Prob > F        =    0.9049
    Residual |   552112352        61  9051022.16   R-squared       =    0.0428
-------------+----------------------------------   Adj R-squared   =   -0.0670
       Total |   576796959        68  8482308.22   Root MSE        =    3008.5

-------------------------------------------------------------------------------
        price | Coefficient  Std. err.      t    P>|t|     [95% conf. interval]
--------------+----------------------------------------------------------------
      foreign |
     Foreign  |   2088.167   2351.846     0.89   0.378     -2614.64    6790.974
              |
        rep78 |
           2  |   1403.125   2378.422     0.59   0.557    -3352.823    6159.073
           3  |   2042.574   2204.707     0.93   0.358    -2366.011    6451.159
           4  |   1317.056   2351.846     0.56   0.578    -3385.751    6019.863
           5  |       -360   3008.492    -0.12   0.905    -6375.851    5655.851
              |
foreign#rep78 |
   Foreign#1  |          0  (empty)
   Foreign#2  |          0  (empty)
   Foreign#3  |  -3866.574   2980.505    -1.30   0.199    -9826.462    2093.314
   Foreign#4  |  -1708.278   2746.365    -0.62   0.536    -7199.973    3783.418
   Foreign#5  |          0  (omitted)
              |
        _cons |     4564.5   2127.325     2.15   0.036      310.651    8818.349
-------------------------------------------------------------------------------

.