Interpretation of Moderation effect

[Jana Schue]

Hello everyone,
I am struggling a bit with interpreting my moderation effect. The questions relate to the effects of my independent variable gender_diversity and is marked in the following.

In my regression the relationship between my variables of interest is positive:
see:
Code:

 xtreg ln_totalplatact educational_diversity gender_diversity background_diversity tenure_diversity itraffic growth  firm_age TMT_size i.year , fe vce(robust)

Results:

Fixed-effects (within) regression               Number of obs     =        703
Group variable: group_id                        Number of groups  =        172

R-squared:                                      Obs per group:
     Within  = 0.6118                                         min =          1
     Between = 0.0967                                         avg =        4.1
     Overall = 0.2024                                         max =          7

                                                F(13,171)         =     138.48
corr(u_i, Xb) = 0.0245                          Prob > F          =     0.0000

                                      (Std. err. adjusted for 172 clusters in group_id)
---------------------------------------------------------------------------------------
                      |               Robust
      ln_totalplatact | Coefficient  std. err.      t    P>|t|     [95% conf. interval]
----------------------+----------------------------------------------------------------
educational_diversity |  -.2721108   .6413871    -0.42   0.672    -1.538167    .9939449
     gender_diversity |   2.533689   1.103264     2.30   0.023     .3559193    4.711459
 background_diversity |   3.274338    1.63843     2.00   0.047     .0401857     6.50849
     tenure_diversity |   1.071932    .551466     1.94   0.054    -.0166253    2.160489
             itraffic |   .1824114   .1852301     0.98   0.326    -.1832205    .5480434
               growth |   .0975318   .0733507     1.33   0.185    -.0472576    .2423213
             firm_age |   .4748669   .0791355     6.00   0.000     .3186586    .6310751
             TMT_size |  -.0077124   .0919141    -0.08   0.933    -.1891448      .17372
                      |
                 year |
                2015  |   .4241251    .160963     2.63   0.009     .1063948    .7418554
                2016  |   .6108117   .1655276     3.69   0.000     .2840711    .9375523
                2017  |   1.097878   .2248011     4.88   0.000     .6541359    1.541621
                2018  |   1.404356   .3116293     4.51   0.000     .7892205    2.019492
                2019  |   1.446496   .3079919     4.70   0.000     .8385399    2.054451
                2020  |          0  (omitted)
                      |
                _cons |   7.330491   2.492535     2.94   0.004     2.410391    12.25059
----------------------+----------------------------------------------------------------
              sigma_u |  3.2468355
              sigma_e |  1.0495547
                  rho |  .90539232   (fraction of variance due to u_i)
---------------------------------------------------------------------------------------

If I am performing my moderation I get confusing results.
The direct effect turns negative and the moderation effect turns positive. Can anyone help me with the interpretation of these results?


Code:

 xtreg ln_totalplatact c.educational_diversity##c.dynamism c.gender_diversity##c.dynamism c.background_diversity##c.dynamism c.tenure_diversity##c.dynamism total_countries growth TMT_size itraffic i.year , fe vce(robust)

Results:

Fixed-effects (within) regression               Number of obs     =        367
Group variable: group_id                        Number of groups  =        152

R-squared:                                      Obs per group:
     Within  = 0.8148                                         min =          1
     Between = 0.4025                                         avg =        2.4
     Overall = 0.3512                                         max =          3

                                                F(15,151)         =     162.92
corr(u_i, Xb) = 0.3930                          Prob > F          =     0.0000

