Compare regression coefficients of two settings with similar samples

Dear Statisticians,

For my thesis I am currently examining whether the accounting principles have an influence on the relation between having a Big 4 auditor and financial statement comparability. Therefore, I compare the same statistical relation for two settings that have different sets of accounting standards. To keep it simple, I use a similar sample for both settings (i.e. the 500 largest companies per fiscal year) and do a cross-sectional analysis. This leads to the following OLS regression for setting 1 and 2 respectively:

COMP1 = a*Big4 + b*controls + e
COMP2 = a*Big4 + b*controls + e

My goal is to compare regression coefficients 'a' in order to examine which type of accounting standard has a bigger influence on said relation. However, by no means am I a statistical genius and thereby this raises the question what would be the best way to compare these two coefficients to answer my research question. Would anyone happen to know this?

The current state of my research
Setting 1: dataset complete - results from OLS regression are all significant at the 1% level
Setting 2: dataset incomplete - still collecting data for a few variables - results expected to be significant as well with said model

Lastly, I would love to ask for some advice on which things to control for (such as heteroskedacity etc) and what the best way to do this would be? This so that I can produce valid and significant results of course! (Here I aim for the statistical tests and checks rather than controls such as firm size etc.)
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