A researcher uses a patient's distance from the nearest hospital as an instrument for whether they received a surgical procedure, aiming to estimate the procedure's causal effect on survival. Distance correlates strongly with procedure receipt. What is the most important assumption that must hold — and cannot be statistically verified?
ADistance must be uncorrelated with the procedure itself, so that it only affects survival directly
BDistance must affect survival only through whether the patient received the procedure, not through any other channel
CDistance must be randomly assigned across patients, like a lottery
DDistance must be measured without error to avoid attenuation bias in the first stage
This is the exclusion restriction — the instrument must affect the outcome only through the treatment. Distance could plausibly violate this: patients farther from hospitals might also live in rural areas with worse diet, less emergency care access, or delayed diagnoses, affecting survival through channels other than whether they received this particular procedure. This assumption cannot be tested statistically — it must be defended with theory and contextual knowledge. The relevance assumption (distance predicts procedure receipt) CAN be tested with an F-statistic.
Question 2 Multiple Choice
A study uses two-stage least squares (2SLS) to estimate the effect of military service on lifetime earnings, using Vietnam draft lottery numbers as the instrument. The IV estimate is 0.15 (15% earnings reduction per year of service). What does this number represent?
AThe average effect of military service on earnings for all men in the draft-eligible cohort
BThe average effect of military service on earnings for men whose service status was actually changed by their lottery number
CThe effect of a one-unit increase in lottery number on lifetime earnings
DThe total earnings effect for the full population of veterans, controlling for observable confounders
IV estimates the Local Average Treatment Effect (LATE) — the causal effect only for 'compliers': men who served because they got a low lottery number and would not have served otherwise. Men who would have served regardless of their number ('always-takers') and men who never served regardless ('never-takers') are not identified by the instrument. The LATE may differ substantially from the Average Treatment Effect (ATE) for the full population, which is why external validity must always be discussed in IV studies.
Question 3 True / False
The relevance assumption in instrumental variables — that the instrument is correlated with the treatment — can be empirically tested.
TTrue
FFalse
Answer: True
Yes — relevance is testable. You regress the treatment variable on the instrument in the first stage and examine the F-statistic. The conventional rule of thumb is F > 10; instruments with F < 10 are 'weak' and produce IV estimates that are biased toward the OLS estimate and have unreliable confidence intervals. The exclusion restriction, by contrast, is fundamentally untestable — you cannot observe all the pathways through which an instrument might affect the outcome, so it must be argued on substantive grounds.
Question 4 True / False
When a valid instrument exists, the IV estimator recovers the same causal parameter as a randomized controlled trial with full compliance — that is, the average treatment effect (ATE) for the entire target population.
TTrue
FFalse
Answer: False
IV estimates the Local Average Treatment Effect (LATE), not the ATE. It identifies the causal effect only for 'compliers' — the subgroup whose treatment status changes in response to the instrument. A randomized trial with full compliance estimates the ATE for the full sample. LATE and ATE coincide only in the special case where treatment effects are homogeneous (the same for everyone). In practice, compliers are often a specific subpopulation (e.g., draft lottery compliers are men with low lottery numbers on the margin of service), limiting generalizability.
Question 5 Short Answer
Why is it problematic to use a variable that is correlated with the treatment as an instrument, even if it is strongly correlated, without verifying the exclusion restriction?
Think about your answer, then reveal below.
Model answer: A strong correlation with treatment only satisfies relevance — one of two requirements. The exclusion restriction requires the instrument to affect the outcome solely through the treatment. If the instrument also affects the outcome through other pathways (direct effects or through unmeasured confounders), the IV estimate is biased — it no longer isolates the causal effect of treatment, instead picking up the instrument's own direct effect on the outcome. Strong relevance with a violated exclusion restriction can give precise but deeply wrong estimates, which may be worse than a confounded OLS estimate.
Many variables are correlated with treatment and seem like convenient instruments without actually satisfying the exclusion restriction. Socioeconomic indicators, geographic variables, and birth-date proxies often have multiple causal pathways to the outcome. The discipline of IV analysis is largely the discipline of constructing a credible argument that the exclusion restriction holds — and recognizing when no such argument is available.