The 2SLS Limiting Distribution Under Weak Instruments

Why the Denominator Matters

In just-identified instrumental variables estimation, the coefficient estimate is a ratio: an instrument-outcome sample moment divided by an instrument-treatment sample moment. With a sufficiently strong first stage, the denominator concentrates around a nonzero value, and the usual normal approximation follows. Under weak identification, that concentration need not occur. The denominator remains random on the same scale as its mean, and may come close to zero. The resulting limit is a ratio of random variables, not generally a normal distribution.

The local-to-zero framework makes this change precise. Instead of holding instrument relevance fixed as the sample grows, it lets the first-stage coefficient shrink at rate \(1/\sqrt{n}\). This tutorial derives the limit for a single endogenous regressor and a single instrument, after any exogenous regressors have been partialled out.

Model and Local-to-Zero Sequence

For observation \(i\), let \(y_i\) be the outcome, \(x_i\) the endogenous regressor, and \(z_i\) the scalar instrument. Write the structural equation and first stage as

y_i=\beta x_i+u_i,\qquad x_i=\pi_n z_i+v_i.

The parameter of interest is \(\beta\). The structural error \(u_i\) may be correlated with \(v_i\), which is why ordinary least squares need not identify \(\beta\). The instrument is required to satisfy the exclusion and exogeneity moment \(E[z_i u_i]=0\), and the first-stage residual satisfies \(E[z_i v_i]=0\). Relevance means that the first-stage coefficient is nonzero at a fixed sample size; the issue here is that its strength changes with \(n\).

Definition: In the local-to-zero sequence, the first-stage coefficient is \(\pi_n=c/\sqrt{n}\), where \(c\) is a fixed finite constant. The case \(c=0\) is allowed. Thus the population first stage becomes weaker as the sample size grows, at exactly the rate that makes its sample fluctuation consequential for 2SLS.

Assume the observations are independent and identically distributed within each sample, with stable moments and finite moments sufficient for a joint central limit theorem. Let \(Q=E[z_i^2]>0\). Define the covariance matrix of the instrument-weighted errors by

\Omega= \begin{pmatrix} \Omega_{uu}&\Omega_{uv}\\ \Omega_{uv}&\Omega_{vv} \end{pmatrix} = E\!\left[ z_i^2 \begin{pmatrix} u_i^2&u_i v_i\\ u_i v_i&v_i^2 \end{pmatrix} \right].

The entries describe the sampling variation in the two IV moments. In particular, \(\Omega_{uv}\) need not be zero: the random numerator and denominator can be correlated. We take \(\Omega_{vv}>0\), so that the limiting denominator has nonzero variance.

Deriving the Limiting Distribution

With one instrument and one endogenous regressor, the just-identified 2SLS estimator equals the IV ratio:

\hat{\beta}_{2SLS} = \frac{\sum_{i=1}^{n}z_i y_i}{\sum_{i=1}^{n}z_i x_i} = \beta+ \frac{\sum_{i=1}^{n}z_i u_i}{\sum_{i=1}^{n}z_i x_i}.

The equality follows by substituting \(y_i=\beta x_i+u_i\) into the numerator. To find the limit, scale the numerator and denominator of the error ratio by \(\sqrt{n}\). Define

U_n=\frac{1}{\sqrt{n}}\sum_{i=1}^{n}z_i u_i,\qquad V_n=\frac{1}{\sqrt{n}}\sum_{i=1}^{n}z_i v_i.

The joint central limit theorem gives \((U_n,V_n)\) a limiting bivariate normal distribution with mean zero and covariance matrix \(\Omega\). The denominator has the decomposition

\frac{1}{\sqrt{n}}\sum_{i=1}^{n}z_i x_i = \frac{\pi_n}{\sqrt{n}}\sum_{i=1}^{n}z_i^2+V_n = c\left(\frac{1}{n}\sum_{i=1}^{n}z_i^2\right)+V_n.

By the law of large numbers, \(n^{-1}\sum z_i^2\) converges to \(Q\), so this scaled denominator converges in distribution jointly with \(U_n\) to \(cQ+Z_v\). Here \((Z_u,Z_v)\) is a mean-zero bivariate normal vector with covariance matrix \(\Omega\). Consequently, provided the limiting denominator is zero with probability zero, the continuous mapping theorem yields

\hat{\beta}_{2SLS} \ \xrightarrow{d}\ \beta+\frac{Z_u}{cQ+Z_v}.

