Weak Instruments and the tF Critical Value

The usual IV \(t\) statistic divides by a quantity estimated using the first stage. When that stage is weak, the resulting null distribution can differ substantially from the normal approximation. The \(tF\) procedure of (Lee et al. 2022) uses the reported first-stage \(F\) to choose a critical value that accounts for this distortion.

1. A useful identity

Let \(\widehat\Gamma\) and \(\widehat\pi\) be the reduced-form and first-stage coefficients, so \(\widehat\beta=\widehat\Gamma/\widehat\pi\). Under \(H_0:\beta=\beta_0\), write

\[ A\triangleq\frac{\widehat\Gamma-\beta_0\widehat\pi}{s_a}, \qquad f\triangleq\frac{\widehat\pi}{s_\pi}, \qquad \rho\triangleq\frac{\operatorname{Cov}(\widehat\Gamma-\beta_0\widehat\pi,\widehat\pi)}{s_as_\pi}. \]

Here \(s_a\) and \(s_\pi\) are the corresponding standard deviations. In the Gaussian limit experiment, \((A,f)\) has unit marginal variances, correlation \(\rho\), and mean \((0,f_0)\). The nuisance parameter \(f_0\) measures first-stage strength.

The delta-method variance used by the conventional IV statistic has the form

\[ s_\beta^2=\widehat\pi^{-2} (s_a^2-2(\widehat\beta-\beta_0)\rho s_as_\pi +(\widehat\beta-\beta_0)^2s_\pi^2). \]

Substituting \(\widehat\beta-\beta_0=s_aA/\widehat\pi\) gives the central identity

\[ t^2=\frac{A^2}{1-2\rho A/f+(A/f)^2},\qquad F=f^2. \]

The numerator looks familiar, but its denominator uses the same Gaussian score together with the noisy first stage. It cannot generally be replaced by one. That dependence is the source of the failure of ordinary normal critical values.

2. The main inference result

Theorem 1. The paper constructs an \(F\)-dependent critical-value function \(c_\alpha\) so that, in the relevant IV limit experiment,

\[ \sup_{f_0,\rho}\Pr_{f_0,\rho}\{t^2>c_\alpha(F)\}\le\alpha. \]

Under the conditions connecting the sample statistics to this experiment, inversion gives a confidence interval of the form

\[ \widehat\beta\ \pm\ \sqrt{c_\alpha(F)}\,\widehat s_\beta. \]

The adjustment uses the conventional coefficient, standard error, and first-stage \(F\). Its guarantee belongs to the calibrated critical-value function from the paper’s tables or implementation. Increasing a critical value by an arbitrary monotone factor does not reproduce that calibration.

3. Why screening on \(F\) is insufficient

A rule that first checks whether \(F\) exceeds a threshold and then uses the ordinary \(t\) critical value changes the joint rejection region. Since \(A\) and \(f\) can be correlated, passing that screen does not turn the remaining \(t\) statistic into an independent standard normal variable.

The calibration instead evaluates the probability of the whole rejection region over the nuisance parameters. For a strong first stage, \(A/f\) becomes small and the identity approaches \(t^2=A^2\). When the first stage is weak, the denominator cannot be replaced by one, and the adjustment can be substantial.

The Anderson–Rubin statistic works directly with the reduced-form null score \(A\). The \(tF\) approach retains the familiar IV estimate and standard error while accounting for the distortion visible in the identity above. This is an inference correction; it does not repair violations of instrument exogeneity or exclusion. For a shift-share instrument, for example, the argument establishing valid identifying variation remains a separate requirement.

4. Recovering the denominator from the regression residual

The conventional standard error estimates the variance of a structural residual formed using \(\widehat\beta\). Under the null, write the reduced-form null error as \(a\) and the first-stage error as \(v\). Replacing \(\beta_0\) by \(\widehat\beta\) changes the relevant error to

\[ a-(\widehat\beta-\beta_0)v. \]

Expanding the variance of this difference gives a covariance term as well as two variance terms:

\[ s_a^2-2(\widehat\beta-\beta_0)\rho s_as_\pi +(\widehat\beta-\beta_0)^2s_\pi^2. \]

The coefficient error is divided by the estimated first stage, and the variance estimate is divided by its square. Substituting \(\widehat\beta-\beta_0=s_aA/\widehat\pi\) cancels those common factors and produces the identity for \(t^2\). The denominator is nonnegative because

\[ 1-2\rho A/f+(A/f)^2 =(A/f-\rho)^2+1-\rho^2. \]

For \(|\rho|<1\) it is strictly positive. Near \(|\rho|=1\), however, it can be very small when \(A/f\) is close to \(\rho\). Large conventional \(t\) values can then reflect joint behavior of the score and estimated residual variance, rather than a large standardized null score alone.

