LIML Under Weak Instrumentation

Why LIML Enters the Weak-Instrument Discussion

Two-stage least squares (2SLS) is familiar, but its finite-sample behavior can be poor when instruments explain little of the endogenous regressor. Limited-information maximum likelihood (LIML) offers a different k-class estimate. In many overidentified designs it has less finite-sample bias than 2SLS, especially when there are many instruments relative to the sample size. That advantage is not a cure for weak identification: under weak instruments, LIML also has a nonstandard sampling distribution and can produce unstable estimates.

The construction below makes the distinction precise. LIML chooses a data-dependent coefficient, determined by a characteristic root, and inserts it into a k-class formula. The characteristic root is part of the estimator; it is not a first-stage strength diagnostic.

Model, Projections, and the LIML Definition

Consider a structural equation with one endogenous regressor \(d\), outcome \(y\), and included exogenous regressors collected in a matrix \(X\). Let \(Z\) contain the excluded instruments. Partial out \(X\) from \(y\), \(d\), and \(Z\), and use the same letters for the resulting residualized variables. Let \(P_Z\) be the orthogonal projection onto the residualized instrument space and let \(M_Z=I-P_Z\). Define \(Q=[d,y]\), so its columns are the residualized endogenous regressor and outcome.

The positive-definiteness assumption ensures that the characteristic roots are finite and positive. Since \(M_Z\) is a projection, \(0\preceq M_Z\preceq I\), and therefore \(Q'M_ZQ\preceq Q'Q\). Consequently every characteristic root is at least one. The projection \(I-\hat kM_Z=P_Z+(1-\hat k)M_Z\) gives negative weight to the part orthogonal to the instruments when \(\hat k>1\).

For comparison, the k-class estimate at \(k=1\) is $$ \hat\beta(1) = \frac{d'P_Zy}{d'P_Zd}, $$ which is the 2SLS estimate after partialling out \(X\). LIML differs by choosing \(\hat k\) from the data rather than fixing \(k=1\).

What the Characteristic Root Measures

The smallest-root characterization has a useful geometric interpretation. It selects the direction in the two-dimensional column space of \(Q\) with the largest fraction of squared length lying outside the instrument space. That direction is the one along which the instruments explain the least.

Proof. The characteristic equation is \(\det(A-kB)=0\). Multiplying on both sides by \(A^{-1/2}\) shows that this equation is equivalent to $$ \det\!\left(I-kA^{-1/2}BA^{-1/2}\right)=0. $$ If \(\lambda_1\ge \lambda_2>0\) are the eigenvalues of the symmetric matrix \(A^{-1/2}BA^{-1/2}\), the roots are \(1/\lambda_1\) and \(1/\lambda_2\). Thus the smallest root is \(1/\lambda_1\). The Rayleigh quotient characterization of the largest eigenvalue gives the stated minimum over \(v\ne0\). Finally, \(B\preceq A\), so every eigenvalue of \(A^{-1/2}BA^{-1/2}\) is at most one. Hence \(\hat k\ge1\). \(\square\)

Computing the Estimate and Comparing It with 2SLS

For any fixed \(k\), the k-class coefficient can be written directly in terms of the two cross-product matrices: $$ \hat\beta(k)= \frac{Q'Q_{12}-k(Q'M_ZQ)_{12}} {Q'Q_{11}-k(Q'M_ZQ)_{11}}. $$ Here the subscripts identify entries: \(11\) is the regressor’s squared length and \(12\) is the regressor-outcome cross-product. Setting \(k=1\) removes the component orthogonal to the instruments and yields 2SLS. Setting \(k=\hat k\) gives LIML.

A just-identified model provides a useful boundary case. With one excluded instrument for one endogenous regressor, the conventional IV estimate and 2SLS estimate coincide because both use the same single instrument moment. LIML also gives that estimate when defined. In this case, changing the k-class weight does not create additional identifying information. If the instrument is weak, the common estimate can still be highly variable.

Weak-Instrument Behavior

Under strong identification, the endogenous regressor has a stable component explained by the instruments. The characteristic-root construction then uses the combination of \(d\) and \(y\) that removes the structural signal, and LIML has the usual large-sample behavior of an instrumental-variables estimator under the relevant regularity conditions.

The weak-instrument setting is different. For example, under a local-to-zero sequence the first-stage coefficient may shrink at rate \(1/\sqrt n\), so the instrument signal and sampling noise remain comparable in the reduced-form limit. With a fixed number of instruments and standard joint reduced-form limit conditions, the LIML root and coefficient then depend on random, nondegenerate limit quantities. The resulting estimator generally does not have a conventional normal limit centered at the structural coefficient. As with 2SLS, conventional Wald inference can therefore be unreliable.

The difference is in finite-sample and limiting behavior, not in whether the problem disappears. LIML often has less finite-sample bias toward OLS than 2SLS in overidentified, homoskedastic designs, particularly when the number of instruments is substantial relative to the sample size. But a smaller bias does not imply a smaller variance, and it does not guarantee a stable estimate in every sample. The weak-IV limit of 2SLS and weak-identification-robust procedures such as Anderson–Rubin inference address different questions from LIML’s point-estimation formula.

Common Pitfalls and Practical Interpretation

• Assuming LIML fixes weak identification. LIML changes the estimator, not the information in the instruments. Its weak-instrument distribution is generally nonstandard.
• Confusing the characteristic root with an F statistic. The root is computed from outcome and endogenous-regressor cross-products. It is not a first-stage strength test.
• Interpreting lower bias as uniformly better performance. Bias, variance, and tail behavior are distinct. A smaller bias can accompany substantial variability or extreme estimates.
• Calling a single estimate a bias comparison. Bias is a repeated-sampling expectation, \(E[\hat\beta]-\beta\), when that expectation exists. A pair of estimates from one sample cannot establish which estimator has lower bias.

LIML is especially useful as a point-estimation alternative to 2SLS in overidentified designs where many instruments may make 2SLS biased toward OLS. Its advantage is conditional, not universal. Report the identification context, examine instrument strength, and use inference designed for weak identification when conventional approximations are not credible.