Identified Sets When First-Stage Strength Is Zero

Why Zero First-Stage Strength Changes the Identification Question

Instrumental variables use variation in an instrument that predicts the endogenous regressor to learn about a structural coefficient. When the population first-stage coefficient is zero, that source of variation is absent. The central question is then not how to calculate a ratio of reduced-form to first-stage coefficients, but which structural values remain consistent with the population reduced forms and the maintained assumptions.

The answer depends on both reduced forms. If the instrument has no population association with the endogenous regressor and also has no population association with the outcome, the reduced forms impose no restriction on the structural coefficient. If the first stage is zero but the outcome reduced form is nonzero, the two cannot be reconciled with the usual linear instrumental-variables model and its exclusion restriction. Parameter-space restrictions can limit the set of compatible coefficients, but they do not create identification from the instrument.

Reduced Forms and the Identified Set

Consider a scalar endogenous regressor \(D\), an outcome \(Y\), and a vector of \(K\) instruments \(Z\). Assume variables have been centered and any included exogenous controls have been partialled out. Write the structural equation and the two population reduced forms as

Here \(\beta\) is the structural coefficient of interest, \(\boldsymbol{\pi}\) is the \(K\)-vector of first-stage coefficients, and \(\boldsymbol{\delta}\) is the \(K\)-vector of reduced-form coefficients for the outcome. The errors \(V\) and \(\varepsilon\) are the population linear-projection residuals, so they are orthogonal to the instruments. The structural error is \(U=Y-\beta D\).

The maintained instrumental-variables assumptions relevant to the argument are that the structural equation holds and the instruments are excluded from it: \(E[ZU]=\boldsymbol{0}\). Instruments must also have enough independent variation for the stated reduced-form coefficient vectors to be defined. This tutorial characterizes the population set implied by these restrictions; it does not treat sample estimates as if they were population coefficients.

The key population implication comes from substituting the first stage into the structural equation. Since \(Y=\beta D+U\), its instrument-related component is \(\beta Z'\boldsymbol{\pi}\), while the exclusion restriction prevents \(U\) from contributing an instrument-related component. Therefore the reduced-form coefficients must satisfy

This is a vector equality: every instrument’s outcome reduced-form coefficient must equal its first-stage coefficient multiplied by the same structural coefficient. Consequently, the identified set based on these reduced-form restrictions is

What Happens When the First Stage Is Zero?

Zero first-stage strength means that every component of the population first-stage vector is zero: \(\boldsymbol{\pi}=\boldsymbol{0}\). It does not merely mean that one instrument has a zero coefficient while another predicts \(D\). Substituting the zero vector into the reduced-form restriction gives

There are two cases. If the outcome reduced-form vector is also zero, every value allowed by \(\mathcal{B}\) satisfies the reduced-form restriction. If the outcome reduced-form vector is not zero, no value of \(\beta\) can satisfy it. Thus the set is not automatically all real numbers: it is all of the admissible parameter space in the compatible case, and empty in the incompatible case.

The result can be justified directly from the definition. In the first case, for any \(b\in\mathcal{B}\), \(b\boldsymbol{\pi}=b\boldsymbol{0}=\boldsymbol{0}=\boldsymbol{\delta}\); hence every admissible value belongs to the set. In the second case, for every \(b\), \(b\boldsymbol{\pi}=\boldsymbol{0}\), which cannot equal a nonzero \(\boldsymbol{\delta}\); hence no admissible value belongs to the set. This argument establishes the characterization without dividing by a first-stage coefficient.

Bounded and Otherwise Restricted Parameter Spaces

The identified set is defined relative to the admissible space \(\mathcal{B}\), which may contain substantive restrictions on \(\beta\). A bounded parameter space can make the set bounded even though the instruments provide no information that selects one coefficient over another. The bound comes from the maintained restriction, not from the zero-strength instrument.

Boundedness should not be confused with identification. A narrow admissible range may be useful in an application, but unless the population restrictions reduce it to one value, the structural coefficient remains set identified. Nor does a parameter restriction repair a conflict between the two reduced forms: if \(\boldsymbol{\pi}=\boldsymbol{0}\) and \(\boldsymbol{\delta}\ne\boldsymbol{0}\), restricting \(\beta\) to a bounded interval still leaves the identified set empty.

Interpretation and Practical Importance

When first-stage strength is zero and both reduced forms are zero, the data implications used here are consistent with every admissible structural coefficient. There is no valid ratio \(\delta/\pi\): it would be \(0/0\), which is undefined and does not determine a value. When the outcome reduced form is nonzero, a nonzero numerator divided by a zero first-stage coefficient is also not a meaningful estimate of \(\beta\); the population restrictions instead conflict.

These are statements about population identification. In an empirical analysis, estimated first-stage and reduced-form coefficients are random and need not equal zero exactly. A small estimated first stage is not by itself proof that the population first stage is exactly zero, and plugging noisy estimates into a ratio does not answer the population set-identification question. Weak-instrument-robust inference addresses uncertainty in settings where conventional ratio-based inference can be unreliable; the population characterization here explains the limiting identification problem when strength is exactly zero.

The vector formulation also clarifies what “zero strength” means with several instruments. All first-stage coefficients must be zero for the result above to apply. If at least one component of \(\boldsymbol{\pi}\) is nonzero, the equation \(\boldsymbol{\delta}=\beta\boldsymbol{\pi}\) can restrict \(\beta\); the zero-strength case is specifically the case where no component supplies that restriction.

Common Pitfall

• Wrong reasoning: “The instrument has no effect on \(D\), so the structural effect \(\beta\) must be zero.” This confuses the first-stage coefficient, which describes how the instrument predicts \(D\), with the structural coefficient describing how \(D\) relates to \(Y\).
• Correction: Use \(\boldsymbol{\delta}=\beta\boldsymbol{\pi}\). When \(\boldsymbol{\pi}=\boldsymbol{0}\) and \(\boldsymbol{\delta}=\boldsymbol{0}\), every admissible \(\beta\) satisfies the restriction. When \(\boldsymbol{\delta}\ne\boldsymbol{0}\), no \(\beta\) does. Neither case identifies \(\beta=0\).
• Another error: Calling a bounded admissible interval a point estimate. A bounded identified set can still contain many values; state its endpoints and the assumptions that impose them.