                                                   (Std. err. adjusted for 152 clusters in group_id)
----------------------------------------------------------------------------------------------------
                                   |               Robust
                   ln_totalplatact | Coefficient  std. err.      t    P>|t|     [95% conf. interval]
-----------------------------------+----------------------------------------------------------------
             educational_diversity |   .0824904   .7078749     0.12   0.907    -1.316128    1.481109
                          dynamism |   -1.18239   2.170082    -0.54   0.587    -5.470036    3.105257
                                   |
c.educational_diversity#c.dynamism |  -.2313091   2.239248    -0.10   0.918    -4.655613    4.192995
                                   |
                  gender_diversity |  -.7022033   .4084872    -1.72   0.088    -1.509292    .1048852
                          dynamism |          0  (omitted)
                                   |
     c.gender_diversity#c.dynamism |   4.326052   2.070952     2.09   0.038     .2342666    8.417837
                                   |
              background_diversity |   .8788531   .7945476     1.11   0.270    -.6910132    2.448719
                          dynamism |          0  (omitted)
                                   |
 c.background_diversity#c.dynamism |   2.179874   2.998382     0.73   0.468    -3.744326    8.104073
                                   |
                  tenure_diversity |   .5108906   .3411474     1.50   0.136    -.1631481    1.184929
                          dynamism |          0  (omitted)
                                   |
     c.tenure_diversity#c.dynamism |   .3655461   1.528332     0.24   0.811    -2.654131    3.385223
                                   |
                   total_countries |   .0261214   .0089363     2.92   0.004     .0084651    .0437778
                            growth |   .0417982   .0596745     0.70   0.485    -.0761065    .1597029
                          TMT_size |   .0089453   .0646107     0.14   0.890    -.1187124     .136603
                          itraffic |   .1905962   .1421331     1.34   0.182    -.0902302    .4714226
                                   |
                              year |
                             2019  |   .2450912   .0985784     2.49   0.014     .0503201    .4398623
                             2020  |  -.8784332   .2218638    -3.96   0.000    -1.316792   -.4400749
                                   |
                             _cons |   11.45734   2.029945     5.64   0.000     7.446572     15.4681
-----------------------------------+----------------------------------------------------------------
                           sigma_u |  3.2363045
                           sigma_e |  .39452096
                               rho |  .98535683   (fraction of variance due to u_i)
----------------------------------------------------------------------------------------------------

I am confused since the coefficients change from positive to negative and back.


Thanks in advance and best regards,
Jana

[Carlo Lazzaro]

Jana:
on your case, -gender_diversity- rules without interaction (1st code; -dynamism- is not included though) and is outperformed by the interaction in your second code. Nothing sinister; two different models, two different set of coefficients.
That said:
1) the number of observations in your second code is about 50% vs first code. I guess it's matter of missing values and/or omission due to perfect collinearity. Be as it may, you cannot safely compare those two regressions;
2) most of your interactions (2nd code) are not that informative and probably you can safely get rid of them;
3) the within R_sq is pretty high in both your regression (and that is good) and there's an apparent evidence of a panel-wise effect (sigma_u vs sigma_e; and that is also good).

[Jana Schue]

Hey Carlo,

thanks for your explanations and help!
You are right the observations for the second code are 50% because of missing values for the moderating variable.

Now my question is: Am I right in interpreting the results: With increasing dynamism (moderator) the positive effect between the basic relationship of gender diversity and firm performance (Model 1) is increased?

or

Increasing dynamism (moderator) positively influences the relationship between gender diversity and firm performance.



Best regards
Jana

[Carlo Lazzaro]

Jana:
I would sponsor your second take that refers to Model 2.
As per you results (Model 2) -gender_diversity- when adjusted for the remaining predictors, does not explain variations in the regressand (and the same holds for -dynamism-, as it is omitted due to collinearity).
However, the interaction with -dynamism- seems to boost things up.
That said:
1) I fail to get how -gender_diversity- is coded as a continuous variable;
2) I assume that -dynamism- is a continuous variable;
3) beware in coefficient interperetation as you ar4e dealing with a log-linear regression:
4) you can use -lincom- and/or -margin- to retrieve more details about your results.

