LIML and Fuller Estimation with Many Instruments

Why Consider LIML and Fuller Estimation?

With a small number of relevant instruments, two-stage least squares (2SLS) is a familiar baseline for estimating the effect of an endogenous regressor. When the instrument set is large relative to the sample, however, the first stage can fit substantial sample-specific noise. In finite samples this can pull 2SLS toward the ordinary least squares (OLS) estimate, which may be badly biased when the regressor is endogenous.

Limited information maximum likelihood (LIML) and Fuller estimation use a different adjustment to the IV estimating equation. They can reduce some of the finite-sample bias associated with many instruments. They do not make an invalid instrument valid, create information when instruments are weak, or automatically provide reliable inference. Their value is comparative and conditional: they can improve an estimator under suitable assumptions, but they do not remove the need to justify the design.

Model and K-Class Estimators

Consider a linear structural equation with one endogenous regressor $x$, outcome $y$, included exogenous covariates $W$, and excluded instruments $Z$. The model is

y = \beta x + W\gamma + u.

Here $u$ is the structural error. The instruments must be relevant for $x$, conditionally on $W$, and satisfy the required exclusion and exogeneity conditions: they must not affect $y$ except through $x$, and must be uncorrelated with $u$. The structural model and instrument set must also be correctly specified, with enough independent instrument variation to identify $\beta$. These conditions concern the design; switching estimators does not relax them.

To describe the estimators compactly, partial out the included covariates: let $\tilde y$ and $\tilde x$ be the residuals from projecting $y$ and $x$ on $W$. Let $P_Z$ be the projection matrix onto the residualized excluded-instrument space, and let $M_Z=I-P_Z$. In the expressions below, all quantities are in this partialled-out sample. For a scalar endogenous regressor, a k-class estimate with parameter $\kappa$ is

\hat\beta(\kappa) = \frac{\tilde x'\left[P_Z+(1-\kappa)M_Z\right]\tilde y} {\tilde x'\left[P_Z+(1-\kappa)M_Z\right]\tilde x}.

This equation shows how the methods are related. At $\kappa=1$, the $M_Z$ terms disappear and the estimate is 2SLS. At $\kappa=0$, the matrix in brackets is $P_Z+M_Z=I$, giving the OLS slope in the partialled-out data. LIML and Fuller use other values of $\kappa$, estimated or adjusted using the instrument-projected data.

Definition: LIML is a k-class IV estimator whose $\kappa$ is determined by a likelihood-based eigenvalue calculation using the outcome and endogenous-regressor data. In the scalar-endogenous-regressor case, under the usual full-rank regularity conditions ensuring a finite root strictly greater than 1, $\kappa_{\text{LIML}}$ is the smallest real root above 1 of the equation below. Otherwise, when it exists, the LIML root is the relevant generalized eigenvalue. $$ \det\!\left(\tilde Y'\left[P_Z+(1-\kappa)M_Z\right]\tilde Y\right)=0, \qquad \tilde Y=[\tilde y,\tilde x]. $$ Fuller($\alpha$) modifies this value to $$ \kappa_F=\kappa_{\text{LIML}}-\frac{\alpha}{n-q}, $$ and uses $\hat\beta(\kappa_F)$. Here $n$ is the sample size, $q$ is the rank of the excluded-instrument space after partialling out $W$, and $\alpha$ is a chosen positive adjustment, commonly 1. This tutorial uses that stated convention for the adjustment.

What the LIML Calculation Changes

The determinant condition selects a value of $\kappa$ at which the transformed cross-product matrix for $\tilde y$ and $\tilde x$ becomes singular. That is the eigenvalue step distinguishing LIML from 2SLS: 2SLS fixes $\kappa=1$, while LIML estimates it from the data. Once $\kappa$ is chosen, the coefficient is calculated from the same k-class equation.

The role of the adjustment can be seen directly. If $\kappa_{\text{LIML}}>1$, subtracting a positive amount moves $\kappa_F$ closer to 1 precisely when $0<\frac{\alpha}{n-q}<2(\kappa_{\text{LIML}}-1)$. When the adjusted value remains above 1, its $M_Z$ weight, $1-\kappa_F$, is less negative than the LIML weight $1-\kappa_{\text{LIML}}$. This correction is designed to improve finite-sample behavior in common many-instrument settings; it is not a general guarantee that every realized estimate is closer to the true effect.

