Unconditional Quantile Regression (rifreg) with an Interaction Term; and an independent percentile score variable

[Carl Groesbrink]

Hi all,
we have a panel data set (defined by firms and years) and are trying to estimate the following model using Unconditional Quantile Regression (via the user-written comand -rifreg-; Stata 15.1) with Year and Industry fixed effects:

Model:

Tobin's Q = Beta_0 + Beta_1 * ESG + Beta_2 * Advertising + Beta_3 * (ESG x Advertising)  ///
              + Gamma * Controls + Year-Dummies + Industry-Dummies + Error_Term

where

• Tobin's Q = Proxy for firm value. A continuous variable.
• ESG = stands for "Environmental, Social, and Governance" and is a proxy for a firms sustainability level. It is a percentile score, i.e. takes values between 0 and 1. A continuous variable.
• Advertising = reflects a firms advertising expenditures scaled with a firms total assets. A continuous variable.
• (ESG x Advertising) = is an interaction term between ESG and Advertising. A continuous variable.
• Controls = vector of several control variables

The coefficients of interest are Beta_1 and Beta_3 and are estimated via the following Stata Code, e.g. for the 10th quantile of Tobin's Q:

Stata Code:

rifreg TobinsQ ESG Advertising Interaction Vector_of_Controls YDum* IndDum*  ,  ///
        bootstrap quantile(.1) kernop(gaussian) width(0.0) reps(100)

where

generate Interaction = ESG * Advertising

and both variables are continuous


Results for the 10th quantile of Tobin's Q:

Variables	Coefficient
ESG	0.211
Advertising	0.429
Interaction	0.260

Questions:

1. How is an interaction term interpreted in an unconditional quantile regression (UQR) model.
   • From OLS I know that we would say: ESG is the stand alone effect when Advertising is zero, and the total effect of ESG is obtained by setting Advertising in the Interaction Term at its mean (for an one unit increase in ESG we would have: 0.211 + 0.260 * Mean_Advertising)
   • But with UQR we are (marginally) shifting the whole distribution of our independent variable ESG (Rios-Avila/Maroto, 2020a), what does this imply for Advertising? So could we say that 0.211 is the effect of a one unit increase in ESG at the 10th quantile of Tobin's Q when Advertising is zero and 0.260 is the additional effect of ESG when ESG is increased by one unit and Advertising is set at its mean given the 10th quantile of Tobin's Q? Or could we use Advertisings "global" mean as we would do in a standard OLS context?
   • It seems that the inlcusion of an interaction term is possible as Rios-Avila (2020b) states on page 62: "This can be easily mended by including higher-order polynomials and interactions that would better capture some of the nonlinear relationships across the independent variables and the RIF{y, v(F_Y)}.".
2. Additional Question:
   • As allready stated above our ESG variable is a percentile score, i.e. all firms obtain a relative rank (percentile) between 0 an 1. This implies that in the population we would observe an uniform distribution of ESG, what is not the case in our sample. Therefore, a marginal change in the distribution of UQR would imply that all firms improve their relative rank (percentile) by an infinitesimal amount, which is actually not possible since, when one firm improves its rank another firm has to worsen its rank. At the moment we think, that for this reasoning the use of a percentile score variable is not possible and are looking for a workaround. Or is it still possible since we are talking only about a marginal change in the distribution of ESG?

Any help is highly appreciated!

Best
Carl




Literature:
Rios-Avila, F., Maroto, M. (2020a). Moving beyond linear regression: implementing and interpreting quantile regression models with fixed effects. Working Paper.
Rios-Avila, F. (2020b). Recentered influence functions (RIFs) in Stata: RIF regression and RIF decomposition. The Stata Journal, 20 (1), 51-94.

Command:
-rifreg- is a user-written command provided by Nicole Fortin on her homepage.

[FernandoRios]

Hi Carl,
These are very interesting questions. So let me provide some comments.

1. whenever you use RIF regressions (UQR among those), you have to think in terms of average marginal effects, so that you capture effects on unconditional distributions (changes that affect everyone in the population).

