[Clyde Schechter]

First, you should not be looking at statistical significance. The American Statistical Association has recommended that this concept be abandoned and has devoted a full issue of its journal, The American Statistician to the topic. See https://www.tandfonline.com/doi/full...5.2019.1583913 for the lead summary article, and
https://www.tandfonline.com/toc/utas20/73/sup1 for the full issue (43 articles).

Your question about interpreting the -margins- output cannot be answered as stated because you have omitted crucial information. There are two different cases.

If your original regression included separate variables, one for the linear term and another for the quadratic, then the output of -margins- is simply incorrect and you have to go back and do it over using correct factor variable notation. The use of factor variable notation is absolutely required in order for -margins- to do correct calculations for models with interactions or with quadratic and higher order terms.

If you did use correct factor variable notation (c.Leverage##c.Leverage) to incorporate linear and quadratic leverage into your regression, then the output of -margins- correctly shows you the average marginal effect of Leverage taking into account both the linear and quadratic terms.

Do bear in mind that with non-linearities, such as quadratic terms, it is not meaningful to speak of "the" marginal effect of the variable: there is a different marginal effect corresponding to each value of the variable. So you can speak of marginal effects conditional on specific values of the variable (which you would have to specify using the -at()- option in -margins-) or you can speak of the average marginal effect, which is what you get from the code you show. But there is no such thing as "the marginal effect" of a variable that enters the model in a non-linear way.

Do I even need to mention the impact of these variables in isolation at all? I guess not!!

That's right. In fact, not only do you not need to mention them in isolation, you can't do so in any meaningful way. Only their joint effect is meaningful.

[mohina saxena]

Many Thanks Sir for giving such a nuanced explanation. Also, the reference that you mentioned is very interesting, I am going through it right now.

I indeed missed out in indicating some important points. Firstly, as you mentioned I have used factor variable notation (c.Leverage##c.Leverage) and secondly, my model is estimated using pooled OLS. And, yes, there cannot be "the" marginal effect in case of non-linearity unless stated with at option.

Therefore, with the current results, I can simply infer that the "average marginal effect" of leverage (as the joint effect) is negative on my outcome variable. Is it correct? I am getting little confused with interpretations in case of non-linear terms.



Thanking You,
regards,
Mohina

[Clyde Schechter]

I can simply infer that the "average marginal effect" of leverage (as the joint effect) is negative on my outcome variable. Is it correct?

Yes, that's correct. But remember that this average is taken over the distribution of all of the variables in your estimation sample. In a different sample, the distributions of the variables might be different, and the average marginal effect there could be different, even of the opposite sign. Average marginal effects can be a useful simple summary when the marginal effect doesn't vary much from one set of values of predictors to the next. But when you genuinely have a quadratic relationship, it implies that the marginal effect is negative for some values and positive for others. So if some future sample has a markedly different mix of values associated with positive effects and values associated with negative effects, then the average marginal effect will be different, possibly even of opposite sign.

In particular, the generalizability of this kind of finding is questionable. If your data sample was representative of the situations to which you would like to apply your conclusions, then noting that the average marginal effect is negative could be useful. But if you have only a convenience sample, attempting to apply your finding about the average marginal effect outside the sample is especially hazardous in this situation.

[mohina saxena]

Many many thanks again Professor. Yes, this conclusion should only be drawn pertaining to the sample and the distribution of underlying variables that I have in my data set. On a related note, I estimated another model where I use both Leverage, Leverage Square and perform interaction of leverage with Industries (3) as well as market structure in my model. I do it using factorial notations by:

reg Y c.Leverage##c.Leverage c.Leverage##c.Market_Structure  i.Industry#c.Leverage, vce(cluster Id)

So, my regression equation is: Y = alpha0 + alpha1Leverage +alpha2Leverage^2 + alpha3Levearge*Market Structure+ alpha4_Industry2*Leverage+alpha5_Industry3*Leverag e


While I perform this regression, I intended to get average marginal effect in each industry and therefore I use the following code:

margins i.Industry, dydx(Leverage)

My question is since the average market structure as well as leverage is different in each of these industries, would this code take into consideration such heterogeneity. I tried solving it manually (say for Industry 1) in which the average marginal effect should be: alpha1 + 2*alpha2*Leverage + alpha3*HHI (all computed at industry specific average). This gives me a different value than that indicated as coefficient in the output table. Am I missing out something in the code? Let me know if I should attach the data and results derived.
Moreover, is it appropriate to combine the square term and industry specific interaction of the same variable in one model?

I would be grateful if you can please clarify this.


regards,
Mohina

[Clyde Schechter]

My question is since the average market structure as well as leverage is different in each of these industries, would this code take into consideration such heterogeneity.

Perhaps not in the way you intend. The command you show takes into account the overall heterogeneity in your data sample, but the calculation for each specific industry uses the entire estimation sample; it is not restricted to that industry's observations. So if you want to have the different distributions of other variables in each industry reflected in the industry-specific marginal effects, you need a different command: -margins, dydx(Leverage) over(Industry)-

I tried solving it manually (say for Industry 1) in which the average marginal effect should be: alpha1 + 2*alpha2*Leverage + alpha3*HHI (all computed at industry specific average). This gives me a different value than that indicated as coefficient in the output table.

I can't really comment without knowing more details about the calculations, but by using industry specific averages you are calculating what sounds like the results for -margins, dydx(Leverage) over(Industry)-, not the results for -margins Industry, dydx(Leverage)-.

Moreover, is it appropriate to combine the square term and industry specific interaction of the same variable in one model?

Yes, it's OK. It's a little bit strange to not also include the industry-specific interactions with the quadratic leverage as well--but if you are pretty confident that the modification of the market structure effect by leverage is strictly linear, then this is OK. On the other hand, the i.Industry#c.Leverage term troubles me. The omission of quadratic leverage from this interaction is equivalent to a very peculiar constraint on how industry modifies the leverage effect: it says that in each industry the width of the outcome:leverage parabola is the same but the location is shifted. That's a very strong constraint and unless you have a really good theory to back it up, I would presume it's wrong. Also, if you're including an interaction with industry, you should also include industry itself. So I would revise this command to

reg Y i.Industry##c.Leverage##c.Leverage c.Leverage##c.Market_Structure, vce(cluster Id)

This allows each industry to have its own parabola without constraining them to all have the same width.

[mohina saxena]

Many Thanks professor, it really helped me a lot. The code that you suggested solves what I intended. Moreover, for sure I would also include the quadratic interaction of leverage with industry.

regards,
Mohina