This post is a quick note as I have been reading papers and books on causal inference recently. I am sure that the materials are extremely intuitive to most social scientists, but I hope some of my notes could help the beginning grad students to quickly understand the notion of causal effect.
I recently came across a paper by Barnow, Cain, and Goldberger, that was published thirty-six years ago. In their paper, they talked about how to operationalize the following equation
where is true treatment effect, is treatment status, is the outcome, and is an unobserved variable, with random term The basic idea is to introduce observable variables that determine the assignment in the equation:
“Assume that an observed variable, , was used to determine assignment into the treatment group and the control group… [S]ince is the only systematic determinant of treatment status, will capture any correlation between and . Thus, the observed could replace the unobserved as the explanatory variable.”
To understand their argument, let’s quickly review the conditional independence assumption (CIA). The CIA states that conditional on the outcomes are independent of treatments, that is,
Suppose we have
However, the selection bias is undesirable in our contexts. Conditional on we obtain
In this way, selection bias disappears! Now, let’s go back to Barnow, Cain, and Goldberger’s equation —
With the CIA, we can decompose into where is a vector of population regression coefficients that is assumed to satisfy
where is the causal effect.