Quantitative Methods

Instrumental Variables and the Exclusion Restriction

Get a causal number out of a decision customers made for themselves, by finding something that pushed some of them into it for a reason that has nothing to do with your outcome, and learn why that something almost never exists.

  • Expert
  • 15 min total
  • 15 chapters

What decision this helps you make: Whether an adoption number you cannot randomize can be given a causal value at all, which customers that value applies to, and when the intent-to-treat figure you already have is the better answer.

What this topic is

An instrumental variable is something that moves whether a customer gets treated, is unrelated to everything else that affects the outcome, and has no path to the outcome except through the treatment. Given one, the causal effect is a ratio: the effect of the instrument on the outcome, divided by the effect of the instrument on treatment. That ratio recovers the effect for one specific group (the customers whose treatment status the instrument actually changed) and for nobody else.

Why it matters

The most valuable questions in a business are about choices customers make themselves. Does enabling auto-renew cause retention, or do loyal customers enable it? Does the annual plan cause higher spend, or do heavy users buy it? A regression on that data answers neither question. Instrumental variables is the standard tool for the case where the choice cannot be randomized but something outside the customer nudged it, and it is also the tool most likely to produce a confident, precise, badly wrong number.

Who should learn it

Analysts asked to value a feature that customers opt into, pricing teams estimating elasticity from historical price variation, growth teams running encouragement designs, and anyone about to defend a causal claim built on observational adoption data.

What you will understand

  • The four assumptions (relevance, independence, exclusion, monotonicity) and what each one buys
  • Why the answer is local to compliers and what goes wrong when you extrapolate it
  • How to compute the estimate and its interval, and why a weak first stage destroys precision faster than anything else
  • When the reduced-form intent-to-treat number you already have is the better business answer

Prerequisites

Common misconception

"We found an instrument, so the estimate is causal." The exclusion restriction requires that the instrument affects the outcome only through the treatment. That is an assumption about every path in the world that you cannot see, and with a single instrument it is mathematically untestable. Overidentification tests do not rescue you: they check whether several instruments give the same answer, so a set of instruments that are all invalid in the same direction passes cleanly. And even when the assumption holds, the answer applies only to the customers the instrument moved. That is often a small, unusual slice of your base, and the number is not the average effect on anyone else.