interaction term in regression

[Guest]

Hello everybody,

I have a question regarding regression's (statistical) interpretation with an interaction term. I find it difficult to see what can ultimately be said from the regression. I want to measure the patent output (Patents_w) of venture capital activities (VC_lag1) under the moderating influence of a founder CEO (dummy variable; 1=Founder CEO, 0=not a founder). For this purpose, I have created the following negative binomial regression.

xtnbreg Patents_w VC_lag1 FounderCEO c.VC_lag1#1.FounderCEO  firm_size_loglag1 xrdat_wlag1 industry_q_wlag1 os_w_lag1 i.fyear i.GSECTOR, fe

From this, I get the following results. Now I know that the interpretation is not easy with interaction terms, and one probably uses the margins command. My question is
a) it is possible to make a general statement regarding the interaction term? Because FounderCEO#c.VC_lag1 is highly non-significant?
b) what is the basic interpretation of margins command?
c) So the interpretation would be that if Founder CEO=0, then the marginal effect would be that innovation output increases by the value .0065886, whereas with FounderCEO no statement is possible because there is no significance and the confidence interval shows no concrete swing in one direction?

        Patents_w | Coefficient  Std. err.      z    P>|z|     [90% conf. interval]
----------------------------+----------------------------------------------------------------
         VC_lag1  |   .0065886   .0032773     2.01   0.044     .0011978    .0119793
       FounderCEO |   .8644112   .1764986     4.90   0.000     .5740968    1.154726
FounderCEO#c.VC_lag1 |
               1  |  -.0003116   .0070244    -0.04   0.965    -.0118657    .0112425

Margins Output

 margins FounderCEO, dydx(VC_lag1) level(90)

Expression: Linear prediction, predict()
dy/dx wrt:  VC_lag1

--------------------------------------------------------------------------------
               |            Delta-method
               |      dy/dx   std. err.      z    P>|z|     [90% conf. interval]
---------------+----------------------------------------------------------------
       VC_lag1 |
    FounderCEO |
            0  |   .0065886   .0032773     2.01   0.044     .0011978    .0119793
            1  |    .006277   .0065364     0.96   0.337    -.0044745    .0170284
--------------------------------------------------------------------------------

Thanks a lot for the help!

[Clyde Schechter]

In assessing an interaction, you want to think about both its magnitude and the precision with which your data have enabled you to estimate it.

Well, your interaction coefficient is about 1/20th the size of the VC_lag1 coefficient, so even if it were statistically significant, it would be a pretty small modification of the VC effect. Whether that ~5% difference would be large enough to write home about, were it established with some real precision, is a substantive matter about which I have no basis to express an opinion. Presumably you, however, do.

But on top of that, that interaction coefficient is even smaller in comparison to its standard error. Or, more intuitively, looking at its 90% CI, your data are compatible with the true effect modification being anywhere between -.012 and +.011 (to three decimal places). Those two endpoints are, in magnitude, roughly twice the magnitude of the VC_lag1 coefficient itself, so the uncertainty here totally swamps that effect. You see this, perhaps, most clearly in the -margins-output where the marginal effect of VC_lag1 when FounderCEO = 1 has a confidence interval ranging from -.004 to +.017, which is a huge range considering that the data are only compatible with a marginal effect from 0.0012 to 0.012 when the FounderCEO = 0. In other words, the uncertainty of the effect modification by Founder CEO is much greater than the precision of the VC_lag1 marginal effect with FounderCEo = 0 itself.

If the goal of the study was to quantify the extent to which Founder CEO modifies the marginal effect of VC, the most appropriate summary would be: inconclusive. The effect modification ranges from much smaller to much larger than the effect (when FounderCEO = 0) itself.

[Guest]

Thank you very much for your reply!
What is still unclear to me is how to interpret the marginal effect in general?

Assumption: significance and confidence interval in the above example fit and allow interpretation. (So shall be disregarded for my question).

In this theoretical example, would the statement be the following?
If there is no Founder CEO (=0), the marginal effect of VC_lag1 on patent output is ≈0.0066 and is higher than the value of a Founder CEO. (Other formulation: if one unit VC_lag1 increases, then for non-founder CEOs patent output increases by ≈0.0066 and is thus higher than for founder CEOs).

Thus, non-founder CEOs achieve a higher effect in terms of innovation output through their venture capital activities?
(even if this is only very minimal in this example).

Thanks for your help!

[Clyde Schechter]

If there is no Founder CEO (=0), the marginal effect of VC_lag1 on patent output is ≈0.0066 and is higher than the value of a Founder CEO. (Other formulation: if one unit VC_lag1 increases, then for non-founder CEOs patent output increases by ≈0.0066 and is thus higher than for founder CEOs).

Well, in a hypothetical example where the confidence interval around the coefficient of the interaction term were narrow enough to reside entirely in negative territory (even if still of very small magnitude), then you would be OK concluding that the marginal effect of VC is greater when there is no Founder CEO.

But in your actual results, that is not the case. The confidence interval around the interaction coefficient extends far into both positive and negative territory. So the only sensible interpretation here is that we cannot tell whether the marginal effect of VC is greater or smaller or no different with a Founder CEO.