[Jana Schue]

Hey Carlo,

thanks for your explanations and clarification.
Dynamism is just committed due to its multiple occurrences as moderator in the data. But your are right that it is insignificant.
Since gender diversity in Model has a p-value of 0.088 it can no explain variations in the regressant. I am right?
Then can I still take may statement: "Increasing dynamism (moderator) positively influences the relationship between gender diversity and firm performance."
Or is it false due to the insignificant direct relationship?

to
1) The gender_diversity is calculated with the Herfindal Hirshman index (also called Blau index)
2) You are right that dynamism is a continuous variable
4) I already plotted the results with margin. Thanks.

Best regards,
Jana

[Carlo Lazzaro]

Jana:
thanks for clarifying.
1) Since gender diversity in Model has a p-value of 0.088 it can no explain variations in the regressant. I am right? Correct.
2) you can still state that: "Increasing dynamism (moderator) positively influences the relationship between gender diversity and, as a consequence, firm performance (when adjusted for the other predictors)."

[Jana Schue]

Hey Carlo,

Sorry for asking again. But I am not sure what you mean by "(when adjusted for the other predictors)". In which way should I adjust the predictors?
Could you clarify this once more for me?

Thanks in advance and best regards,
Jana

[Carlo Lazzaro]

Jana:
no worries: it's a traditional phrasing.
In all the regressions:
1) the expected value of the regressand is conditional on the predictors (different predictors produce different values of the fitted values):
2) the value of each coefficient is conditional on the other predictors:

. use "https://www.stata-press.com/data/r17/nlswork.dta"
(National Longitudinal Survey of Young Women, 14-24 years old in 1968)


. xtreg ln_wage c.age##c.age, fe vce(cluster idcode)

Fixed-effects (within) regression               Number of obs     =     28,510
Group variable: idcode                          Number of groups  =      4,710

R-squared:                                      Obs per group:
     Within  = 0.1087                                         min =          1
     Between = 0.1006                                         avg =        6.1
     Overall = 0.0865                                         max =         15

                                                F(2,4709)         =     507.42
corr(u_i, Xb) = 0.0440                          Prob > F          =     0.0000

                             (Std. err. adjusted for 4,710 clusters in idcode)
------------------------------------------------------------------------------
             |               Robust
     ln_wage | Coefficient  std. err.      t    P>|t|     [95% conf. interval]
-------------+----------------------------------------------------------------
         age |   .0539076    .004307    12.52   0.000     .0454638    .0623515
             |
 c.age#c.age |  -.0005973    .000072    -8.30   0.000    -.0007384   -.0004562
             |
       _cons |    .639913   .0624195    10.25   0.000     .5175415    .7622845
-------------+----------------------------------------------------------------
     sigma_u |   .4039153
     sigma_e |  .30245467
         rho |  .64073314   (fraction of variance due to u_i)
------------------------------------------------------------------------------

. xtreg ln_wage c.age, fe vce(cluster idcode)

Fixed-effects (within) regression               Number of obs     =     28,510
Group variable: idcode                          Number of groups  =      4,710

R-squared:                                      Obs per group:
     Within  = 0.1026                                         min =          1
     Between = 0.0877                                         avg =        6.1
     Overall = 0.0774                                         max =         15

                                                F(1,4709)         =     884.05
corr(u_i, Xb) = 0.0314                          Prob > F          =     0.0000

                             (Std. err. adjusted for 4,710 clusters in idcode)
------------------------------------------------------------------------------
             |               Robust
     ln_wage | Coefficient  std. err.      t    P>|t|     [95% conf. interval]
-------------+----------------------------------------------------------------
         age |   .0181349   .0006099    29.73   0.000     .0169392    .0193306
       _cons |   1.148214   .0177153    64.81   0.000     1.113483    1.182944
-------------+----------------------------------------------------------------
     sigma_u |  .40635023
     sigma_e |  .30349389
         rho |  .64192015   (fraction of variance due to u_i)
------------------------------------------------------------------------------

.

[Jana Schue]

Hey Carlo,

thanks for the explanation.
So if I understand correctly I do not have to change anything. It is just a phrase you would add to my original statement?

Best regards,
Jana

[Carlo Lazzaro]

Correct!