2. For UQR, if ESG would just be a regular continuous variable, I argue that interpreting ESG on its own is not correct when you have this interaction. Rather, you should interpret by considering both the coefficient on ESG as well as its interaction with Advertisement. This will give you the effect of what if ESG increases in 1 unit, keeping the level of advertisement for "everyone" fixed.
Again, you need to think in terms of unconditional moments, average marginal effects, or as you say, global means.

I do also argue that interactions can be added because you account for other moments of the distribution of the explanatory variable.

3. Now, since ESG ranges from 0-1, I would also argue that interpreting the coefficient as is (0.211+0.26*Mean(AD)) is also incorrect. The reason for that this marginal effect assumes a change from 0->1 for everyone. This isn't a reasonable "thought experiment". What I would say instead is to "scale" the coefficient down, by multiplying it by say 0.1 or 0.01. Since that change for everyone would be more reasonable. This is similar to my discussion on the interpretation of Dummy variables.

4. What if ESG is a relative Score? In this case, I would say you cannot use this variable to make direct interpretations in your model.
As you already indicate ESG is a relative score. Thus, its distribution never changes. It will always be between 0-1, with a variance 1/12, and mean 0.5. Thus, you cannot make interpretations of what if the ranking of every firm increases in 0.01.
What could change, however, is the effect through the interaction. If instead of ESG*AD you use (ESG-mean(ESG)) * (ad - mean(AD)), you could interpret the coefficient of this last component as how changes in the covariance between ESG and AD that would affect the distribution of the TOBIT Q.

In other words, Mean(ESG) and VAR(ESG) will not change, thus you cannot say anything about how that will affect the tobit Q. However COV(ESG,other variables) could change, and that could affect Tobit's Q distribution.

HTH
F

[Carl Groesbrink]

Hi Fernando,

Thank you very much for your quick and comprehensive answer, that helped a lot.

Regarding point 2: Does this mean that with a UQR ESG and the interaction term can always only be interpreted together, since Advertising was set to the average of the entire sample and it is not possible to theoretically set Advertising to 0, so that the interaction term is omitted and ESG can be interpreted alone? Because the idea to set Advertising to 0 in order to be able to interpret the coefficient of ESG itself would be quite possible with a standard OLS regression.
At least that is how I have understood the issue so far and this is also addressed in the educational paper by Richard Williams (2015), for example.


Regarding point 3: You are of course absolutely right, ESG would be increased by one unit, i.e. 0.01 and not by 1 as implied before, so the effect would be (0.211+0.260 * Mean(AD))*0.01 (I hope I did not make another mistake this time).


Regarding point 4:

As you already indicate ESG is a relative score. Thus, its distribution never changes. It will always be between 0-1, with a variance 1/12, and mean 0.5. Thus, you cannot make interpretations of what if the ranking of every firm increases in 0.01.

Thank you for clarifying the issue.
But what if our sample does not have these characteristics, is not representative of the population in this respect, so to say? The attached screenshot shows in the first row (R1) the descriptive statistics of ESG. If ESG were representative of the population, the median (P50) and also the mean would have to be 0.50, but in our case the values are only 0.33 and 0.37. Does this change the problem, in the sense that one now examines the effect of ESG on Tobin's Q in the context of the sample compared to the population? So to speak, a subsample (our entire sample) is examined, whereby it is now possible that the distribution of the ESG score in this subsample is shifted, while the distribution in the population remains unchanged? I don't think such an analysis makes economically sense, since it would only be possible if the relevant variable (ESG percentile score) is not representative of the population, which would be a strange assumption/condition. But from a statistical point of view, would it now be possible to apply a UQR?

And one last question regarding the use of subsamples to solve the problem.
Is it possible to apply a UQR if we calculate the change in ESG_t and ESG_t-1 and divide our dataset into two subsamples, the companies whose ESG score increased and the companies whose ESG score worsened and perform a separate UQR for both subsamples? According to our understanding, it should be possible to shift the distribution of the ESG score if only the companies are considered that have experienced a positive (negative) change in ESG score. Rows 2 (R2) and 3 (R3) give the descriptive statistics of ESG under the condition that the change is positive and negative, respectively.


