Bounds from Minimum Instrument-Strength Restrictions

Why a Minimum-Strength Restriction Produces Bounds

With a valid instrument, the reduced-form coefficient links the instrument to the outcome, while the first-stage coefficient links it to the endogenous regressor. In the scalar model, their ratio is the structural coefficient. If the first-stage coefficient is not known exactly, that ratio is not generally a point. A known positive lower bound on first-stage strength can nevertheless limit how large the structural coefficient can be.

The resulting bounds depend on two ingredients: the reduced-form coefficient and the strength restriction. If the reduced form is known exactly, the restriction produces an identified set. If the reduced form is estimated with uncertainty, its interval must also be carried through the ratio. The resulting set can be wider and may include structural coefficients of either sign.

Scalar Setup and Assumptions

Let \(Y\) be an outcome, \(X\) an endogenous regressor, and \(Z\) a scalar instrument. Consider the structural equation \(Y=\theta X+U\), where \(\theta\) is the scalar coefficient of interest and \(U\) is the structural error. Define \(\pi\) as the population first-stage coefficient of \(Z\) in the linear projection of \(X\) on \(Z\), and \(\rho\) as the population reduced-form coefficient of \(Z\) in the linear projection of \(Y\) on \(Z\). For example, without additional controls, these coefficients are \(\pi=\operatorname{Cov}(Z,X)/\operatorname{Var}(Z)\) and \(\rho=\operatorname{Cov}(Z,Y)/\operatorname{Var}(Z)\), provided \(\operatorname{Var}(Z)>0\).

The bound calculation requires the following structural relation and restriction. The instrument has no direct effect on \(Y\) apart from its effect through \(X\), and it is uncorrelated with the structural error: \(\operatorname{Cov}(Z,U)=0\). The first-stage coefficient is positive and satisfies the substantive restriction \(\pi\geq c\), where \(c>0\) is specified in the same units as \(\pi\). There is no assumed upper bound on \(\pi\).

To see the reduced-form relation, substitute the structural equation into its covariance with \(Z\). Then \(\operatorname{Cov}(Z,Y)=\theta\operatorname{Cov}(Z,X)+\operatorname{Cov}(Z,U)\). The last term is zero by instrument exogeneity, so division by \(\operatorname{Var}(Z)\) gives \(\rho=\theta\pi\). The exclusion and exogeneity assumptions are substantive: a numerical bound cannot make an invalid instrument valid.

The positive sign restriction matters. It treats the direction of the first stage as known, as well as imposing a minimum magnitude. The value of \(c\) also depends on the scale of \(Z\): rescaling the instrument rescales \(\pi\), and therefore requires rescaling \(c\) consistently. The ratio \(\rho/\pi\), in contrast, has the units of \(Y\) per unit of \(X\).

Bounds When the Reduced Form Is Known

For any admissible first stage, \(\theta=\rho/\pi\), with \(\pi\geq c>0\). If \(\rho>0\), increasing \(\pi\) decreases the positive ratio, so the largest value is \(\rho/c\). As \(\pi\) grows without bound, the ratio approaches zero but never equals zero for finite \(\pi\). Thus the identified set is \((0,\rho/c]\). If \(\rho<0\), the ratio is negative; its smallest value is \(\rho/c\), and it approaches zero from below. The set is \([\rho/c,0)\). If \(\rho=0\), the ratio is zero for every admissible \(\pi\), so the identified set is \(\{0\}\).

The open endpoint at zero is not a technicality: no finite value of \(\pi\) produces \(\theta=0\) when \(\rho\neq 0\). However, zero is the infimum or supremum of the set, depending on the sign of \(\rho\). If reporting a closed interval for convenience, label it as the closure of the identified set rather than claiming its zero endpoint is attained.

Carrying Reduced-Form Uncertainty into the Bounds

In applications, \(\rho\) is usually estimated rather than known. Suppose an interval \([L,U]\) is available for \(\rho\), with \(L\leq U\), and retain the restriction \(\pi\geq c>0\). The set of coefficient values compatible with the interval and the strength restriction is

When the reduced-form interval is entirely positive, the largest ratio is \(U/c\), attained by \(r=U\) and \(p=c\); ratios can approach zero from above as \(p\) increases. When the interval is entirely negative, the smallest ratio is \(L/c\), and ratios approach zero from below. If the interval contains zero, both extreme endpoints are attained at \(p=c\), while zero is also attainable by setting \(r=0\).

If \([L,U]\) is a confidence interval for \(\rho\) with coverage \(1-\alpha\), the projected set above is a confidence set for \(\theta\), under the stated model and the known restriction \(\pi\geq c\). To justify the coverage claim, whenever the interval contains the true \(\rho\), the true coefficient satisfies \(\theta=\rho/\pi\) for some \(\pi\geq c\). It must therefore belong to the projected set. Thus the projected set covers at least as often as the original interval covers \(\rho\). The shape of the set is important: an interval for the reduced form that includes zero generally yields a confidence set containing zero, even though the first stage is bounded away from zero.

What the Bounds Do—and Do Not—Establish

A lower bound on first-stage strength limits the magnitude of the ratio when the reduced form is fixed: \(|\theta|\leq|\rho|/c\). It does not provide a positive lower bound on \(|\theta|\) unless there is also a finite upper bound on \(\pi\). A first stage can be arbitrarily large under the stated restriction, making the ratio arbitrarily close to zero.

This is a partial-identification result, not a test of whether an observed first stage is strong. The restriction \(\pi\geq c\) must be justified as a constraint on the population coefficient; it is not automatically true because a sample estimate exceeds \(c\). Also, a bound on \(\pi\) is scale-dependent, so its units and instrument normalization must be reported. The result concerns a scalar coefficient under a valid-instrument relation. It does not, by itself, establish a causal interpretation when the exclusion or exogeneity assumptions fail.

• Exact reduced form: the identified set is generally one-sided and has zero as an unattained endpoint when \(\rho\neq 0\).
• Uncertain reduced form: project the entire interval, not only its point estimate, through the ratio restriction.
• Interval includes zero: the projected set includes zero and can include both positive and negative structural coefficients.

Common mistake: “The reduced-form estimate is positive, and the first stage is positive, so the structural coefficient must be positive.” The correction is that the sign conclusion requires the reduced-form information to support positivity. If a confidence interval for \(\rho\) includes zero, the projected confidence set can include zero and negative values despite the positive first-stage restriction.