Choosing Weak-Instrument-Robust Inference Procedures

Why the Choice of Procedure Matters

A conventional 2SLS confidence interval can look precise even when the instruments provide little information about the endogenous regressor. In that setting, its usual standard-error approximation may not describe the sampling uncertainty well: a test can reject too often, and a confidence interval can have lower coverage than its stated level. A weak-instrument-robust procedure instead aims to control the rejection probability under the null without requiring the first stage to be strong.

These procedures do not repair invalid instruments. Their conclusions still depend on the exclusion restriction and instrument exogeneity, as well as an appropriate model for the outcome and treatment. Their purpose is narrower: to make inference about a structural coefficient reliable when relevance is weak or uncertain.

Model and the Meaning of Weak-Instrument Robustness

Consider one endogenous regressor \(x\), an outcome \(y\), and \(q\) excluded instruments collected in \(Z\). Any included exogenous regressors are understood to have been partialled out of all variables. Write the structural equation as \(y=\beta x+u\), and the reduced forms as \(x=Z\pi_x+v_x\) and \(y=Z\pi_y+v_y\). The coefficient of interest is \(\beta\).

The maintained IV assumptions include instrument exogeneity, \(E[Z'u]=0\), and exclusion: the instruments affect \(y\) only through \(x\). Relevance requires that the instruments predict \(x\), represented by \(\pi_x\ne 0\). Weak identification means that this prediction is too small, relative to sampling uncertainty, for standard strong-identification approximations to be dependable. Weak-instrument robustness concerns inference that remains valid even when \(\pi_x\) is small; it does not mean the data will necessarily provide a narrow or informative answer.

For a candidate value \(\beta_0\), the structural null \(H_0:\beta=\beta_0\) implies a reduced-form restriction. Substituting the structural equation into the reduced form gives \(\pi_y=\beta\pi_x\), so under the null, \(\pi_y-\beta_0\pi_x=0\). The procedures below differ in how they test this restriction and account for uncertainty in first-stage strength.

Four Procedures and Their Trade-Offs

Wald Inference: Useful Under Strong Identification

The familiar Wald statistic for testing \(H_0:\beta=\beta_0\) is

With strong identification and suitable regularity conditions, this statistic has an approximately standard normal null distribution. A two-sided test rejects at level \(\alpha\) when \(|W|\) exceeds the corresponding standard normal critical value. A confidence interval is then formed by inverting those tests. Robust 2SLS standard errors can address certain forms of heteroskedasticity or clustering, but that adjustment alone does not make the Wald procedure robust to weak identification.

Anderson–Rubin Inference

The AR procedure tests the null-implied restriction directly. For each proposed \(\beta_0\), form \(y-\beta_0x\) and test whether its coefficient on the excluded instruments is zero, while accounting for included exogenous regressors if present. In the homoskedastic linear model with independent, normally distributed errors and fixed instruments, the usual regression \(F\)-test has an exact finite-sample reference distribution under the null. Without those distributional assumptions, heteroskedasticity-robust or other appropriate versions typically use asymptotic calibration.

The key point is that this test does not divide by an estimate of the first-stage coefficient. Under \(H_0\), the instrument restriction holds whether the first stage is strong or weak. Thus, when the stated error assumptions and IV restrictions hold, weak relevance does not invalidate the null calibration. If the instruments carry little information, however, the test may rarely reject a wide range of candidate values.

Score and Conditional Likelihood Ratio Procedures

The score procedure, often implemented as a Lagrange multiplier or Kleibergen-type test, evaluates the likelihood score for \(\beta\) under the null while treating the first-stage coefficients as nuisance parameters. The nuisance parameters are not simply fixed at their estimated values: the score is adjusted to account for estimation of those nuisance quantities. With the appropriate weak-identification-robust reference distribution, the test can control size under weak first stages. Its calibration is not automatically exact in every finite sample; the model, covariance assumptions, and asymptotic method used for the reference distribution must be stated.

The CLR procedure starts from the likelihood ratio for the candidate structural value, comparing the fit under the null with the unrestricted fit. In the Gaussian reduced-form model, the first-stage strength is a nuisance parameter. CLR conditions on a statistic that carries information about this nuisance parameter, then uses the resulting conditional null distribution to determine critical values or a p-value. This conditional calibration is why the procedure can retain size control under weak identification. It also commonly has better power than AR in the canonical model, though no method is uniformly most powerful in every design and model.

For both score and CLR, the label alone is not enough to justify a result. A procedure derived for a homoskedastic Gaussian model should not be treated as automatically valid under arbitrary heteroskedasticity or clustering. Use a version whose variance assumptions and reference distribution match the design. Under strong identification, Wald inference may be simpler and comparably informative; under uncertain strength, a robust procedure protects size at a potential cost in interval width.

Inverting Tests: Why Confidence Sets Can Look Unusual

A confidence set at level \(1-\alpha\) is obtained by testing every candidate value \(\beta_0\) at level \(\alpha\) and retaining the values that are not rejected. Under weak identification, the data may not sharply distinguish among many possible effects. Consequently, an AR, score, or CLR confidence set can be very wide, unbounded, disconnected, or even equal to the whole real line. Such a set is not a computational defect: it communicates that the data and assumptions do not rule out a broad range of effects at the chosen level.

For an AR test with one instrument, suppose the reduced-form instrument coefficients are estimated as \(\hat{\pi}_y=0.60\) and \(\hat{\pi}_x=0.20\). Suppose their estimated variances are \(0.0144\) and \(0.0064\), with covariance \(0.0060\). Under a joint normal approximation, the reduced-form contrast for a candidate value is \(\hat{\pi}_y-\beta_0\hat{\pi}_x\), with estimated variance \(0.0144+0.0064\beta_0^2-0.0120\beta_0\). A two-sided 95% test retains values satisfying

Expanding and collecting terms gives approximately

The roots are approximately \(1.84\) and \(10.7\). Since the quadratic coefficient is positive, the inequality holds between the roots, giving the illustrative confidence set \([1.84,10.7]\). This simplified calculation uses an estimated covariance and a normal critical value, so it illustrates inversion rather than claiming exact finite-sample coverage. In other data, the coefficient pattern can instead yield an unbounded set or no finite interval.

A Practical Choice and a Common Pitfall

Choose the procedure by asking what kind of validity is needed and which assumptions can be defended. If relevance is convincingly strong and the standard asymptotic approximation is appropriate, Wald inference is straightforward. If weak relevance is plausible, report an appropriate weak-IV-robust test and confidence set. AR offers a direct, transparent test and can be a useful benchmark; score and CLR can improve power, but require their own valid calibration. Reporting more than one robust procedure can reveal whether conclusions depend on the test’s power properties.

A common mistake is to use heteroskedasticity-robust 2SLS standard errors and conclude that Wald inference is therefore protected against weak instruments. Robust standard errors address the estimated variance under their stated conditions; they do not remove the strong-identification approximation behind the usual Wald reference distribution. The correction is to distinguish variance robustness from identification robustness and use a weak-IV-robust test when first-stage strength is in doubt.