Conditional Likelihood-Ratio Inference for Weak Instruments

Why condition in weak-instrument inference?

When an instrument is weak, the reduced-form first stage contains little information about the structural parameter. Conventional normal approximations for 2SLS can then be unreliable. The Anderson–Rubin test avoids that problem by testing a structural value through the reduced form. The conditional likelihood-ratio (CLR) test uses the same reduced-form information but combines it in a way that can improve power while preserving validity under weak identification.

The construction below uses a fixed design with $k$ instruments and one endogenous regressor. It assumes Gaussian, homoskedastic reduced-form errors and initially treats their covariance matrix as known. These assumptions make the conditioning and its reference distribution explicit. The essential weak-instrument property is that validity does not require the first-stage coefficient to be large or nonzero.

Null-specific reduced-form statistics

Write the reduced form as $y_1=Z\pi_1+v_1$ and $y_2=Z\pi_2+v_2$, where $y_1$ is the outcome, $y_2$ is the endogenous regressor, $Z$ is an $n$-by-$k$ instrument matrix of full column rank, and the error pairs are independent $N(0,\Omega)$. Let $\hat\pi_1$ and $\hat\pi_2$ be the reduced-form least-squares estimates. The covariance matrix $\Omega$ has entries $\Omega_{11},\Omega_{12},\Omega_{22}$ and is positive definite. Under the structural null $H_0:\beta=\beta_0$, the reduced-form coefficients satisfy $\pi_1=\beta_0\pi_2$.

Definition: For a proposed value $\beta_0$, define $A=\Omega_{11}-2\beta_0\Omega_{12}+\beta_0^2\Omega_{22}$, $B=\Omega_{12}-\beta_0\Omega_{22}$, and $D=\Omega_{22}-B^2/A$. Set $$ S=\frac{(Z'Z)^{1/2}(\hat\pi_1-\beta_0\hat\pi_2)}{\sqrt{A}}, \qquad T=\frac{(Z'Z)^{1/2}\left[\hat\pi_2-\frac{B}{A}(\hat\pi_1-\beta_0\hat\pi_2)\right]}{\sqrt{D}}. $$ Here $A>0$ and $D>0$ follow from positive definiteness of $\Omega$.

The first statistic measures the reduced-form discrepancy from the null restriction. The second is an orthogonalized first-stage statistic. Under the null, their means are $0$ and $(Z'Z)^{1/2}\pi_2/\sqrt D$, respectively; their covariance matrices are both $I_k$, and they are independent. In particular, the mean of $T$ can be arbitrarily small, so this setup does not build in strong identification.

Worked Example: Checking the Null Transformation

Take $k=1$, $Z'Z=1$, $\beta_0=1$, and $\Omega=\begin{pmatrix}4&1\\1&2\end{pmatrix}$. Then $A=4-2+2=4$, $B=1-2=-1$, and $D=2-(-1)^2/4=7/4$. If the null holds with $\pi_2=3$, then $\pi_1=3$. The mean of $\hat\pi_1-\hat\pi_2$ is zero, while the mean of $\hat\pi_2-\frac{B}{A}(\hat\pi_1-\hat\pi_2)$ is $3$. Thus $E[S]=0$ and $E[T]=3/\sqrt{7/4}=6/\sqrt7$.

To check orthogonality, let $e_1=v_1-v_2$ and $e_2=v_2-\frac{B}{A}e_1$. Their covariance is $\operatorname{Cov}(e_1,e_2)=B-(B/A)A=0$. Their variances are $A=4$ and $D=7/4$. Joint normality therefore gives independent, unit-variance errors after the two scalings.

The CLR statistic and its conditional reference distribution

For $S,T\in\mathbb R^k$, write $Q=S'S$, $R=T'T$, and $C=S'T$. The CLR statistic is the larger eigenvalue of the matrix of their inner products. The formula below is useful because it calculates the statistic directly from three scalar quantities.

Definition: The conditional likelihood-ratio statistic for testing $H_0:\beta=\beta_0$ is $$ \operatorname{CLR}(\beta_0) =\lambda_{\max}\begin{pmatrix}Q&C\\C&R\end{pmatrix} =\frac{Q+R+\sqrt{(Q-R)^2+4C^2}}{2}. $$ Its conditional p-value is the probability, under $S^*\sim N_k(0,I_k)$ with the observed $T$ held fixed, that the same statistic calculated from $S^*,T$ is at least as large as the observed statistic.

The eigenvalue expression follows from the quadratic characteristic equation. The characteristic polynomial of the displayed matrix is $\lambda^2-(Q+R)\lambda+(QR-C^2)$. Its larger root is $\{Q+R+\sqrt{(Q+R)^2-4(QR-C^2)}\}/2$, and the expression under the square root simplifies to $(Q-R)^2+4C^2$.

The conditioning is the key to nuisance removal. Under the null, $S$ is standard normal and independent of $T$, while the mean of $T$ contains the unknown first-stage coefficient $\pi_2$. Given the observed $T$, the null distribution of the statistic is obtained by drawing only $S^*\sim N_k(0,I_k)$. It therefore depends on $T$ through its observed value, not through an unknown first-stage mean. Rotational symmetry of $S^*$ means the conditional distribution depends on $T$ only through $R=T'T$.

Theorem: In the stated Gaussian model, the conditional p-value has size at most $\alpha$ under $H_0$, for every value of $\pi_2$, including $\pi_2=0$.

