# 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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