Local Power of Weak-Instrument-Robust Tests

Why Compare Local Power?

Weak-instrument-robust tests are designed to control rejection probability even when the first stage is weak. That protection does not make their power identical. When the parameter is not exactly at its null value, a test’s local power is the probability that it rejects under a specified alternative in the limiting experiment that retains weak-instrument uncertainty. Comparing these probabilities shows where a test can gain sensitivity without relying on a strong-first-stage approximation.

The Anderson–Rubin (AR), Kleibergen LM (KLM), and conditional likelihood-ratio (CLR) tests use the same data but exploit it differently. AR is a useful benchmark because its null statistic has a distribution that does not depend on the unknown first-stage signal. KLM and CLR use additional structure in the joint reduced-form information. Their power gains are not automatic or uniform across all configurations; their appeal is that they can use this information while retaining weak-identification-robust calibration.

A Canonical Local Experiment

Use the following Gaussian canonical representation for testing one structural coefficient at a null value \(\beta_0\) with \(k\) instruments. The vector \(U\) is the standardized reduced-form outcome after subtracting \(\beta_0\) times the reduced-form treatment; \(V\) is the standardized first-stage reduced form. The parameter \(\eta\) represents the first-stage signal, and \(h\) is the local structural departure from the null. The correlation \(\rho\) describes dependence between the two reduced-form errors.

\begin{pmatrix}U\\V\end{pmatrix} \sim N\left[ \begin{pmatrix}h\eta\\\eta\end{pmatrix}, \begin{pmatrix}I_k&\rho I_k\\\rho I_k&I_k\end{pmatrix} \right], \qquad |\rho|<1.

Here \(I_k\) is the \(k\)-dimensional identity matrix. Under \(H_0:\beta=\beta_0\), \(h=0\), so \(U\) has mean zero, while \(V\) may have any mean \(\eta\). The weak-instrument asymptotic framework keeps \(\eta\) fixed as the sample size grows; it does not require the first-stage signal to diverge. A local alternative keeps \(h\) fixed and nonzero in this experiment. The model is a device for comparing limiting rejection probabilities, not a claim that a particular finite sample is exactly Gaussian.

Definition: For a test \(j\) with rejection rule \(\phi_j\), its local power at \((h,\eta,\rho)\) is \(\pi_j(h,\eta,\rho)=P_{h,\eta,\rho}(\phi_j=1)\), using the test’s weak-identification-robust critical rule. Its null rejection probability is \(\pi_j(0,\eta,\rho)\). A level-\(\alpha\) test has null rejection probability no greater than \(\alpha\) over the nuisance values covered by its validity guarantee.

Anderson–Rubin: A Benchmark Power Curve

In this representation the AR statistic is \(U'U\). Its validity follows from the marginal distribution of \(U\): under the null, \(U\sim N(0,I_k)\) for every first-stage signal \(\eta\). Thus an upper-tail critical value from a central chi-square distribution gives a test whose null rejection probability does not depend on \(\eta\) or \(\rho\). Under a local alternative, the distribution becomes noncentral chi-square.

Result: Under \(H_0\), \(U'U\sim\chi^2_k\). Under the local alternative, \(U'U\sim\chi^2_k(\lambda)\), where \(\lambda=h^2\eta'\eta\). At nominal level \(\alpha\), the AR local power is \(P\{\chi^2_k(\lambda)>\chi^2_{k,1-\alpha}\}\), where \(\chi^2_{k,1-\alpha}\) is the \(1-\alpha\) quantile of the central chi-square distribution with \(k\) degrees of freedom.

The noncentrality parameter separates the sources of AR power: the farther the structural parameter is from its null value in the local experiment, the larger \(|h|\); and the more first-stage signal is available, the larger \(\eta'\eta\). The error correlation \(\rho\) does not enter this marginal AR power expression. This does not mean \(\rho\) is irrelevant to all tests: procedures that use the joint information in \(U\) and \(V\) can depend on it.