Stata Code for R2 and R3 in the screenshot:

xtset id Year

generate D_ESG
replace D_ESG = D.ESG

// ESG if D.ESG positive
tabstat ESG if D_ESG > 0 & D_ESG != . , ///
           statistics (count mean sd min max p1 p5 p10 p25 p50 p75 p90 p95 p99) save

// ESG if D.ESG negative
tabstat ESG if D_ESG < 0 & D_ESG != . , ///
           statistics (count mean sd min max p1 p5 p10 p25 p50 p75 p90 p95 p99) save

Many thanks in advance for your time and effort!

Best
Carl



Literature:
Williams, R. (2015). Interaction effects between continous variables (optional). https://www3.nd.edu/~rwilliam/stats2/l55.pdf

[FernandoRios]

Hi Carl

Regarding point 2: Does this mean that with a UQR ESG and the interaction term can always only be interpreted together, since Advertising was set to the average of the entire sample and it is not possible to theoretically set Advertising to 0, so that the interaction term is omitted and ESG can be interpreted alone? Because the idea to set Advertising to 0 in order to be able to interpret the coefficient of ESG itself would be quite possible with a standard OLS regression.
At least that is how I have understood the issue so far and this is also addressed in the educational paper by Richard Williams (2015), for example.

Correct, when thinking about RIF regressions, you always have to think in terms of average marginal effects, and changes that affect everyone will affect the overall distribution.
RIFregressions are really a different way of thinking about analysis. you are not finding effects for particular units, but for the distribution as a whole.

Thank you for clarifying the issue.
But what if our sample does not have these characteristics, is not representative of the population in this respect, so to say? The attached screenshot shows in the first row (R1) the descriptive statistics of ESG. If ESG were representative of the population, the median (P50) and also the mean would have to be 0.50, but in our case the values are only 0.33 and 0.37. Does this change the problem, in the sense that one now examines the effect of ESG on Tobin's Q in the context of the sample compared to the population? So to speak, a subsample (our entire sample) is examined, whereby it is now possible that the distribution of the ESG score in this subsample is shifted, while the distribution in the population remains unchanged? I don't think such an analysis makes economically sense, since it would only be possible if the relevant variable (ESG percentile score) is not representative of the population, which would be a strange assumption/condition. But from a statistical point of view, would it now be possible to apply a UQR?

This is an interesting point. So, as i described above ESG cannot have a direct effect on the distribution of tobit Q. ESG distribution will cannot change, only its relationship with other variables will. (you can still control for it, but the effects could not be reasonably described)

What if your sample is different from the population? Say, your sample contains only firms in the Manufacturing Industry (Exogenous sample selection). In this case, you could try to interpret what happens if ESG increases, because that means Manufacturing is experiencing an overall increase in its ranking compared to all other industries. Alternatively, what if ESG was created considering Rules for say 1990, and your data is from 2000. Again, ESG could capture how an improvement in ranking compared will affect the tobit Q.

I would argue that in these two cases, ESG is no longer a relative measure, but absolute, compared to a different comparison point.

What about your example? If you want to interpret ESG in your sample just because their distribution is different from the true population, what does it mean? is your sample biased? why is it bias?
for Whom are you are analyzing the distribution of the Tobit Q?

At the end of the day, you can apply any variable to UQR, because the estimation is just an OLS. My point is more on interpretation. If you cannot make a clear story of how could the distribution of ESG change, then you cannot really interpret it from the UQR perspective.

And one last question regarding the use of subsamples to solve the problem.
Is it possible to apply a UQR if we calculate the change in ESG_t and ESG_t-1 and divide our dataset into two subsamples, the companies whose ESG score increased and the companies whose ESG score worsened and perform a separate UQR for both subsamples? According to our understanding, it should be possible to shift the distribution of the ESG score if only the companies are considered that have experienced a positive (negative) change in ESG score. Rows 2 (R2) and 3 (R3) give the descriptive statistics of ESG under the condition that the change is positive and negative, respectively.

This is a harder question. You can certainly do this, and estimate UQR. But im not sure how could you compare both groups. Perhaps a kind of OB decomposition looking at distributional differences between firms that improve vs those that got worse in ESG score.

It is still unclear to me, however, how wouldyou interpret ESG within each subsample.

HTH

[Carl Groesbrink]

Hi Fernando,

Thanks again for your extensive help, you have helped me a lot. I will think about the points again and get back to you if more questions have popped up.

Best
Carl