This is the local-to-zero limiting distribution. Its numerator and denominator are generally correlated, and the denominator has both a nonzero mean \(cQ\) and random variation. Since \(\Omega_{vv}>0\), \(cQ+Z_v\) is a continuous normal random variable and equals zero with probability zero, as required for the ratio limit.

Worked Example: Writing the Limiting Ratio

Suppose, hypothetically, that \(Q=1\), \(c=1\), \(\Omega_{uu}=4\), \(\Omega_{vv}=1\), and \(\Omega_{uv}=0.5\). The limiting vector has variances 4 and 1 and covariance 0.5. The local-to-zero limit is therefore

\hat{\beta}_{2SLS} \ \xrightarrow{d}\ \beta+\frac{Z_u}{1+Z_v}, \qquad \begin{pmatrix}Z_u\\Z_v\end{pmatrix} \sim N\!\left( \begin{pmatrix}0\\0\end{pmatrix}, \begin{pmatrix}4&0.5\\0.5&1\end{pmatrix} \right).

For one illustrative draw, suppose \(Z_u=1.1\) and \(Z_v=-0.2\). The ratio is \(1.1/(1-0.2)=1.1/0.8=1.375\), so the corresponding value of the estimator is \(\beta+1.375\). This calculation is an illustration of the random ratio, not a claim that this one draw characterizes its full distribution.

Why the Ratio Is Not Generally Normal

A ratio of normal variables is not generally normal. Even if the numerator and denominator are independent, a random denominator can make the ratio highly dispersed and give it substantial probability in the tails. When the two variables are correlated, their joint movement also affects the shape of the ratio distribution. When the denominator’s mean \(cQ\) is small relative to its standard deviation \(\sqrt{\Omega_{vv}}\), realizations near zero are especially consequential: dividing by a small value can produce a large estimate in either direction.

This differs from the fixed-strength, or strong-instrument, asymptotic result. If \(\pi_n=\pi\ne 0\) stays fixed, then \(n^{-1}\sum z_i x_i\) converges to \(\pi Q\), a nonzero constant. The numerator’s fluctuations are then small relative to the denominator, yielding

\sqrt{n}(\hat{\beta}_{2SLS}-\beta) \ \xrightarrow{d}\ N\!\left(0,\frac{\Omega_{uu}}{\pi^2 Q^2}\right).

Under local-to-zero asymptotics, by contrast, the denominator and numerator of the estimation error are both of order \(\sqrt{n}\) before taking their ratio. Neither can be replaced by a fixed nonzero limit in that ratio. The estimator is generally not root-\(n\) consistent around \(\beta\) under this sequence: its estimation error has a nondegenerate limit on the original scale.

Worked Example: A Small Denominator Creates a Large Estimate

Consider a hypothetical local-to-zero model with \(Q=1\), \(c=0.5\), \(\Omega_{uu}=\Omega_{vv}=1\), and \(\Omega_{uv}=0.8\). The two limiting normal variables each have variance 1 and correlation 0.8. One way to represent them is \(Z_u=0.8Z_v+0.6E\), where \(Z_v\) and \(E\) are independent standard normal variables. The limiting estimator is

\beta+\frac{0.8Z_v+0.6E}{0.5+Z_v}.

For the hypothetical values \(Z_v=-0.45\) and \(E=0.30\), the numerator is \(0.8(-0.45)+0.6(0.30)=-0.36+0.18=-0.18\), while the denominator is \(0.5-0.45=0.05\). The ratio is \(-0.18/0.05=-3.6\), giving \(\beta-3.6\). A modest numerator has produced a large deviation because the denominator is close to zero. The values illustrate the mechanics; they are not a data-based estimate of any real effect.

Worked Example: Computing a Hypothetical Sample IV Estimate

Suppose a hypothetical sample has \(n=400\), and, after partialling out any exogenous regressors, its relevant sums are \(\sum z_i^2=400\), \(\sum z_i x_i=3.6\), and \(\sum z_i y_i=9.0\). The just-identified 2SLS estimate is

\hat{\beta}_{2SLS} = \frac{9.0}{3.6} = 2.50.

