# non stochastic regressors and causation

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?

• If you read the very book you mention, Brooks (2014), he has a session on "omission of important variables", in which he says "The consequence would be that the estimated coefficients on all the other variables will be biased and inconsistent unless the excluded variable is uncorrelated with all the included variables. ". – Carlos Cinelli Nov 14 '18 at 0:50
• I known that Brooks (2014), as any other econometrics textbook, speak about omitted variables bias and others source of bias. However the problem is exactly on his interpretation. Surely omitted variables bias is related on causal problems. In Brooks(2014) seems that basic regression parameters have causal meaning, therefore bias, independently of his source, have causal meaning too. In others textbooks this point is not clear, in some seems that bias not necessarily have causal meaning. – markowitz Nov 14 '18 at 15:00
• Brooks take the assumption as $corr(X,u)=0$ for demonstrate that there isn’t bias (pag 179). Brooks don’t use here the term exogeneity but usually this condition is referred to it; or better a certain form of it, let me say form 1. But Brooks speak about this (exogeneity) also in the form of independence between u and X, let me say form 2, and in the form of non-stochastic X, let me say form 3 (pag 208/9). – markowitz Nov 14 '18 at 15:00
• Shortly, probably the main problem of Brooks(2014), as other econometrics textbooks, is that is not so clear what the error term (u) exactly is. This is true also for omitted variables example; see pag 224 where true model and an underspecified model have the same notation error. – markowitz Nov 14 '18 at 15:01
• I think that in his assumptions Brooks have in mind “true” error. However, true model/error or not, since the fact above are the origins of my guesses of think about form 2 and/or 3 as conditions related on causality (form 1 is surely not enough at least with estimated error). Form 2 is not enough and you helped me about convince me about that. If form 3 is essentially the same of form 2, the same is the result. – markowitz Nov 14 '18 at 15:01

Fixed non stochastic regressors assumption produce (by costruction) independence between errors and regressors.

This is not true, fixed regressors assumptions just mean that the regressors are not "random". You can still have bias, as showed in here.

If you read a couple pages further in the very book you mention, Brooks (2014), he has a session on "omission of important variables", in which he mentions omitted variable bias (although I would not recommend this book if you want to learn causality, if you want to learn causality--and not just linear regression--start here).

The hypothesis of fixed non stochastic regressors is equal to known the “true model / true data generating process” ?

No, this hypothesis has more to do with the mathematics of the probabilistic manipulations. If regressors are fixed, you will treat them as "parameters", not as random variables. But this has no bearing on causal effects.

Known this “true model” is, at least in certain sense, equal to construct an (idealized) randomized controlled experiment?

Most authors do not really define what "fixed regressor" means except that it is treated as a fixed known quantity. Even when some authors do make the analogy to "fixed by experiment" (such as Seber, in a parenthetical remark), they do not offer any mathematics of causality.

In short, to make causal conclusions you need to make explicit causal assumptions, as in a causal model.

• My confusion about this point coming from the fact that, as you said, some authors make the analogy between "non stochastic regressors" and "regressors fixed by experiment" and or "regressors fixed in repeated samples". I can mention at least two econometric books about that. However non stochastic regressors and the so called "as if" (an experiment) condition are completely different things. Thanks. – markowitz Apr 20 '19 at 9:10