The Scalar Weak-Instrument Model

The Scalar IV Setup

A single instrument and a single endogenous regressor give the simplest setting in which instrument strength can be discussed. The key distinction is between relevance—whether the instrument predicts the endogenous regressor—and strength—how precisely that predictive relationship is estimated. A nonzero first-stage coefficient is relevant in the population, but may still be too small relative to its sampling uncertainty to provide much information in a given sample.

Use centered variables, or equivalently suppose an intercept has been removed from each equation. For observation \(i\), let \(y_i\) be the outcome, \(x_i\) the endogenous regressor, \(z_i\) the instrument, and \(u_i\) the structural error. The scalar structural and first-stage equations are

Here \(\beta\) is the structural coefficient of interest, \(\pi\) is the population first-stage coefficient, and \(v_i\) is the first-stage disturbance. We consider independent and identically distributed observations unless otherwise stated. Instrument validity requires the instrument to be orthogonal to the structural error: \(E[z_i u_i]=0\). The first-stage condition \(E[z_i v_i]=0\) makes \(\pi\) the population coefficient in the linear projection of \(x_i\) on \(z_i\). The instrument is relevant when \(\pi\ne 0\), provided \(E[z_i^2]>0\).

The regressor is endogenous when it is correlated with the structural error, \(E[x_i u_i]\ne 0\). The model permits this because \(x_i\) contains \(v_i\), which may be related to \(u_i\). Validity instead requires \(z_i\) to be uncorrelated with \(u_i\). These are distinct conditions: endogeneity motivates the use of an instrument, while validity and relevance are the conditions that let it identify \(\beta\).

Why Relevance Identifies the Structural Coefficient

The reduced-form relationship between the instrument and outcome follows by substituting the first-stage equation into the structural equation:

Taking moments with \(z_i\), validity gives \(E[z_i u_i]=0\), and the first-stage projection gives \(E[z_i v_i]=0\). Thus the instrument-outcome moment is \(\beta\) times the instrument-regressor moment.

Proof. From \(x_i=\pi z_i+v_i\),

Similarly, \(y_i=\beta x_i+u_i\) and validity imply

Since \(\pi\ne0\) and \(E[z_i^2]>0\), \(E[z_i x_i]\ne0\). Dividing the last equality by this nonzero moment yields the stated ratio. \(\square\)

This argument also shows what fails when \(\pi=0\): the instrument has no population covariance with \(x_i\), so the ratio has a zero denominator. That is a failure of identification, not merely imprecise estimation. When \(\pi\) is nonzero but small, the ratio exists, yet its denominator may be difficult to estimate precisely.

Sampling Uncertainty in the First Stage

The sample first-stage coefficient, using the centered no-intercept notation, is

Substituting \(x_i=\pi z_i+v_i\) gives the exact error identity

This identity makes the role of the sampling design visible: the numerator is first-stage noise, and the denominator measures the sample variation in the instrument. A sampling statement requires assumptions about how observations are generated; marginal facts about each \(z_i\) alone are not enough.

Proof. Let \(Q_n=n^{-1}\sum_{i=1}^n z_i^2\). The iid assumption and finite \(E[z_i^2]\) imply, by the law of large numbers, \(Q_n\xrightarrow{p}Q\). Let \(W_i=z_i v_i\). These are iid, have mean zero, and have finite variance \(\Omega\). The central limit theorem therefore gives

The exact error identity can be written as \(\sqrt n(\hat\pi-\pi)=[n^{-1/2}\sum_i z_i v_i]/Q_n\). Since \(Q_n\xrightarrow{p}Q>0\), Slutsky’s theorem yields the claimed limit with variance \(\Omega/Q^2\). Under the additional conditional variance assumption,

Substitution gives \(\Omega/Q^2=\sigma_v^2/Q\), as stated. \(\square\)

The asymptotic standard error is consequently \(\sqrt{\Omega/(nQ^2)}\); under the stated homoskedastic condition it is \(\sigma_v/\sqrt{nQ}\). The first-stage signal-to-uncertainty comparison is the magnitude of \(\pi\) relative to this standard error. This is a way to describe the strength of the first-stage signal, not an absolute definition with a universal numerical cutoff.

Interpreting Weakness and Avoiding Common Errors

A weak instrument is not necessarily an invalid instrument. Validity concerns the relation between \(z_i\) and \(u_i\); weakness concerns how much information the first stage provides about \(x_i\). Likewise, a first-stage coefficient can be nonzero but weak relative to its sampling uncertainty. Conversely, the same nonzero coefficient can be estimated more precisely with a larger sample under the iid conditions used above.

Common mistake: “The first-stage coefficient is nonzero, so the instrument is strong.” This confuses a population relevance condition with a strength assessment. The correction is to compare the coefficient’s magnitude with its sampling uncertainty, which depends on the variation in \(z_i\), the variation in \(v_i\), and the sample size under an appropriate sampling model.

Another error is to infer that \(\sum_i z_i^2\) grows with \(n\) merely because each instrument has a finite positive second moment. Without a condition such as iid sampling, the observations could be dependent in a way that prevents the sample sum from growing typically. The law of large numbers, not the marginal moment alone, supports the proportional-growth statement.

The distinction matters because the scalar IV ratio divides by the sample instrument-regressor association. When that association is estimated imprecisely, the ratio can be unstable even though the instrument satisfies the population validity condition. The appropriate first question is therefore not only whether the instrument is relevant, but also whether its first-stage signal is large relative to the uncertainty with which that signal is measured.