What assumptions must an instrumental variable satisfy to identify a causal effect?

## What an instrument is for Suppose you want the causal effect of a treatment `D` on an outcome `Y`, but the people who take `D` differ from the people who do not in ways you cannot measure. Motivation, health, prior skill, whatever it is, it sits in the error term and is correlated with `D`. A straight comparison of treated and untreated units then mixes the effect you want with the effect of that hidden difference, and no amount of adding the covariates you happen to have observed fixes it. An instrumental variable `Z` is a third variable that shoves `D` around for reasons that have nothing to do with the hidden difference. If you can find one, you can throw away all the variation in `D` that you do not trust and keep only the slice of it that `Z` created. The causal effect is then read off that trustworthy slice. That plan only works if `Z` has three properties. ## 1. Relevance `Z` must actually move `D`. Formally `Cov(Z, D)` is non-zero, and conditional on whatever covariates you include, `Z` must retain real partial explanatory power for `D`. This is the one assumption the data can speak to: regress the treatment on the instrument and the covariates, and look at how strongly the instrument predicts. This regression is called the first stage. Relevance is not a yes-or-no property. An instrument can be statistically significant in the first stage and still be far too weak to support a usable estimate, because the estimate divides by the first-stage strength. A near-zero denominator makes everything unstable. That is the weak-instrument problem, and it is why the first-stage F statistic gets reported alongside every instrumental-variables estimate. ## 2. Independence, or as-good-as-random assignment `Z` must be unrelated to the unmeasured confounders and to potential outcomes. If the instrument is itself assigned by the same hidden forces that drive treatment, it inherits exactly the problem you were trying to escape. The cleanest way to earn this assumption is for `Z` to be physically randomized: a lottery draw, a randomly sent invitation, a coin flip in an administrative process. When the instrument is merely a feature of the world rather than a designed randomization, independence becomes an argument you have to make and an interviewer will press. ## 3. The exclusion restriction `Z` must affect `Y` **only** through `D`. No direct effect, no effect through a side channel. This is the assumption that fails most often and the one that cannot be checked. In a just-identified design, with one instrument for one treatment, the data are consistent with any amount of exclusion violation; nothing in the numbers reveals it. With more instruments than treatments you can run an overidentification test, but that only asks whether the instruments agree with each other, and it is uninformative if they are all invalid in the same direction. The exclusion restriction is a statement about mechanism, and it is defended with mechanism talk. Draw the path you claim does not exist and say why it does not exist. ## 4. Monotonicity, if you want to interpret the number The three conditions above buy you identification. A fourth, monotonicity, says the instrument pushes every unit in the same direction or not at all, never the opposite way. Without it the estimate is a difference of subgroup effects that need not correspond to anything. With it, the estimate is the average effect for the units whose treatment the instrument actually changed. ## A worked example The Vietnam-era draft lottery assigned each birth date a random sequence number, and low numbers meant a high chance of being drafted. Take military service as `D`, later civilian earnings as `Y`, and the lottery number as `Z`. - Relevance holds and is visible in the data: men with low numbers served at a much higher rate. - Independence holds by construction: the numbers were drawn at random, so they cannot be correlated with anyone's underlying earning capacity. - Exclusion is the arguable one. If a low number caused a man to stay in school longer to avoid the draft, then the instrument reached earnings by a route other than service, and the restriction is violated. Notice that this critique is about mechanism, not about the data. ## What interviewers are listening for Candidates who name relevance and exclusion but forget independence, or who claim exclusion is testable, get marked down. So does anyone who proposes an instrument on the grounds that it correlates strongly with the outcome. Correlation with the outcome is not a qualification; it is a warning sign, because a variable that predicts `Y` directly is exactly the variable that violates exclusion. The qualification is a credible story about why the instrument moved the treatment for reasons unrelated to everything else.