Randomized controlled experiment is base case for causality (also) in regression.
However currently I’m analyzing the role of causality in linear regression as shown in many econometrics textbook. For example in Brooks 2014 – introductory econometrics for finance 3rd edition pag 76-83 the fixed (non-stochastic) regressors are the base scenario and causal interpretation is explicitly offered. In this book the causal interpretation of regression coefficients seems the basic scenario too.
The example (pag 83) is about the CAPM and in this setting experiment, also ideal, and/or potential outcome language, at least in my experience, don’t play any role.
I have several doubt but my questions are primarily three:
Fixed non stochastic regressors assumption produce (by costruction) independence between errors and regressors. Then, hypothesis of stochastic independence between errors and regressors is equal to hypothesis of fixed non stochastic regressors ?
If no (as I think see also here regression and causation) the hypothesis of fixed non stochastic regressors is equal to known the “true model / true data generating process” ?
Known this “true model” is, at least in certain sense, equal to construct an (idealized) randomized controlled experiment?