Proof. Under $H_0$, $S$ and $T$ are independent, and $S\sim N_k(0,I_k)$. Conditional on $T=t$, the observed statistic is therefore distributed exactly as the statistic formed from $S^*\sim N_k(0,I_k)$ and the fixed vector $t$. The conditional p-value is the upper-tail probability under this conditional null distribution. For a continuous conditional distribution, its conditional rejection probability at level $\alpha$ is $\alpha$; if there are ties, using the upper-tail probability gives rejection probability at most $\alpha$. Averaging the conditional rejection probability over the distribution of $T$ gives unconditional rejection probability at most $\alpha$. None of these steps restricts the mean of $T$, which proves the assertion for every $\pi_2$. $\square$

Worked Example: Calculating a CLR Statistic with Two Instruments

Suppose the transformed statistics for a proposed null are $S=(1,0)'$ and $T=(1,1)'$. Then $Q=1$, $R=2$, and $C=1$. The statistic is

\operatorname{CLR} =\frac{1+2+\sqrt{(1-2)^2+4(1)^2}}{2} =\frac{3+\sqrt5}{2} \approx 2.618.

For the conditional p-value, hold $T=(1,1)'$ fixed and generate $S^*=(S_1^*,S_2^*)'$ with independent standard normal components. For each draw, calculate $Q^*=(S_1^*)^2+(S_2^*)^2$, $C^*=S_1^*+S_2^*$, and

\frac{Q^*+2+\sqrt{(Q^*-2)^2+4(C^*)^2}}{2}.

The conditional p-value is the probability that this quantity is at least $2.618$. It can be evaluated by integration or simulation from the stated standard normal distribution. Crucially, no value of the unknown first-stage mean is needed.

Worked calculations in the one-instrument case

With one instrument, $S$ and $T$ are scalars, and the inner-product matrix has rank one. Its nonzero eigenvalue is $S^2+T^2$, so $\operatorname{CLR}=S^2+T^2$. Conditional on $T$, the test rejects at level $\alpha$ exactly when $S^2$ exceeds the $1-\alpha$ quantile of a $\chi^2_1$ distribution. This special case makes test inversion particularly transparent.

Worked Example: A Bounded Confidence Set

Assume $\Omega=I_2$, $Z'Z=1$, and observed reduced-form estimates $\hat\pi_1=6$, $\hat\pi_2=3$. For any proposed $\beta$, the definitions give $A=1+\beta^2$, $B=-\beta$, and $D=1/(1+\beta^2)$. The scalar CLR test rejects when

S^2=\frac{(6-3\beta)^2}{1+\beta^2}>\chi^2_{1,0.95}\approx3.841.

Inverting this inequality means retaining values satisfying $36-36\beta+9\beta^2\le3.841(1+\beta^2)$, or $5.159\beta^2-36\beta+32.159\le0$. The roots are approximately $1.05$ and $5.93$, so the 95% confidence set is approximately $[1.05,5.93]$. The set was obtained by testing candidate values, not by applying a normal approximation to a ratio estimator.

Worked Example: An Unbounded Confidence Set

Again let $\Omega=I_2$ and $Z'Z=1$, but now suppose $\hat\pi_1=0.8$ and $\hat\pi_2=0.4$. For any real $\beta$,

S^2=\frac{(0.8-0.4\beta)^2}{1+\beta^2} \le 0.8^2+0.4^2=0.8.

The inequality follows from Cauchy–Schwarz applied to $(0.8,0.4)$ and $(1,-\beta)/\sqrt{1+\beta^2}$. Since $0.8<3.841$, no value of $\beta$ is rejected by the 5% test, and the inverted confidence set is the entire real line. This is a legitimate expression of weak information, not a computational failure.

Inverting the test and interpreting the result

For each candidate structural value $\beta_0$, form its null-specific $S$ and $T$, calculate the CLR statistic, and obtain the conditional p-value using the observed $T'T$. The level-$\alpha$ confidence set is

\mathcal C_{1-\alpha} =\{\beta_0\in\mathbb R:\ p_{\mathrm{CLR}}(\beta_0)>\alpha\}.

Because the transformation and conditional distribution vary with $\beta_0$, the calculation must be repeated for each candidate value. The resulting set need not be a single bounded interval: it may be unbounded, disconnected, or all of $\mathbb R$. These shapes can signal that the data do not rule out distant parameter values when the first stage is weak.

The guarantee proved above applies to the specified Gaussian, homoskedastic model with known $\Omega$. In applications where $\Omega$ is estimated, the appropriate finite-sample or asymptotic calibration must account for that estimation; simply substituting an estimate while retaining the known-covariance reference distribution is not justified by this proof. Heteroskedastic errors likewise require a different calibration.

Common pitfall

Wrong reasoning: “The observed $T$ is large, so I can use a fixed chi-square cutoff for the CLR statistic.” The conditional distribution of CLR depends on $T'T$, so its critical value is not generally a fixed chi-square quantile.
Correction: Hold the observed $T$ fixed and calculate the null distribution by drawing $S^*\sim N_k(0,I_k)$. The nuisance first-stage mean is removed from this conditional reference distribution, but the observed value of $T$ still affects the cutoff.
Wrong reasoning: “An unbounded confidence set means the test failed.” A set that includes distant values can correctly reflect limited information about the structural parameter.

Check Your Understanding

Use the definitions and conditioning argument above to answer these questions.

Under the null, what are the mean and covariance matrix of $S$, and why is $S$ independent of $T$?
Which unknown parameter affects the mean of $T$, and why does conditioning on the observed $T$ remove it from the conditional null distribution?
For $Q=4$, $R=1$, and $C=1$, calculate the CLR statistic.
Why can inversion of a valid weak-instrument test produce the entire real line as a confidence set?
What additional issue must be addressed before using the known-covariance conditional reference distribution when $\Omega$ is estimated?