The denominator is small relative to \(\sum z_i^2\): the sample first-stage coefficient implied by these moments is \(3.6/400=0.009\). To see the local-to-zero scaling, let the hypothetical sequence have \(c=1\), so \(\pi_n=1/\sqrt{400}=0.05\). The first-stage component of \(\sum z_i x_i\) is \(\pi_n\sum z_i^2=0.05(400)=20\); the observed denominator of 3.6 therefore corresponds to a first-stage residual contribution of \(3.6-20=-16.4\). Dividing the observed numerator and denominator by \(\sqrt{400}=20\) gives \(9.0/20=0.45\) and \(3.6/20=0.18\). Their ratio remains \(0.45/0.18=2.50\). Under the local sequence, both scaled moments retain sampling variation, so the sample ratio cannot be understood by treating its denominator as fixed.

The estimate is in units of outcome per unit of \(x_i\), according to the measurement units of \(y_i\) and \(x_i\). The calculation alone does not establish that 2.50 is near the structural coefficient; weak first-stage information can make the ratio unstable.

Worked Example: Probability of a Negative Limiting Denominator

Let \(Q=1\), \(c=0.5\), and \(\Omega_{vv}=1\). The limiting denominator is \(0.5+Z_v\), with \(Z_v\sim N(0,1)\). Its probability of being negative is

P(0.5+Z_v<0) = P(Z_v<-0.5) = \Phi(-0.5) \approx 0.3085.

Thus in this hypothetical limiting distribution the denominator is negative about 30.85% of the time. A negative denominator does not by itself make the estimator undefined, but it shows that the random denominator is not behaving like a reliably positive fixed quantity. The approximation is for the limiting distribution under the specified local sequence, not a universal probability for weak instruments.

Common Mistake: Substituting a Strong-Instrument Standard Error

A common mistake is to derive a normal approximation under a fixed nonzero first-stage coefficient and then apply it unchanged when the first stage is weak. The reasoning is that the sample size is large, so the denominator must be close to its population value. But under \(\pi_n=c/\sqrt{n}\), the population first-stage moment itself shrinks at the same rate that the denominator’s sampling fluctuation matters after scaling. The derivation above shows that the scaled denominator converges to \(cQ+Z_v\), not simply \(cQ\). Replacing it by its mean discards a component of the limiting distribution.

A strong-instrument normal approximation can be informative when the first-stage strength is fixed and nonzero in the asymptotic sequence. It is not generally justified by sample size alone under local-to-zero asymptotics. The distinction matters because inference based on the wrong limiting distribution can misstate uncertainty. Weak-identification-robust inference requires methods designed for this setting; the ratio limit explains why those methods are needed, without itself prescribing a particular test.

What the Limit Establishes

The local-to-zero result is a distributional statement about a sequence of estimators under a specified sequence of first stages. It does not say that every weak-instrument estimate will be extreme, nor does it determine whether a particular instrument is valid. Rather, it identifies the key mechanism: with weak relevance, the same random sample variation affects both the IV numerator and its denominator, and the denominator does not become effectively constant. This is why the usual strong-instrument normal approximation and the weak-identification limit are different.

Key takeaway: For just-identified 2SLS with \(\pi_n=c/\sqrt{n}\), the limit is \(\beta+Z_u/(cQ+Z_v)\), where \((Z_u,Z_v)\) is jointly normal with covariance determined by the instrument-weighted structural and first-stage errors. The random denominator is central to the result.

Check Your Understanding

Use the definitions and limiting result to answer these questions.

In the local-to-zero sequence, what happens to the first-stage coefficient as \(n\) increases, and why does the denominator’s sampling variation remain important?
Write the limiting distribution of just-identified 2SLS in terms of \(c\), \(Q\), \(Z_u\), and \(Z_v\).
Which covariance-matrix entry determines whether the limiting numerator and denominator are correlated?
For \(Q=1\), \(c=1\), and \(\Omega_{vv}=1\), what is the probability that the limiting denominator is negative?
Why does the fixed-strength normal approximation not follow merely from having a large sample under a local-to-zero sequence?