Worked Example: AR Power With One Instrument

Suppose there is one instrument, the local first-stage signal is \(\eta=1\), and the local alternative is \(h=0.8\). Use a two-sided AR test at \(\alpha=0.05\). In this one-dimensional case, the test rejects when \(|U|>1.96\).

Under the alternative, \(U\sim N(h\eta,1)=N(0.8,1)\). Therefore

\begin{aligned} \pi_{\mathrm{AR}} &=P(U<-1.96)+P(U>1.96)\\ &=\Phi(-1.96-0.8)+1-\Phi(1.96-0.8)\\ &=\Phi(-2.76)+1-\Phi(1.16)\\ &\approx 0.0029+0.1230\\ &\approx 0.1259. \end{aligned}

The local rejection probability is about 12.6%. The AR test maintains its null calibration for weak first stages, but in this configuration it has limited power against the specified local departure.

What KLM and CLR Add

AR uses \(U\) but does not use the observed first-stage vector \(V\) to tailor its rejection rule. KLM is score-based: it uses information in the joint reduced forms to construct a weak-identification-robust test of the structural restriction. CLR uses a likelihood-ratio comparison in the canonical experiment and conditions on an appropriate statistic for the nuisance first-stage information. The conditioning is essential: it permits a critical rule whose null rejection probability is controlled without treating the first stage as known to be strong.

These descriptions identify the source of potential power gains, not a universal ranking. In some configurations, the joint or conditional information can distinguish alternatives more effectively than the AR statistic does. CLR is often especially competitive in the homoskedastic canonical setting because its conditional likelihood construction uses the nuisance information while preserving the conditional null calibration. KLM can also improve power relative to AR in relevant configurations. Neither statement says that every KLM or CLR rejection rule has higher power for every \((h,\eta,\rho)\), nor that nominal critical values from an unrelated reference distribution may be substituted.

A valid comparison uses each procedure’s own weak-identification-robust critical rule and evaluates all tests at the same \((h,\eta,\rho)\). In an application, this can be done by computing rejection probabilities under the canonical alternative, usually by numerical integration or simulation. Comparisons should report the first-stage signal, error correlation, instrument dimension, local departure, and target size; a power number without those settings is incomplete.

Worked Example: AR Power With Two Instruments

Consider \(k=2\), \(\eta=(1,2)'\), and local departure \(h=0.5\). The AR noncentrality parameter is

\lambda=h^2\eta'\eta =(0.5)^2(1^2+2^2) =0.25(5) =1.25.

At \(\alpha=0.05\), the central \(\chi^2_2\) critical value is approximately \(5.991\). The local power is therefore

P\{\chi^2_2(1.25)>5.991\}\approx 0.155.

Thus the AR test rejects about 15.5% of the time under this specified local alternative. This calculation is a benchmark for a KLM or CLR comparison at the same \(k\), \(\eta\), \(h\), and \(\rho\). It does not itself determine either competing test’s power: those probabilities require their own rejection rules and the joint distribution of \(U\) and \(V\).

Why Weak-Identification Validity Matters

A power gain is useful only if the test still controls its null rejection probability when the first stage is weak. In the canonical experiment, AR’s marginal null distribution makes this protection transparent. KLM and CLR obtain their own weak-robust calibration through their score or conditional construction. The common principle is that the test’s null behavior must be justified across weak first-stage values, not only at a convenient strong-instrument approximation.

This distinction matters when comparing with ordinary Wald inference. A Wald test based on a 2SLS estimate and its conventional standard error can have poor size under weak identification. A procedure that first screens for a large first-stage statistic and then reports ordinary Wald inference does not automatically inherit the weak-identification validity of AR, KLM, or CLR. Screening changes which samples proceed to inference; it does not establish that the conditional or unconditional rejection probability is controlled.

Power should also be interpreted as a property of a specified alternative, not as a general measure of test quality. A test that is more powerful at one first-stage signal or error correlation can be less powerful elsewhere. For decisions where weak-identification validity is essential, the relevant question is whether a proposed gain occurs under the alternatives of interest while the test retains its stated size guarantee across the nuisance configurations that matter.

