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