What Defiers Do to the Wald Interpretation

Why Defiers Change the Wald Interpretation

The usual interpretation of the Wald ratio as a complier average treatment effect relies on a substantive restriction: no one responds to the instrument by changing treatment receipt in the direction opposite to the encouragement. A person who does respond in that opposite direction is a defier. When defiers are present, the instrument-induced change in treatment is a net contrast between people pushed into treatment and people pushed out of it. The outcome contrast is netted in the same way, so the resulting ratio can place a negative weight on one group’s effect.

This is not merely a technical change to a formula. The ratio may fall outside the range of the complier and defier effects, or it may be undefined even though the instrument changes outcomes. The calculations below show exactly why.

Potential Treatment Receipt and the Signed Mixture

Let \(Z\in\{0,1\}\) be a binary instrument and \(D\in\{0,1\}\) a binary treatment. For each unit, \(D(1)\) and \(D(0)\) are the treatment receipts that would occur under the two instrument values. The four compliance types are defined by this pair:

• Compliers: \(D(1)=1,\ D(0)=0\).
• Defiers: \(D(1)=0,\ D(0)=1\).
• Always-takers: \(D(1)=D(0)=1\).
• Never-takers: \(D(1)=D(0)=0\).

Write their population shares as \(p_C,p_F,p_A,p_N\), respectively, so \(p_C+p_F+p_A+p_N=1\). Let \(Y(1)\) and \(Y(0)\) be the potential outcomes under treatment and no treatment, and define the individual treatment effect as \(Y(1)-Y(0)\). The average effects within the two types that change treatment receipt are \( \tau_C=E[Y(1)-Y(0)\mid C] \) and \( \tau_F=E[Y(1)-Y(0)\mid F] \).

For the population contrasts, assume instrument independence of the potential treatment receipts and outcomes: \(Z\) is independent of \(\{D(0),D(1),Y(0),Y(1)\}\). Also assume exclusion, so the instrument affects the outcome only through treatment, and consistency, so observed treatment and outcome equal the potential values corresponding to the observed instrument and treatment. These conditions make the contrasts by \(Z\) identify contrasts in potential values. Finally, the Wald ratio requires a nonzero first-stage contrast. Monotonicity is not assumed.

The type-by-type outcome calculation explains the signs. Compliers move from \(D=0\) to \(D=1\) as \(Z\) changes from 0 to 1, so their contribution to the reduced form is \(p_C\tau_C\). Defiers move from \(D=1\) to \(D=0\), so their contribution is \(p_F\{Y(0)-Y(1)\mid F\}=-p_F\tau_F\). Always-takers and never-takers do not change treatment receipt, so exclusion makes their contributions zero.

When \(p_C\ne p_F\), dividing the reduced form by the first stage gives

The coefficients on \(\tau_C\) and \(\tau_F\), when written as \(p_C/(p_C-p_F)\) and \(-p_F/(p_C-p_F)\), sum to one. But they need not both be positive. If \(p_C>p_F>0\), the defier coefficient is negative and the complier coefficient is greater than one. If \(p_F>p_C\), the first stage is negative and the signs reverse: the complier coefficient is negative and the defier coefficient is greater than one. A sum of one does not make these coefficients ordinary averaging weights.

Worked Examples

How the Result Relates to Monotonicity

Under monotonicity, \(p_F=0\). Provided \(p_C>0\), the first stage is \(p_C\), the reduced form is \(p_C\tau_C\), and their ratio is \(\tau_C\). That is why the complier interpretation follows when monotonicity holds alongside the other assumptions. When defiers are present, even if the first stage is positive, the ratio generally has a negative defier weight instead.

There are special cases in which the defier presence does not change the ratio’s value. If \(\tau_C=\tau_F=\tau\) and \(p_C\ne p_F\), then \((p_C\tau-p_F\tau)/(p_C-p_F)=\tau\). This is a consequence of equal effects in the two groups, not evidence that monotonicity holds or that the ratio identifies a complier-specific effect. If the effects differ, the ratio generally depends on both shares and both effects.

Diagnostics, Limitations, and Common Pitfalls

The compliance type of an individual is not observed: each person reveals treatment receipt under only the instrument value actually received. The first stage identifies the net difference \(p_C-p_F\) under instrument independence, not the two shares separately. Consequently, a nonzero first stage cannot establish that defiers are absent. Nor does observing a positive first stage show that every unit responds in the same direction.

A negative estimated first stage may prompt careful review of how the instrument and treatment are defined, the data, and the substantive response pattern. By itself, it does not tell an analyst the separate complier and defier shares or their effects. With a zero population first stage, the Wald ratio is unavailable; when the first stage is close to zero, the ratio can also be unstable to sampling variation. These are reasons to report the two contrasts and their uncertainty, not just a ratio.

• Pitfall: treating the weights as shares. In the positive-first-stage example, \(5/3\) and \(-2/3\) sum to one, but neither is an ordinary pair of nonnegative averaging weights. The correction is to call them signed coefficients and avoid describing the ratio as an average across compliance types.
• Pitfall: inferring monotonicity from a first stage. A positive net change can occur when both compliers and defiers exist. The correction is to distinguish the net first stage from the unobserved type-specific responses.
• Pitfall: assuming the reduced form must vanish when the first stage is zero. This is not true when complier and defier effects differ. The correction is to calculate \(p_C\tau_C-p_F\tau_F\) separately from \(p_C-p_F\).

Compared with the monotonicity-based LATE interpretation, the signed-mixture expression is more general but less directly interpretable as a single population subgroup average. Assuming equal effects across compliers and defiers can yield a common-effect interpretation, but that is a different substantive restriction. Neither algebraic result removes the need for instrument independence and exclusion; without those conditions, the observed contrasts need not have the potential-outcome interpretation used in the derivation. The contrasts alone also do not identify all four compliance-type shares or both group-specific effects.