Strong and Local-to-Zero Instrument Sequences

Why the Instrument’s Sequence Matters

Instrument strength is not determined by whether a first-stage coefficient is nonzero in one particular sample. In asymptotic analysis, the coefficient may itself depend on the sample size \(n\). The key question is how quickly the first-stage signal changes relative to sampling noise. A coefficient that stays fixed as \(n\) grows leads to a different inferential regime from one that shrinks at rate \(1/\sqrt{n}\).

We use a scalar instrumental-variable model, with controls understood to have been partialled out if present: \(y_i=\beta x_i+u_i\) and \(x_i=\pi_n z_i+v_i\). Here \(z_i\) is the instrument, \(x_i\) is the endogenous regressor, and \(\pi_n\) is its first-stage coefficient. The errors \(u_i\) and \(v_i\) may be correlated; validity requires \(E[z_i u_i]=0\), while the first-stage equation requires \(E[z_i v_i]=0\). For simplicity, suppose observations are iid within each sample, the distribution of \((z_i,v_i,u_i)\) does not change with \(n\), and the moments used below are finite. Assume \(E[z_i]=0\), \(q=E[z_i^2]>0\), and \(0<\omega=E[z_i^2v_i^2]<\infty\).

The first-stage least-squares estimator is \(\hat{\pi}_n=\sum_{i=1}^n z_i x_i/\sum_{i=1}^n z_i^2\). Its sampling error is governed by the same noise term under both asymptotic regimes. What differs is the size of the signal \(\pi_n\).

What Happens to the First-Stage Estimator?

Under the stated assumptions, the first-stage estimator can be written as \(\hat{\pi}_n=\pi_n+\{\sum_{i=1}^n z_i v_i\}/\{\sum_{i=1}^n z_i^2\}\). The law of large numbers gives \(n^{-1}\sum z_i^2\xrightarrow{p}q\). The central limit theorem gives \(n^{-1/2}\sum z_i v_i\xrightarrow{d}N(0,\omega)\), since \(E[z_i v_i]=0\) and \(\operatorname{Var}(z_i v_i)=\omega\).

Proof. From the estimator identity, \[ \sqrt{n}(\hat{\pi}_n-\pi_n) = \frac{n^{-1/2}\sum_{i=1}^n z_i v_i}{n^{-1}\sum_{i=1}^n z_i^2}. \] The numerator converges in distribution to \(N(0,\omega)\), and the denominator converges in probability to the positive constant \(q\). Slutsky’s theorem therefore gives \(\sqrt{n}(\hat{\pi}_n-\pi_n)\xrightarrow{d}N(0,\omega/q^2)\). For fixed \(\pi_n=\pi\), this also implies \(\hat{\pi}_n-\pi\xrightarrow{p}0\), so the estimator is consistent. For \(\pi_n=c/\sqrt{n}\), write \(\sqrt{n}\hat{\pi}_n=c+\sqrt{n}(\hat{\pi}_n-\pi_n)\). Adding the constant \(c\) gives the stated shifted normal limit. \(\square\)

The distinction is visible in the scale of \(\hat{\pi}_n\). With fixed strength, its standard error shrinks at rate \(1/\sqrt{n}\) while its target remains away from zero. With a local-to-zero coefficient, the target itself is of that same order. Thus, multiplying the estimate by \(\sqrt{n}\) leaves both signal and noise in the limit; the estimate does not become large relative to its sampling uncertainty.

Identification and the Concentration Parameter

The population IV moment at a candidate coefficient \(b\) is \(E[z_i(y_i-bx_i)]=(\beta-b)E[z_i x_i]\), because \(E[z_i u_i]=0\). From the first-stage equation, \(E[z_i x_i]=\pi_n q\). Thus, when \(\pi_n\ne0\), the population moment has a unique zero at \(b=\beta\). In that finite-sample-size population sense, the parameter is identified. But the sensitivity of the moment to \(b\) is \(-\pi_n q\), which approaches zero under a local-to-zero sequence. Identification becomes weak in the asymptotic sense: small changes in \(b\) produce increasingly small changes in the population moment.

In the homoskedastic first-stage setting, with \(\operatorname{Var}(v_i)=\sigma_v^2\), the concentration parameter has the familiar population-scale expression \(\lambda_n=n\pi_n^2q/\sigma_v^2\). This expression makes the rate distinction explicit: it measures the squared first-stage signal accumulated across \(n\) observations relative to first-stage noise.

Derivation. Substituting \(\pi_n=c n^{-\alpha}\) into the expression for \(\lambda_n\) gives \[ \lambda_n=\frac{n(c n^{-\alpha})^2q}{\sigma_v^2} =\frac{c^2q}{\sigma_v^2}n^{1-2\alpha}. \] The exponent \(1-2\alpha\) is positive, zero, or negative according as \(\alpha\) is below, equal to, or above \(1/2\). Since the leading constant is positive, these three cases imply divergence, a positive constant, or convergence to zero, respectively. \(\square\)

Why the Local-to-Zero Case Changes Inference

A useful way to see the boundary rate is to compare the first-stage signal with its sampling error. In the model above, \(\hat{\pi}_n-\pi_n\) is of order \(1/\sqrt{n}\). If \(\pi_n\) is fixed and nonzero, the signal is larger than this error as \(n\) increases. If \(\pi_n=c/\sqrt{n}\), the two have the same order. If \(\pi_n\) shrinks faster than \(1/\sqrt{n}\), the signal is smaller still. The local-to-zero case is therefore not merely a small-coefficient version of the fixed-strength case; it changes which terms remain relevant asymptotically.

The same issue appears in the sample IV denominator, \(n^{-1}\sum z_i x_i\). Its population counterpart is \(\pi_n q\). Under fixed strength, the sample denominator converges to the nonzero value \(\pi q\). Under local-to-zero strength, its population value is of order \(1/\sqrt{n}\), while sampling variation in \(n^{-1}\sum z_i v_i\) is also of order \(1/\sqrt{n}\). The denominator is no longer approximated by a stable nonzero constant. The resulting weak-instrument behavior is why conventional strong-instrument approximations cannot simply be carried over. The joint reduced-form limit and the resulting IV estimator limit require separate treatment.

Common Mistake

Incorrect reasoning: “For every \(n\), \(\pi_n\ne0\), so the instrument is strong and the usual large-sample IV approximation applies.” This confuses finite-\(n\) population identification with the strength of identification along a sequence.

Correction: A nonzero \(\pi_n\) gives a unique population moment zero at that \(n\), but it says nothing by itself about how sharply that moment distinguishes nearby values of \(\beta\). For \(\pi_n=c/\sqrt{n}\), the moment’s slope approaches zero and the signal remains comparable to sampling noise. Check the sequence’s rate, or equivalently the behavior of the concentration parameter, rather than checking only whether each coefficient is nonzero.