The Local Average Treatment Effect

A brokerage randomly offers a fee discount voucher to some clients to nudge them into a new trading platform, using voucher receipt as an instrument for platform adoption in order to estimate the platform's effect on trading frequency (see endogeneity and instrumental variables). The instrumental-variables estimate comes back: platform adoption increases trading frequency by 3 trades/month. It's tempting to say "the platform boosts trading by 3/month for our clients." That's not quite what was estimated — the number applies specifically to the clients whose adoption decision was actually changed by getting the voucher, not to clients who'd have adopted anyway or who'd never adopt regardless of the voucher. That specific subgroup, and the effect measured for it, is the local average treatment effect.

An analogy: a coupon that only sways some shoppers

A store mails discount coupons to a random half of its mailing list to study whether buying a loyalty card increases spending, using coupon receipt as the instrument. Some customers were always going to buy the card, coupon or not (always-takers). Some were never going to buy it no matter what (never-takers). A third group buys the card because of the coupon, but wouldn't have without it (compliers). Comparing spending between coupon-recipients and non-recipients, scaled by the coupon's effect on card purchase, only tells you about the compliers — the always-takers' and never-takers' behavior doesn't change either way, so they contribute no information about the causal effect of the card itself, only noise that has to be netted out.

The estimand, one symbol at a time

Let $Z_i$ be the instrument (voucher offered), $D_i$ treatment received (platform adopted), and $Y_i$ the outcome. Under the standard IV assumptions — relevance, exclusion (see the exclusion restriction and instrument validity), and monotonicity (no one is swayed the "wrong way": no client who'd adopt without the voucher is talked out of adopting by receiving it) — the Wald/IV estimator identifies:

$$\text{LATE} = \mathbb{E}[Y_i(1) - Y_i(0) \mid \text{complier}] = \frac{\mathbb{E}[Y_i \mid Z_i=1] - \mathbb{E}[Y_i \mid Z_i=0]}{\mathbb{E}[D_i \mid Z_i=1] - \mathbb{E}[D_i \mid Z_i=0]}$$

In plain English: the numerator is the instrument's effect on the outcome (the "reduced form"), the denominator is the instrument's effect on treatment take-up (the "first stage"), and their ratio is the average causal effect — but only among compliers, the people whose treatment status the instrument actually moved. Always-takers and never-takers never change $D_i$ in response to $Z_i$, so they contribute nothing to the denominator and the estimator is silent about their treatment effect entirely.

Worked example 1: computing LATE by hand

Suppose 40% of vouchered clients adopt the platform versus 15% of non-vouchered clients ($\mathbb{E}[D\mid Z=1]=0.40$, $\mathbb{E}[D\mid Z=0]=0.15$) — a first stage of 0.25. Average trading frequency is 8.2 trades/month among vouchered clients and 7.4 among non-vouchered ($\mathbb{E}[Y\mid Z=1]=8.2$, $\mathbb{E}[Y\mid Z=0]=7.4$) — a reduced form of 0.8.

$$\text{LATE} = \frac{8.2 - 7.4}{0.40 - 0.15} = \frac{0.8}{0.25} = 3.2 \text{ trades/month.}$$

This 3.2 is the average effect specifically for the 25% of clients (the first-stage gap) who adopted because of the voucher — not for the 15% who'd have adopted regardless (always-takers) or the 60% who never adopted either way (never-takers).

Worked example 2: why LATE can mislead if compliers are unusual

Suppose compliers here skew toward younger, more active traders who were on the fence — precisely the type most responsive to a small discount. If the firm wants to know the effect of a full platform mandate (forcing everyone onto it, including never-takers who might be older, less active clients resistant to any change), the LATE of 3.2 trades/month, measured on responsive fence-sitters, is likely a poor proxy for what a mandate would do to reluctant never-takers — their true individual treatment effect could be much smaller (or even negative, if the new platform frustrates them). Using a voucher-based LATE to justify a universal mandate implicitly assumes compliers and never-takers respond similarly, an assumption the design itself provides no evidence for.

What this means in practice

LATE is genuinely useful — it's often the only causal quantity an instrument can credibly deliver — but it's an answer to a narrower question than "what is the effect for our population," and which subgroup counts as compliers depends entirely on which instrument you used. A different instrument (say, a in-app pop-up nudge instead of a mailed voucher) would sway a different set of fence-sitters and generally produce a different LATE, even estimating the exact same underlying treatment on the exact same population. Reporting an IV estimate without naming and characterizing the complier subgroup — even informally, by comparing their observed covariates to the full sample — leaves the reader unable to judge how far the result should be extrapolated.