Trying to figure out multiple regression analysis progressive taxation

I am trying to figure out how to run a multiple regression analysis with gini index as the dependent variable. Originally I planned to use top income tax rates (individual and corporate) as independent variables for a time series analysis.

However, upon doing lit review, it became clear that accepted methods to analyze tax progressivity’s impact on inequality involves using other indices such as the suits index. Unfortunately, I cannot find any research that explains how to calculate the suits index clearly. Various notation forms shown are unfamiliar to me and I can’t make sense of it.

If someone could maybe explain how to calculate the suits index, I would appreciate it. I understand logarithms and calculus well enough, but again the notation systems used are either something I forgot or never learned.

In discrete form that can be used for analysis the Suits index is given by (Suits 1977):

$$S= \sum_i \frac{1}{2} \left( T_x(y_i) + T_x(y_{i-1})\right)(y_i-y_{i-1})$$

where $$T_x$$ is share of total accumulated tax burden of tax $$x$$, $$y$$ is the cumulative income of individual (or population %, decile etc depending on data availability) $$i$$ (so it can only be between 0-100). Accumulated burden and income has to be calculated separately from some micro data such as tax data. Accumulated income per decile or quintile will appear in some publicly available datasets but burden is more difficult to find.

In addition, if you want to analyze progressiveness of total tax system you will also have to combine this index calculated for different taxes. Combined index can be calculated using:

$$S_{xz} = \frac{(r_xS_x + r_zS_z)}{(r_x+r_z)}$$

where $$S_x$$ would be Suits index for some tax $$x$$ (e.g. income tax), and $$S_z$$ would be index for some tax $$z$$ (e.g. VAT). $$r$$ is the average tax rate for $$x$$ and $$z$$ respectively.

PS:

You can't just run a naive multivariate OLS when dependent variable is inequality and independent variable some tax structures as taxes are not exogenously selected but tax legislature might be passed to combat inequality and so on. Hence you need to pick some more reasonable identification strategy that can handle endogeneity (e.g. VAR, GMM), simple multivariate OLS can't. In addition, in time series context there are other issues (e.g. these variables won't be necessarily stationary).