Dependent variable as a delta, control variable as total of the same variable

[Ole Tomas]

I am writing my thesis about how tax avoidance affects next year's ESG scores (Environmental, Social, Governance). So this variable ESG is measured as the change from year t to year t+1. The actual ESG scores are a measurement score ranging from 0-100. Should i include as a control variable the total score of ESG, because if a company already scores 95 on ESG in year t, than improving this score is very difficult (maxed by 5). It seemed sensible to include ESG score as a control variable.

This is my model before adding ESG total score:

ESGΔ = tax avoidance (t-1) + Size + ROE + Leverage + i.Year, fe

Should i just straightforward add ESG total score as a control, which would the regression results still interpretable?

[Rich Goldstein]

including or excluding the first year gives an answer to different questions so whether to include depends on what question you are answering; see Fitzmaurce, G (2001), "A conundrum in the analysis of change", Nutrition, 17: 360-361

[Ole Tomas]

I want to answer the following quesion: Does tax avoidance in year t affect change in ESG for year t+1. So if a company already has really high ESG lets say 99, than increasing this is almost impossible as 100 is the maximum score one could get. That is why I am contemplating whether ESG score for year t should be included as a control variable

[Rich Goldstein]

you think your question is clear but it is not with respect to the issue of whether to use the year t variable - read the paper I cited

[Clyde Schechter]

Rich Goldstein Thanks for that reference in #2. I've been looking for a reference that gives a clear, simple explanation of this difference for a while--you've ended my search!

Ole Tomas The occurrence of ceiling effects such as the one you describe for delta ESG is one of the reasons that change score analysis is controversial. See, for example, https://www.fharrell.com/post/errmed/#change. By pointing out that if you are starting from a score of 99 there is little room for increase, you are, in effect, saying that it is not realistic to posit that a given level in tax avoidance, size, ROE, leverage, or year will correspond to some level of delta ESG. Rather that change will be dependent on the prior level of ESG. You recognize that in your description of your problem, but including the prior level of ESG as a covariate isn't going to solve that problem because the effect of prior ESG on delta ESG can't be linear in the presence of a ceiling effect, just as the effects of the other variables can't be linear in the presence of a ceiling effect. I think a reasonable approach in this situation is to instead transform ESG so that the transformed values do not have a ceiling effect. One particularly simple way is to use the -logit()- function. The value of logit(x) grows to infinity as x approaches 100% (i.e. x approaches 1). So even if you are starting out at 99% (logit(.99) = 4.6), you still have plenty of room to grow the logit! So you might consider something like:

gen ESG_transform = logit(ESG/100)
xtset firm year
regress ESG_transform L1.tax_avoidance size roe leverage i.year

The difficulty with this approach lies in the interpretation of the results. Your coefficient of L1.tax_avoidance will be the expected difference in logit(ESG/100) associated with a unit difference in tax avoidance, which is not a natural or intuitive thing to work with. You can get an estimate of the average effect of tax avoidance on the ESG itself by following that with:

margins, dydx(L1.tax_avoidance) expression(100*invlogit(predict()))

But do bear in mind that this would just be the average marginal effect because, as you have already observed and we have discussed at length, the effect on ESG itself is not a constant effect of L1.tax avoidance but depends on the starting value of ESG.