Kleibergen LM Tests for Weakly Identified Models

Why Use a Kleibergen LM Test?

A conventional Wald test for an instrumental-variables coefficient can be unreliable when the instruments predict the endogenous regressor only weakly. Its usual approximation relies on the coefficient estimator behaving approximately normally, an approximation that can fail when the first stage is weak. The Anderson–Rubin test avoids that particular problem by testing the instrument moment implied by a candidate coefficient. The Kleibergen Lagrange-multiplier (LM) test offers another weak-instrument-robust test: it uses the component of the reduced-form score aligned with the direction associated with the first stage.

The key feature is that the test’s null calibration does not require the first-stage coefficient to be large. Its interpretation still depends on the structural model and instrument assumptions, and weak instruments can limit the test’s power. A valid p-value is not evidence that the coefficient is precisely estimated.

Model and Construction

Consider one endogenous regressor \(x\), an outcome \(y\), and \(q\) excluded instruments collected in the \(n\times q\) matrix \(Z\). Any included exogenous controls are partialled out of \(y\), \(x\), and \(Z\) before forming the reduced-form quantities. Assume \(Z\) has full column rank. Write the two reduced forms as

where \(\pi_y\) and \(\pi_x\) are \(q\)-vectors and \(u,v\) are reduced-form errors. The structural equation is \(y=\beta x+u_s\), where \(u_s\) is the structural error. Instrument validity requires the excluded instruments to be uncorrelated with the structural error; the usual IV model also requires the structural and reduced-form specifications to be appropriate. For the standard homoskedastic version developed here, assume independent observations, a common nonsingular covariance matrix \(\Omega\) for the two reduced-form errors, and either Gaussian errors or regular conditions supporting the large-sample normal approximation. Keep \(q\) fixed as the sample size grows.

The null \(H_0:\beta=\beta_0\) implies a restriction on the two reduced-form coefficient vectors:

Let \(Y=[y\ \ x]\), let \(\hat B=(Z'Z)^{-1}Z'Y\), and let \(\hat\Omega\) estimate the covariance matrix of the reduced-form errors. For exposition, first treat \(\Omega\) as known. Define \(a=(1,-\beta_0)'\), \(c=(\beta_0,1)'\), and \(b=\Omega^{-1}c\). Then \(a'c=0\), so \(a'\Omega b=0\). The two standardized \(q\)-vectors are

In an application, replace \(\Omega\) with a consistent estimate \(\hat\Omega\), and use the corresponding estimated \(S\) and \(T\). Under the null, \(S\) has zero mean: the reduced-form coefficient matrix has rows proportional to \((\beta_0,1)\), so multiplying it by \(a\) gives zero. By contrast, \(T\) contains the first-stage direction. The covariance-orthogonal choice of \(a\) and \(b\) makes \(S\) and \(T\) independent in the Gaussian model. That separation is central to the null distribution.

Why the Null Distribution Is Chi-Square

Under \(H_0\), with known \(\Omega\) and Gaussian errors, \(S\sim N(0,I_q)\), while \(T\) may have any mean determined by the first-stage coefficients. Because \(S\) and \(T\) are independent, condition on the observed \(T\). If \(T\ne0\), \(P_T\) is a projection matrix onto a one-dimensional space: it is symmetric, idempotent, and has rank one. A standard normal vector projected onto a fixed one-dimensional space has a squared length distributed as chi-square with one degree of freedom. Therefore,

The argument does not require a large mean for \(T\). A weak first stage changes the distribution of \(T\), but does not change the null distribution of \(S\) conditional on \(T\). With an estimated covariance matrix and non-Gaussian errors, the exact finite-sample statement generally becomes an asymptotic result: under the stated regularity conditions and fixed \(q\), the null reference distribution remains \(\chi^2_1\), including when first-stage coefficients are local to zero. The test is thus robust to weak identification for null size; it does not guarantee good power or identification of \(\beta\).

The score interpretation also explains the projection. Near the null, a change in the structural coefficient shifts the null-restricted reduced-form statistic in the direction of the first-stage signal. The score for that scalar change is proportional to \(S'T\), and its information is proportional to \(T'T\). Squaring the score and dividing by its information gives

Thus the statistic uses the part of \(S\) relevant to a local change in \(\beta\), rather than its entire length. The rank-one projection is why the reference degrees of freedom are one, even when there are several instruments.

What the Projection Keeps—and Discards

The Anderson–Rubin statistic in this standardized representation is \(S'S\), with a \(\chi^2_q\) null reference distribution. It uses all \(q\) components of the null-restricted statistic. The KLM statistic instead uses only the component of \(S\) in the direction \(T\). Consequently, its reference distribution has one degree of freedom. The tests are not interchangeable: they use different functions of the reduced-form evidence and can produce different p-values. Both can be calibrated without assuming a strong first stage under their respective conditions.

Checking the Calculation with Several Instruments

With \(q\) instruments, the same calculation uses \(q\)-vectors. An equivalent computational route is to form \(P_T\) and calculate \(S'P_T S\). For a single endogenous regressor, the projection always has rank one when \(T\ne0\). A very small but nonzero \(T\) is not a reason to switch the degrees of freedom; numerical computation should use the projection or ratio carefully to avoid avoidable rounding error.

Assumptions, Limitations, and a Common Pitfall

The stated construction relies on a correctly specified linear reduced-form setup, valid instruments under the null, full-rank instrument variation, and the homoskedastic covariance structure used to standardize the statistics. For the exact conditional argument, Gaussian errors and known \(\Omega\) suffice. In practice, covariance estimation yields an asymptotic test under suitable regularity conditions. A heteroskedastic setting requires an appropriate robust construction rather than simply reusing the homoskedastic formula.

The weak-identification robustness claim concerns the null distribution, not all aspects of inference. Failure to reject a candidate coefficient does not show that the coefficient is identified, nor does it imply that a confidence set formed by inverting tests will be narrow. Under very weak identification, test inversion can produce a wide or unbounded set. At a first stage with no identifying information, the data may provide little or no power to discriminate among structural coefficients.

• Pitfall: treating a weak first stage as a reason to use a Wald p-value anyway. The usual Wald approximation can be distorted precisely when the first stage is weak. Use a test designed to retain its null calibration under weak identification.
• Pitfall: using \(S'S\) but assigning it one degree of freedom. \(S'S\) is the Anderson–Rubin statistic and has a \(\chi^2_q\) reference distribution. KLM is \(S'P_T S\) and has a \(\chi^2_1\) reference distribution in this one-coefficient test.
• Pitfall: interpreting non-rejection as proof of the null. A p-value is calculated assuming the null; it is not the probability that the null is true. Weak instruments can also make rejection difficult even when the candidate coefficient is wrong.