At \(\rho=0\), the denominator is at least one, so \(t^2\leq A^2\) pathwise. Thus weakness does not create the same distortion for every nuisance parameter. A uniform critical value must also protect the correlated cases, which is why calibrating at a convenient value of \(\rho\) would be insufficient.

5. Why a first-stage screen changes the score distribution

The Gaussian experiment admits the representation

\[ f=f_0+V,\qquad A=\rho V+\sqrt{1-\rho^2}\,E, \]

with independent standard normals \(V,E\). Conditional on \(f=u\),

\[ A\mid f=u\sim \mathcal{N}(\rho(u-f_0),1-\rho^2). \]

When the true first stage \(f_0\) is small, an unusually large observed \(f\) can be caused by a large \(V\). If \(\rho\ne0\), the same \(V\) shifts the null score. Conditioning on a large first-stage statistic therefore selects a different distribution of \(A\). Screening on \(F=f^2\) mixes the two signs of this effect; it does not remove it.

The unconditional probability of any proposed rejection rule \(R(A,f)\) is, for \(|\rho|<1\),

\[ \int_{\mathbb R}\int_{\mathbb R} \mathbf1\{R(a,u)\} \frac{\phi((a-\rho(u-f_0))/\sqrt{1-\rho^2})} {\sqrt{1-\rho^2}} \phi(u-f_0)\,da\,du. \]

For the \(tF\) procedure, use the rejection event \(R(a,u)=\{a^2/[1-2\rho a/u+(a/u)^2]>c_\alpha(u^2)\}\). Calibration bounds this double integral for all allowed nuisance parameters. A few selected first-stage values cannot establish that uniform probability bound.

The rejection region can be written as a quadratic inequality, which helps explain what calibration must accomplish. At a realized first stage \(f\ne0\), put \(c=c_\alpha(f^2)\). Multiplying by the positive variance denominator gives, for \(|\rho|<1\),

\[ t^2>c \quad\Longleftrightarrow\quad (f^2-c)A^2+2c\rho fA-cf^2>0. \]

For a fixed \(f\), this inequality can describe two tails or a bounded interval in \(A\), depending on its leading coefficient and discriminant. It is not generally the event \(|A|>\sqrt c\). The conditional normal law of \(A\) is also shifted by \(\rho(f-f_0)\), so both the rejection region and the probability assigned to it change with the first stage.

At \(\rho=0\), the formula simplifies to

\[ t^2=\frac{A^2f^2}{A^2+f^2}<f^2. \]

In this uncorrelated case, an \(F\) below the positive squared critical value precludes rejection. Correlation changes the cross-term in the denominator, which can instead become small. A calibration based only on the uncorrelated case would miss this source of overrejection.

The \(F\)-dependent curve controls the unconditional rejection probability uniformly over true first stages and score correlations. It does not assert exact conditional size at each observed \(F\). Conditioning on the first stage selects noise correlated with the null score, as the conditional Gaussian formula shows.

6. The strong-first-stage limit

If \(|f_0|\to\infty\), then \(f/f_0\to1\) in probability while \(A=O_p(1)\). Consequently \(A/f=o_p(1)\), and

\[ t^2-A^2=o_p(1). \]

For a more detailed approximation, put \(x=A/f\). Around \(x=0\),

\[ \frac1{1-2\rho x+x^2} =1+2\rho x+(4\rho^2-1)x^2+O(x^3). \]

The leading correction to \(A^2\) is \(2\rho A^3/f\). A strong first stage makes it vanish. In weak-IV asymptotics, \(f_0\) can remain bounded while \(f\) approaches zero, so \(A/f\) need not be a small expansion parameter. The series then provides no justification for the ordinary delta-method approximation.

7. Inverting Anderson–Rubin for comparison

The null score \(A(\beta)\) uses \(\widehat\Gamma-\beta\widehat\pi\) and its null variance. Squaring an Anderson–Rubin inequality gives

\[ (\widehat\Gamma-\beta\widehat\pi)^2 \leq z_{1-\alpha/2}^2 (s_\Gamma^2-2\beta s_{\Gamma\pi}+\beta^2s_\pi^2). \]

This is a quadratic inequality in \(\beta\). The coefficient on \(\beta^2\) after moving the right side to the left is

\[ \widehat\pi^2-z_{1-\alpha/2}^2s_\pi^2. \]

A nonpositive coefficient can leave the accepted set unbounded, reflecting limited information about a ratio with a weak denominator. The \(tF\) method uses a different rejection region expressed through the conventional estimate and standard error. Its validity still depends on the calibrated critical values and their weak-first-stage qualifications.

Lee, David S., Justin McCrary, Marcelo J. Moreira, and Jack Porter. 2022. “Valid t -Ratio Inference for IV.” American Economic Review 112 (10): 3260–90. https://doi.org/10.1257/aer.20211063.