7 – Instrumental Variables

What is an instrumental variable (IV)?

A variable used to isolate exogenous variation in an endogenous regressor to identify causal effects.

What is the IV relevance condition?

Cov(Z, X) neq 0 — the instrument must be correlated with the endogenous explanatory variable.

What is the IV exclusion restriction?

Cov(Z, e) = 0 — the instrument affects the outcome only through the endogenous regressor, not directly or through omitted variables.

Why is exclusion restriction less certain?

It is less certain because Z may affect Y through channels other than X or be correlated with unobserved factors in e, violating Cov(Z, e) = 0.

How do you explain the exclusion restriction in an exam?

This is less certain because [instrument] may be correlated with unobserved factors affecting [outcome], or may affect [outcome] through channels other than [endogenous variable].

What is the purpose of the first-stage regression?

To test whether the instrument explains variation in the endogenous variable.

What are the hypothesis in the first stage?

H0: n1 = 0 ; HA: n1 neq 0

What is the first-stage test statistic?

t = n1 / se(n1), which follows a t-distribution under the null hypothesis.

What is the first-stage decision rule?

Reject H0 if |t| > tc

Conclusion first-stage |t| > tc

Since |t| < tc, we reject H0. There is sufficient evidence that the IV has explanatory power in the first stage.

Conclusion first-stage |t| < tc

Since |t| < tc, we fail to reject H0. There is insufficient evidence that the IV has explanatory power in the first stage.

What is the rule of thumb for weak instruments?

The first-stage F-statistic should be greater than 10.

What is a weak instrument?

An instrument that has a weak correlation with the endogenous variable, providing little explanatory power in the first stage.

What is the consequence of weak instruments?

Cause IV estimates to be biased toward OLS and lead to unreliable standard errors and invalid hypothesis tests due to non-normal sampling distributions.

What is the purpose of the reduced form?

To show whether the instrument affects the endogenous variable.

What does the second stage measure?

The causal effect of the predicted (instrumented) X on Y.

What is identification in IV?

The ability to isolate causal variation in X using variation in Z.

How do you compare OLS and IV estimates?

Compare sign and magnitude to infer the direction of OLS bias due to endogeneity.

If IV estimate has larger absolute value than OLS, what does it imply?

OLS is likely biased toward zero (attenuation-type bias).

If IV estimate has smaller absolute value than OLS, what does it imply?

OLS is likely upward biased.