A quick note on causal effect

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

y=\alpha z+w+\varepsilon,

where \alpha is true treatment effect, z is treatment status, y is the outcome, and w is an unobserved variable, with random term \varepsilon. The basic idea is to introduce observable variables that determine the assignment in the equation:

“Assume that an observed variable, t, was used to determine assignment into the treatment group and the control group… [S]ince t is the only systematic determinant of treatment status, t will capture any correlation between z and w. Thus, the observed t could replace the unobserved w as the explanatory variable.”

To understand their argument, let’s quickly review the conditional independence assumption (CIA). The CIA states that conditional on t, the outcomes are independent of treatments, that is, \{y_{0},y_{1}\}\perp\!\!\!\perp z|t.

Suppose we have
\underbrace{ E[y|z=1] - E[y|z=0]}_{\text{observed difference}} = \underbrace{ E[y_{1}-y_{0}|z=1]}_{\text{treatment effect}} + \underbrace{ (E[y_{0}|z=1] - E[y_{0}|z=0])}_{\text{selection bias}}.

However, the selection bias is undesirable in our contexts. Conditional on t, we obtain
{ E[y|t,z=1] - E[y|t,z=0]} = { E[y_{1}-y_{0}|t}].

In this way, selection bias disappears! Now, let’s go back to Barnow, Cain, and Goldberger’s equation —

y=\alpha z+w+\varepsilon.

With the CIA, we can decompose w into w=\beta t + \varepsilon^*, where \beta is a vector of population regression coefficients that is assumed to satisfy

E[w|t]=\beta t .

That is,

y=\alpha z +\beta t + \varepsilon,

where \alpha is the causal effect.

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