# How to choose the control variables in the conditional expectation to hold fixed when studying a causal relationship

I'm reading the introductory chapter of the wooldridge's book, "Econometric analysis of cross section and panel data". The chapter begins by highlighting the role and importance of conditional expectation for econometrics. In short, if we want to establish a causal relationship betwwen two variables - $$y$$ and $$w$$-, we have to consider the notion of ceteris paribus.

The best way to do this analysis (ceteris paribus) is by considering the conditional expectation $$E[y| w,c]$$ and $$\frac{\partial E[y| w,c]}{\partial w}$$, where the vector $$c$$ denotes a set of control variables that we would like to explicitly hold fixed when studying the effect of $$w$$ on the expected value of $$y$$.

In the author's own words: "Deciding on the list of proper controls is not always straightforward, and using different controls can lead to different conclusions about a causal relationship between y and w. This is where establishing causality gets tricky: it is up to us to decide which factors need to be held fixed."

And he goes on to cite a few examples. For example: "it is widely agreed among labor economists that experience and ability are two factors we should hold fixed to obtain the causal effect of education on wages. Other factors, such as years with the current employer, might belong as well. We can all agree that something such as the last digit of one’s social security number need not be included as a control, as it has nothing to do with wage or education".

In addition, he alerts us to the problem of encountering control variables that can not be observed. In the above example: $$E[wage | educ, exper , abil ]$$, abil is innate ability and it necessarily has to belong to the control group to be kept fixed, but we can not collect data from this variable.

We can say that intuition makes us assume that the variable abil has to be kept fixed in order to obtain the causal effect of education on wages. And consequently, we have to include abil in the group of control variables.

But what to do when intuition does not help us? That is, what criteria can we use to know what control variables we must include in order to obtain the causal effect? Remember the wooldridge's words: "it is up to us to decide which factors need to be held fixed"

• This matter is well discussed among authors writing about causal directed acyclic graphs, so that might be a good place to start. Elwert (2013) "Graphical Causal Models" is a fantastic introductory text. basically, you need to control for a set of variables that blocks all "backdoor" paths and doesn't block any "frontdoor" paths. Substantive considerations are required to decide which variables lie on these paths. – Noah May 1 at 3:51