Instrumental Variables

Here are slides.

Instrumental variables is a strategy to identify causal effects in the presence of unmeasured confounding. We illustrate instrumental variables through the example of a randomized controlled trial with noncompliance.

For a set of patients enrolled in a medical trial, let \(Z\) indicate whether they are randomized to receive an experimental drug or a placebo. We will call \(Z\) the instrument. Suppose that only some of the patients assigned the experimental drug actually take the drug. Let \(A\) be the treatment: whether they take the drug. We are interested in the causal effect of the drug on a health outcome \(Y\), which may also be shaped by unmeasured variable \(U\) that cause compliance with the randomized treatment. The DAG below visualizes this causal structure.

Instrumental variables is a strategy to learn the causal effect of the drug (\(A\rightarrow Y\)) despite the unmeasured confounding \(A\leftarrow U\rightarrow Y\). Two key facts about the DAG are essential.

  1. Exchangeability. The instrument \(Z\) is exchangeable with respect to the potential outcomes under each instrument value. \[Z⫫ \{Y^{z=1},Y^{z=0}\}\]
    • This assumption holds when the only open paths between \(Z\) and \(Y\) are causal paths.
  2. No direct effect. The instrument \(Z\) affects the outcome \(Y\) only through the treatment \(A\).
    • This assumption holds when the only causal paths from \(Z\) to \(Y\) include the treatment \(A\).

For estimation purposes, we also need

Under these assumptions, IV allows inference by estimating:

  1. The effect of \(Z\) on \(Y\)
  2. The effect of \(Z\) on \(A\)
  3. Then estimate the effect of \(A\) on \(Y\) by (1) / (2).

In class, we will discuss the intuition behind this approach to identification and estimation, and how to reason about its plausibility in various settings.

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