Worked Example: Zero First-Stage Signal

Suppose \(\eta=0\). Under the null, \(U\) and \(V\) both have mean zero. Under any fixed local departure \(h\), their means are \((h\eta,\eta)=(0,0)\) as well. The covariance matrix is unchanged, so the joint null and alternative distributions coincide.

Consequently, for any fixed rejection rule \(\phi\),

P_{h,0,\rho}(\phi=1)=P_{0,0,\rho}(\phi=1).

If the test has exact size \(\alpha\) at this nuisance value, its power here is \(\alpha\). If it is only a level-\(\alpha\) test, its power equals its null rejection probability and is therefore no greater than \(\alpha\). For example, a test with null rejection probability \(0.03\) at nominal level \(0.05\) has power \(0.03\) in this case, not \(0.05\). No test can extract information about the structural departure from these identical distributions.

A Short Derivation and a Practical Comparison

The AR result follows directly from the distribution of a shifted standard normal vector. Under the local alternative, write \(U=h\eta+\varepsilon\), where \(\varepsilon\sim N(0,I_k)\). The squared Euclidean norm of a \(k\)-dimensional standard normal vector shifted by a fixed vector \(a\) has a noncentral chi-square distribution with \(k\) degrees of freedom and noncentrality \(a'a\). Setting \(a=h\eta\) gives

U'U=(h\eta+\varepsilon)'(h\eta+\varepsilon) \sim\chi^2_k\big((h\eta)'(h\eta)\big) =\chi^2_k(h^2\eta'\eta).

The derivation also explains why AR power is unchanged when \(\rho\) changes while \(h\) and \(\eta\) remain fixed: the marginal law of \(U\) does not contain \(\rho\). By contrast, the joint law used by KLM and CLR does contain \(\rho\), so their local power can respond to how \(U\) and \(V\) move together.

A disciplined comparison follows four steps:

Fix the local configuration.
Specify \(k\), \(h\), \(\eta\), and \(\rho\). Keep these the same for every test being compared.

Use each test’s valid calibration.
Apply its weak-identification-robust critical rule; do not replace it with a standard Wald critical value or an unjustified chi-square approximation.

Evaluate rejection probabilities.
For AR, use the noncentral chi-square probability. For KLM and CLR, evaluate rejection under the corresponding joint canonical distribution and their specified rules.

Separate power from validity.
Report any power gain together with the null size guarantee; a higher rejection probability under an alternative alone does not establish valid inference.

Common Pitfalls

Assuming every valid test has the same power. Validity constrains null rejection probabilities, not alternative rejection probabilities. Compare power at stated local configurations.
Claiming that CLR or KLM always dominates AR. The relative power depends on first-stage signal, error correlation, and the alternative. State comparisons for specified settings, or describe them as tendencies rather than universal rankings.
Confusing nominal level with exact size. A level-\(\alpha\) test may reject less often than \(\alpha\) under a particular null. When the null and alternative distributions coincide, its power equals that actual null rejection probability, not necessarily the nominal level.
Using a strong-instrument approximation to claim weak validity. A conventional Wald calibration does not become weak-instrument-robust simply because a first-stage diagnostic is reported.

Key takeaway: Local power compares rejection probabilities in a weak-instrument limiting experiment. AR provides a transparent benchmark with noncentrality \(h^2\eta'\eta\); KLM and CLR can use joint or conditional information for power gains while maintaining weak-identification-robust calibration. Those gains are configuration-dependent, and at zero first-stage signal power equals the actual null rejection probability.

Check Your Understanding

Use the canonical model and distinctions developed above.

For the AR test, how does the local noncentrality parameter change if \(h\) is doubled while \(\eta\) remains fixed?
Why does the AR local power expression not depend on \(\rho\), and why might KLM or CLR power depend on it?
At \(\eta=0\), a level-\(0.05\) test has null rejection probability \(0.02\). What is its power against the local alternative, and why?
Why is a reported KLM-versus-AR power comparison incomplete if it omits \(\eta\), \(h\), and \(\rho\)?
Why does first-stage screening followed by conventional Wald inference not by itself establish weak-identification-robust size control?