Sufficient number of observations in poisson with FE

[Klaus Klausen]

Hi,

I am analyzing what determines the number of press releases published by an acquiring firm around the announcement of an acquisition.
Each observation of my datasets relates to one deal.
I am estimating the following model including industry and time dummies (i.Acq_FF12 and i.Deal_Announced_Year accordingly):

nbreg IM_Offsetting ///
CEO_Age CEO_tenure CEO_Gender CEO_duality CEO_No_Boardsitze CEO_own_rel1 /// CEO controls
Deal_AllCash Deal_Stock Deal_Value_rel Targ_Listing_bin FF12_Div /// Deal level controls
Acq_Size_MV42_ln Acq_Lev3 Acq_TobinsQ_WWU Acq_Size_MB_ratio Acq_Cash_hold Acq_ROA Acq_MA_exp Acq_Boardsize /// Acquirer level controls
c.CEO_Opt_ratio_ln##c.CNS ///
i.Acq_FF12 i.Deal_Announced_Year if CEO_tenure >= 1, vce(robust)

Negative binomial regression                           Number of obs =     258
                                                       Wald chi2(41) = 1482.20
Dispersion: mean                                       Prob > chi2   =  0.0000
Log pseudolikelihood = -140.29351                      Pseudo R2     =  0.2415

-------------------------------------------------------------------------------------------------------------------------------------
                                                                    |               Robust
                                                      IM_Offsetting | Coefficient  std. err.      z    P>|z|     [95% conf. interval]
--------------------------------------------------------------------+----------------------------------------------------------------
                                                            CEO_Age |   .0114709   .0214966     0.53   0.594    -.0306616    .0536035
                                                         CEO_tenure |   .0308632   .0282695     1.09   0.275    -.0245439    .0862704
                                                         CEO_Gender |  -1.913658   .4889472    -3.91   0.000    -2.871977   -.9553396
                                                        CEO_duality |   .3984847   .3088063     1.29   0.197    -.2067645    1.003734
                                                  CEO_No_Boardsitze |   .0899909   .0484129     1.86   0.063    -.0048967    .1848785
                                                       CEO_own_rel1 |   .1441658   .0541012     2.66   0.008     .0381294    .2502022
                                                       Deal_AllCash |  -1.247625   .5766768    -2.16   0.031    -2.377891   -.1173597
                                                         Deal_Stock |   .0038307   .6212151     0.01   0.995    -1.213729     1.22139
                                                     Deal_Value_rel |  -1.701057   .8456556    -2.01   0.044    -3.358512   -.0436024
                                                   Targ_Listing_bin |  -.2799149   .3092228    -0.91   0.365    -.8859804    .3261507
                                                           FF12_Div |   .6632142   .2639036     2.51   0.012     .1459727    1.180456
                                                   Acq_Size_MV42_ln |   .2066513   .1731989     1.19   0.233    -.1328124    .5461149
                                                           Acq_Lev3 |   1.219381   .8676654     1.41   0.160    -.4812125    2.919974
                                                    Acq_TobinsQ_WWU |   .0168431   .1312263     0.13   0.898    -.2403558     .274042
                                                  Acq_Size_MB_ratio |   1.254243   1.447275     0.87   0.386    -1.582365    4.090851
                                                      Acq_Cash_hold |  -.1102091   .8738334    -0.13   0.900    -1.822891    1.602473
                                                            Acq_ROA |    1.80115   2.912082     0.62   0.536    -3.906425    7.508725
                                                         Acq_MA_exp |   -.006235   .0390778    -0.16   0.873    -.0828261    .0703561
                                                      Acq_Boardsize |  -.1411429   .1003425    -1.41   0.160    -.3378106    .0555248
                                                   CEO_Opt_ratio_ln |  -.1097507   .1204086    -0.91   0.362    -.3457473    .1262459
                                                                CNS |   .9912636   .5385647     1.84   0.066    -.0643039    2.046831
                                                                    |
                                           c.CEO_Opt_ratio_ln#c.CNS |   .4086796   .2167709     1.89   0.059    -.0161836    .8335428

with:
IM_Offsetting : Number of acquirer published press releases in a -1+1 window around the announcement date of a deal
CNS: A CEO narcissism indicator (Variable of interest)
CEO_Opt_ratio_ln: Measurement for a CEO's confidence in the deal
In the model above I am interested in the interaction terms effect.

I am applying a negative binomial regression model as the analysis features a nonnegative count dependent variable with a large number of zeros (deals in which no offsetting press release was published). A negative binomial model is chosen over a Poisson model as the data of the IM variable displays overdispersion.

