Interpreting negative binomial coefficient with log IV

[Jack Smith]

Hello,

I'm having a hard time trying to interpret the results of a negative binomial regression (xtnbreg) where the independent variable is log transformed (natural). Here it is L_grant_stockoptions.

I'm hesitating between two interpretations.

-A 1% change in L_grant_stockoptions leads to a -0.017 in log count of Acquisitions.
OR
-A 1% change in L_grant_stockoptions leads to a -0.017/100 change in Acquisitions unit.

Any help is appreciated

-------------------------------------------------------------------------------------
        Acquisitions |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
---------------------+----------------------------------------------------------------
L_Grant_stockoptions |  -.0173741   .0098044    -1.77   0.076    -.0365904    .0018422
         L_firm_size |   .2318761   .0536278     4.32   0.000     .1267676    .3369847
   L_stock_ownership |   .0799546   .0347109     2.30   0.021     .0119224    .1479867
        L_other_comp |   .0677387   .0455417     1.49   0.137    -.0215213    .1569987
             CEO_age |   -.017233   .0115617    -1.49   0.136    -.0398936    .0054276
          CEO_gender |   1.481501    .874864     1.69   0.090    -.2332005    3.196203
          CEO_tenure |   .0093143   .0106577     0.87   0.382    -.0115745    .0302031
             Company |   .0003142   .0019597     0.16   0.873    -.0035268    .0041552
               _cons |  -5.340459   1.612631    -3.31   0.001    -8.501157   -2.179761
---------------------+----------------------------------------------------------------
               /ln_r |   2.452073   .3523153                      1.761548    3.142599
               /ln_s |   .3758031    .188786                      .0057893     .745817
---------------------+----------------------------------------------------------------
                   r |    11.6124   4.091227                      5.821442    23.16399
                   s |    1.45616   .2749028                      1.005806    2.108163
--------------------------------------------------------------------------------------

[Clyde Schechter]

Your model is log-linear in L_Grant_stockoptions. So there is no answer to the question "what happens when L_Grant_stockoptions changes by 1%." That will depend on the value of L_Grant_stockoptions you start from.

I suspect what you really mean to ask is, "what happens when Grant_stockoptions (the non-log-transformed variable) changes by 1%?" That has an answer. When Grant_stockoptions differs by 1%, L_Grant_stockoptions, it's log transform, differs by log(1.01) = 0.00995 (approx.). Consequently log count Acquisitions differs by -0.017 * .00995, which, for practical purposes, is -0.017/100.

This result corresponds closely to the rule of thumb that when the independent variable is log-transformed in the model, a change of 1% in the non-log-transformed variable is associated with a beta% change in the outcome (in this case the outcome is log count acquisitions) where beta is the regression coefficient. As the above calculations show, this rule of thumb is only an approximation. If the regression coefficient is very large, the amount of error in the approximation can be appreciable. But for small coefficients such as in your case, it is good to at least 3 decimal places.

Finally, be careful about the use of causal language here. Unless your data come from a randomized experiment, you should not speak of anything "leading to" anything else. Rather, use causally neutral wording: a 1% difference in Grant_stock options is associated with a -0.017/100 difference in log count acquisitions.

[Jack Smith]

Thank you so much! And yes indeed I meant the Grant_stock!


Have an amazing day!