Weak First Stages in Fuzzy RD

Why the First Stage Matters

In a fuzzy regression discontinuity (RD) design, crossing the cutoff changes the probability of receiving treatment, but does not determine treatment perfectly. The causal estimate is built from two local discontinuities: the jump in the outcome and the jump in treatment receipt. When the treatment-probability jump is small or imprecisely estimated, dividing by it can make the ratio estimate highly sensitive to sampling error. A conventional confidence interval can then give a misleading impression of precision.

This tutorial focuses on diagnosing that problem and on inference that remains informative about uncertainty when the first stage is weak. The key distinction is between a small estimated ratio standard error under a strong-first-stage approximation and a confidence set that allows for uncertainty in the denominator itself.

The left and right limits describe conditional means arbitrarily close to the cutoff, not averages over the entire sample. In estimation, local polynomial methods approximate these limits using observations near \(c\). The estimated jumps are denoted \(\hat{\Delta}_Y\) and \(\hat{\Delta}_D\), and the usual ratio estimate is \(\hat{\tau}=\hat{\Delta}_Y/\hat{\Delta}_D\).

A causal interpretation requires more than a nonzero denominator. In particular, the cutoff must generate a discontinuous change in treatment receipt, while relevant potential outcomes and treatment statuses would otherwise vary continuously through the cutoff. The cutoff must affect the outcome through treatment rather than through a separate discontinuous channel, and treatment assignment must satisfy the usual monotonicity condition for a complier-effect interpretation. Under these conditions, the ratio identifies a local effect for compliers at the cutoff. Weak-first-stage methods address uncertainty in estimating that ratio; they do not establish these design assumptions.

Diagnosing a Weak First Stage

The first-stage estimate is not just a denominator to report after computing the outcome ratio. Inspect \(\hat{\Delta}_D\), its standard error, its confidence interval, and the local treatment-probability pattern on both sides of the cutoff. A useful basic signal is the standardized first-stage estimate \(\hat{\Delta}_D/SE(\hat{\Delta}_D)\). A small value relative to its uncertainty means the data do not sharply distinguish the estimated jump from zero.

This standardized value is a diagnostic, not a universal pass-or-fail rule. A fixed threshold such as a particular first-stage statistic cannot replace design-specific judgment: effective sample size, local variation in treatment, dependence or clustering, bandwidth, and the estimator all affect uncertainty. In RD, the local polynomial specification and bias correction used for inference also matter. A first-stage confidence interval that includes zero is a clear warning that ratio-based normal approximations may be unreliable, but a confidence interval that excludes zero does not by itself guarantee a well-behaved ratio estimate.

The numerator and denominator estimates are generally correlated because they are computed from the same observations. That covariance matters both for the uncertainty of the ratio and for weak-identification-robust inference. Therefore, retain the joint covariance estimate for \((\hat{\Delta}_Y,\hat{\Delta}_D)\), rather than treating the two jumps as independent unless independence is justified.

Why Ratio Inference Can Fail

The usual delta method approximates the ratio near the estimated denominator. That approximation works best when the denominator is sufficiently well separated from zero relative to its uncertainty. If plausible first-stage values include values close to zero, small changes in the denominator can produce very large changes in the ratio. The ratio's distribution can be skewed or have heavy tails, and a symmetric normal interval around \(\hat{\tau}\) may miss important values or have poor coverage.

A more direct strategy tests candidate values of the causal ratio without dividing by the estimated first stage. For a proposed value \(\tau_0\), the null implication is that the discontinuity in \(Y-\tau_0D\) is zero:

The estimated contrast is \(\hat{\Delta}_Y-\tau_0\hat{\Delta}_D\). Its variance must account for the covariance between the two estimated jumps. If their estimated variances are \(\hat{V}_Y\) and \(\hat{V}_D\), and their estimated covariance is \(\hat{C}_{YD}\), then

An Anderson–Rubin (AR) test compares this contrast with its standard error. Under an appropriate joint large-sample normal approximation for the estimated jumps, and with valid RD inference for those jumps, a two-sided 5% test rejects when the squared standardized contrast exceeds \(1.96^2\), approximately \(3.841\). In an RD application, the jump estimates and their covariance should use an inference procedure appropriate to the chosen local specification, including any required bias correction. The weak-identification robustness comes from testing the contrast directly, not from ignoring bias or other RD inference problems.

How AR Inversion Produces a Confidence Set

The AR set is found by inverting the test: evaluate the test inequality across candidate values of \(\tau_0\), and retain every value that is not rejected. Rearranging the inequality gives a quadratic expression. This explains why the result need not be a conventional interval.

If the quadratic opens upward, its nonpositive values can lie between two roots, producing a bounded interval. If it opens downward, its nonpositive values can lie outside the roots, producing two unbounded pieces. If there are no real roots, the inequality may hold everywhere or nowhere, depending on the sign. These possibilities are features of the information in the data, not computational errors.

Practical Interpretation and Common Pitfall

A weak-identification-robust confidence set answers a specific question: which ratio values are consistent with the joint uncertainty in the reduced-form and first-stage discontinuities under the test's assumptions? It does not answer whether the RD design is credible, whether the cutoff was manipulated, or whether the local polynomial approximation is adequate. Those questions require their own design checks and sensitivity analyses. Nor does an unbounded set mean the effect is infinite; it means the data and assumptions used for the test do not rule out arbitrarily large values in those directions.

For an applied analysis, report the first-stage and reduced-form jumps with standard errors, the covariance used for ratio inference, the ratio estimate with any conventional interval clearly labelled, and the weak-identification-robust set. Explain the set's shape and units. Use the same defensible local estimation and bias-handling choices for the jumps and their joint covariance; a weak-stage method cannot repair an invalid choice of bandwidth or an unaddressed discontinuity bias.

• Mistake: “The first-stage estimate is positive, so the denominator is safe to divide by.” Correction: A positive point estimate can still be imprecise. Examine its uncertainty and use inference that does not rely on a well-separated denominator.
• Mistake: “The AR set is disconnected or unbounded, so the calculation failed.” Correction: Such a set can be the correct inversion result when the first stage is weak.
• Mistake: “A weak-identification-robust set validates the causal design.” Correction: It addresses uncertainty in the ratio under the maintained RD and instrumental-variable assumptions, not whether those assumptions hold.