As a robustness test, I would like to run a model using firm fixed effects in order to control for unobserved time-invariant firm-level factors. Existing literature strongly suggests not to use xtnbreg with the fe option but go with a poisson fe model instead. For a recent discussion see: https://www.statalist.org/forums/for...cave-iteration

Following this advice, I would construct the following fe model instead

xtset Acq_Firm_ID ///Set Acquiring firm ID as panel variable

xtpoisson IM_Offsetting ///
CEO_Age CEO_tenure CEO_Gender CEO_duality CEO_No_Boardsitze CEO_own_rel1 ///
Deal_AllCash Deal_Stock Deal_Value_rel Targ_Listing_bin FF12_Div ///
Acq_Size_MV42_ln Acq_Lev3 Acq_TobinsQ_WWU Acq_Size_MB_ratio Acq_Cash_hold Acq_ROA Acq_MA_exp Acq_Boardsize ///
c.CEO_Opt_ratio_ln##c.CNS ///
if CEO_tenure >= 1, fe robust

Conditional fixed-effects Poisson regression        Number of obs    =      76
Group variable: Acq_Boardex_ID                      Number of groups =      27

                                                    Obs per group:
                                                                 min =       2
                                                                 avg =     2.8
                                                                 max =       9

                                                    Wald chi2(22)    = 8409.58
Log pseudolikelihood = -23.430424                   Prob > chi2      =  0.0000

                                     (Std. err. adjusted for clustering on Acq_Boardex_ID)
------------------------------------------------------------------------------------------
                         |               Robust
           IM_Offsetting | Coefficient  std. err.      z    P>|z|     [95% conf. interval]
-------------------------+----------------------------------------------------------------
                 CEO_Age |    .341687   .6398362     0.53   0.593    -.9123689    1.595743
              CEO_tenure |   1.006754   .4366841     2.31   0.021     .1508688    1.862639
              CEO_Gender |  -7.330783   1.964203    -3.73   0.000    -11.18055   -3.481017
             CEO_duality |   19.07419   11.68457     1.63   0.103    -3.827141    41.97553
       CEO_No_Boardsitze |  -1.934068   1.254498    -1.54   0.123    -4.392839    .5247028
            CEO_own_rel1 |   .2044255   1.006215     0.20   0.839     -1.76772    2.176571
            Deal_AllCash |  -2.585404   2.598665    -0.99   0.320    -7.678694    2.507885
              Deal_Stock |  -1.142069    3.09576    -0.37   0.712    -7.209646    4.925508
          Deal_Value_rel |  -7.321167   2.488111    -2.94   0.003    -12.19778   -2.444559
        Targ_Listing_bin |  -1.303236   .7242266    -1.80   0.072    -2.722694    .1162222
                FF12_Div |   .7240413   .9343113     0.77   0.438    -1.107175    2.555258
        Acq_Size_MV42_ln |  -8.072457   4.735665    -1.70   0.088    -17.35419    1.209275
                Acq_Lev3 |  -14.61711   8.362594    -1.75   0.080    -31.00749    1.773272
         Acq_TobinsQ_WWU |    3.85544   3.861568     1.00   0.318    -3.713093    11.42397
       Acq_Size_MB_ratio |  -1.780407   1.662121    -1.07   0.284    -5.038104    1.477291
           Acq_Cash_hold |  -7.416765   7.661377    -0.97   0.333    -22.43279    7.599258
                 Acq_ROA |  -96.71658   72.00123    -1.34   0.179    -237.8364    44.40322
              Acq_MA_exp |   .8768758   .3977481     2.20   0.027     .0973039    1.656448
           Acq_Boardsize |  -.1360496   .4086773    -0.33   0.739    -.9370424    .6649432
        CEO_Opt_ratio_ln |  -.4200439   .7934073    -0.53   0.597    -1.975094    1.135006
                     CNS |  -6.696853   2.802306    -2.39   0.017    -12.18927   -1.204434
                         |
c.CEO_Opt_ratio_ln#c.CNS |   1.478188   1.318108     1.12   0.262    -1.105256    4.061632
------------------------------------------------------------------------------------------

According to the possion fe model there is no significant relationship between IM_Offsetting and the interaction term (which is contradictory to my main model).
However, I am concerned whether a model including firm fixed effects is valid in the first place as I might have too few observations (or more specifically too few firms in my sample with more than one deal) leaving me with only 76 observations. I would be happy to receive feedback on whether this is a valid concern. In this case, I would write in my work that a model including firm fixed effects can't be applied due to non-sufficient within variation.

