Interpretation Magnitude Interaction Two Continuous Variables Economic Significance

[Maxence Morlet]

Hi all,

I have read a few threads regarding this topic (e.g. https://www.statalist.org/forums/for...uous-variables) but have not yet found an answer to my question. Apologies if the identical question has already been answered.

I have a count dependent variable (Y) (linear TWFE results are shown, but marginal effects from Poisson are virtually identical), which I have regressed on two continuous variables (X1 and X2) and their interaction.

The summary statistics of the variables are:

    Variable |        Obs        Mean    Std. dev.       Min        Max
-------------+---------------------------------------------------------
  Y |    373,763    .1933016    2.363418          0        502
X1 |    373,763    .5695304    .6594848          0      4.364
X2 |    373,763   -2246.553     773.906   -3651.53    316.896

These are the results:

HDFE Linear regression                            Number of obs   =    373,763
Absorbing 3 HDFE groups                           F( 141,   7317) =    1955.95
Statistics robust to heteroskedasticity           Prob > F        =     0.0000
                                                  R-squared       =     0.0578
                                                  Adj R-squared   =     0.0292
                                                  Within R-sq.    =     0.0016
Number of clusters (token1)  =      7,318         Root MSE        =     2.3286

                                                       (Std. err. adjusted for 7,318 clusters in ID)
--------------------------------------------------------------------------------------------------------
                                       |               Robust
                            Y | Coefficient  std. err.      t    P>|t|     [95% conf. interval]
---------------------------------------+----------------------------------------------------------------
                        X1 |  -.5015123   .1355351    -3.70   0.000    -.7672003   -.2358244
                                       |
c.X1#c.X2 |  -.0002419   .0000586    -4.13   0.000    -.0003567    -.000127

//The coefficient on X2 alone is perfectly collinear with the fixed-effects

I am fully aware that the economic significance of a coefficient depends on the field, research question, etc. However, what is the methodology / logic to assess the economic significance of an interaction term between two continuous variables?

For instance, one often compares the magntiude of the coefficient on a dummy variable to the mean of the dependent variable to assess economic significance, and whether the effect is large enough to be "interesting". Similarly, what would one compare the coefficient on c.X1#c.X2 to in order to assess its magnitude?

Please let me know if my question is unclear, I would happy to rephrase it.

[Maarten Buis]

The not very helpful answer is that you just understand what the coefficient means, and once you understand what it means its (ir)relevance becomes self-evident.

For the interaction effect, it tells that for a unit change in X2, the effect of X1 changes by-.0002. That does not look like a lot, but X2 ranges from -3652 to 317, so a unit change in X2 is a very small changes. It is not surprising that small changes have small effects. You might want to consider rescaling X2, such that the unit is not so small. At this point we need to know more about X2 to interpret if this is big or small. That is where you know more than we do.

[Maxence Morlet]

Thanks for the helpful response!

So you would use the range of the variable in question to assess economic significance?

I gave a specific example for concreteness, my next question would be: in general methodology, what moments of which variables' distributions / variables' ranges would you use to evaluate economic significance of such an interaction term (two continuous variables)?

[Maarten Buis]

So you would use the range of the variable in question to assess economic significance?

No, in general I use is the meaning of the unit. Since I obviously don't know what the unit of X2 is, I used the range as an indicator that the unit is probably small. But you know what X2 is, so you can, and should, decide for yourself whether the unit is small or large. Than taking into account the unit of X1 and Y, you can decide whether the interaction effect is large or small. As I said before, you "just" understand what the coefficient means, then the meaning becomes self-evident.

Remember, that the term "economic significance" is just a fancy word for "your subjective assessment of whether you think and effect is large or small". People don't like to admit that something is subjective in a paper, so they hide that behind this word. However, that does not change the cold hard fact that deciding whether something is "large" or "small" is at its very core subjective. So don't look for some magic statistic (like a range or a moment) that will give you THE answer; it does not exist. You just look, you understand, you know about the context, and than you decide. If some gatekeeper (like an advisor, referee, editor) makes a different decision, then that is tough luck for you.

[Maxence Morlet]

Thanks for the guidance and recommendation!