[George Ford]

As long as you use robust, you can use Poisson (it addresses the dispersion issue).

Note the big change in CNS coef between them. I bet CNS is nearly colinear with the fixed effect.

Margins might be useful in this case since both terms are continuous.

[Klaus Klausen]

Thank you.

Note the big change in CNS coef between them. I bet CNS is nearly colinear with the fixed effect.

I have not much experience with fixed effects yet and am therefore not sure about the consequences of this statement. Does this imply a misspecified model or that fixed effects estimation is not sufficient here (or both)?

Margins might be useful in this case since both terms are continuous.

Would you mind telling me how the margins command helps in this regard?

[George Ford]

If CNS was perfectly colinear, then you would not get a coef on CNS. But I bet in almost all cases it is the same, which is why the coef is so sensitive.

as a start, try

tabstat CNS, by(Acq_Firm_ID) stats(mean sd min max)

on margins, see

https://www.stata.com/stata-news/news32-1/spotlight/

[Klaus Klausen]

Thanks. This is the output for the 27 groups (firms) in my sample considered in the model.

Summary for variables: CNS
Group variable: Acq_PermID (Acquiror PermID)

Acq_PermID |      Mean        SD       Min       Max
-----------+----------------------------------------
4295899452 | -.4227434  .0103658 -.4300731 -.4154136
4295903382 |  .6335967  .1898018  .4104524  1.044316
4295903594 | -.2737113  .0951125 -.4603471 -.2017401
4295903637 |  .3134485  .4247876 -.1761779   .583639
4295903707 |  .3467153  .6592092  -.119416  .8128466
4295903798 | -.3379688  .2445955 -.5109239 -.1650136
4295904307 |  1.356645  .1027358  1.268339  1.469399
4295904886 |  -.208847         0  -.208847  -.208847
4295905055 |  .3461898   .301389  .1330756   .559304
4295906238 | -.4417125  .1276466 -.5319723 -.3514528
4295906509 | -.4890333  .1689981  -.608533 -.3695337
4295906803 | -.3667303  .2052953  -.511896 -.2215646
4295907365 |   1.83837  .2826245  1.638524  2.038216
4295907485 | -1.174897  .0108392 -1.182561 -1.167232
4295908005 |  .7147319   .029279  .6940285  .7354352
4295908174 | -.0154643  .0889538 -.0783641  .0474356
4295912155 |  1.083852  .1660258  .8926109  1.191077
4295912188 | -.2370997   .013375 -.2465573 -.2276422
4295912239 |  1.123284         0  1.123284  1.123284
4295912319 |  .5653847  .2025683  .4221472  .7086221
4296658618 | -.0505468  .0482556 -.0857877  .0023146
5000014741 |  1.005158  .0206244  .9905743  1.019742
5000072036 |  .6893137  .1649859   .534082   .862573
5037613143 |  .1975406   .170624   .015561  .3539149
5052137706 |  .7511886         0  .7511886  .7511886
5060689053 | -.1156397  .1709074 -.2364895  .0052101
5080018615 | -.4668972  .1389773 -.5668625 -.2829587
-----------+----------------------------------------
     Total |  .2389086  .6612735 -1.182561  2.038216
----------------------------------------------------

Considering the mean of .239 and a SD .661 doesn't really seem that CNS is the same in most cases or do I misunderstand something?

[George Ford]

not in most cases. 3 cases for sure. small sd in several cases.

run the poisson without the fe.

[Klaus Klausen]

My results remain the same for all 4 hypotheses when applying a poisson model instead of nbreg.
I was wondering what a convincing argument is for why I only apply poisson with fe as a robustness test for three out of my four models. For the model discusses here, could one argue that there is not sufficient within variation for a firm fixed effect model as there are only 27 firms that had more than one deal?

[George Ford]

if you've got a bunch of firms with only 1 observation, then don't use FE. You are essentially dropping those observations.

https://www.statalist.org/forums/forum/general-stata-discussion/general/1338614-country-fixed-effect-identified-inspite-just-one-observation

[Klaus Klausen]

I am aware that these single observations will be dropped when applying fe. However, I am curious whether it is a common approach to apply a fe model as a robustness test even if one is not able to apply it to every model (due to too many single observations). In my case, I would simply state that I use fe to test the robustness of three out of my four models but can't use it on the fourth due to data limitations.

[George Ford]

Not sure it's a check on robustness when you drop most of your observations (explaining, in part, the loss of significance). Or, run the model without FE and exclude the singularities and see how different things look between those two, which is comparable. You can run the FE first then restrict the non-FE model to if e(sample).

I'd just go with the Poisson in your situation.