The Fuller adjustment depends on the sample size and instrument rank, as well as on the chosen $\alpha$. It is not a new instrument and does not change which variation identifies the effect. Software can use different counting conventions or finite-sample options, so an applied analysis should report the Fuller version and its adjustment convention.

Worked Example: Calculating the Three Estimates

Suppose a hypothetical analysis has $n=220$ observations and $q=20$ independent excluded-instrument directions after partialling out the included covariates. The LIML calculation gives $\kappa_{\text{LIML}}=1.20$. The relevant sample cross-products are $\tilde x'P_Z\tilde y=24$, $\tilde x'P_Z\tilde x=80$, $\tilde x'M_Z\tilde y=4$, and $\tilde x'M_Z\tilde x=40$. These are illustrative summaries, not findings from a real study.

2SLS. Since 2SLS uses $\kappa=1$, the $M_Z$ contributions have weight zero:

\hat\beta_{\text{2SLS}} = \frac{24}{80} =0.3000.

LIML. Here $1-\kappa_{\text{LIML}}=-0.20$, so

\hat\beta_{\text{LIML}} = \frac{24+(-0.20)(4)}{80+(-0.20)(40)} = \frac{23.2}{72} \approx 0.3222.

Fuller(1). Its adjustment is $1/(n-q)=1/200=0.005000$, giving $\kappa_F=1.20-0.005000=1.195$. Thus $1-\kappa_F=-0.195$, and

\hat\beta_F = \frac{24+(-0.195)(4)}{80+(-0.195)(40)} = \frac{23.22}{72.2} \approx 0.3216.

In this hypothetical calculation, both LIML and Fuller give larger estimates than 2SLS, while Fuller is slightly closer to 2SLS than LIML. That pattern is an arithmetic consequence of these cross-products, not a rule about the direction of bias. The coefficients have the units of $y$ per unit of $x$; no causal interpretation follows from the calculations alone.

Why They Can Help with Many Instruments

As the instrument count grows, the first stage can fit more of the sample variation in $x$, including noise that happens to be related to the structural error in that sample. This is the source of the familiar many-instrument concern for 2SLS. Under suitable many-instrument asymptotic conditions, 2SLS can retain non-negligible finite-sample-type bias even as the sample grows, whereas LIML is often less affected. Fuller’s correction is intended to improve LIML’s finite-sample behavior while retaining this useful comparative property in many common settings.

These are conditional results, not universal rankings. They rely on a correctly specified linear IV model, valid instruments, and an identification structure adequate for the estimator and asymptotic argument being used. The extent of bias reduction depends on instrument strength, instrument count relative to sample size, and the data-generating process. In a particular sample, LIML or Fuller can be less precise, produce a more extreme estimate, or differ little from 2SLS.

Worked Example: Fuller Adjustment When Instruments Are Numerous

In a second hypothetical sample, suppose $n=120$, $q=30$, $\kappa_{\text{LIML}}=1.25$, and the cross-products are $\tilde x'P_Z\tilde y=36$, $\tilde x'P_Z\tilde x=120$, $\tilde x'M_Z\tilde y=10$, and $\tilde x'M_Z\tilde x=80$. The instrument rank is one quarter of the sample size, so the many-instrument issue merits attention, though the count alone does not establish that the instruments are weak or invalid.

The 2SLS estimate is

\hat\beta_{\text{2SLS}}=\frac{36}{120}=0.3000.

For LIML, $1-\kappa_{\text{LIML}}=-0.25$:

\hat\beta_{\text{LIML}} = \frac{36+(-0.25)(10)}{120+(-0.25)(80)} = \frac{33.5}{100} =0.3350.

For Fuller(1), the adjustment is $1/(120-30)=1/90$, not a rounded substitute. Therefore,

\kappa_F=1.25-\frac{1}{90}=1.238888\ldots, \qquad 1-\kappa_F=-0.238888\ldots.

Using those values consistently,

\hat\beta_F = \frac{36-(0.238888\ldots)(10)} {120-(0.238888\ldots)(80)} = \frac{33.61111\ldots}{100.88889\ldots} \approx 0.3331.

Fuller is between 2SLS and LIML in this example. The arithmetic illustrates the adjustment; it does not show that the estimate is unbiased or that the instruments meet the causal assumptions. Those questions require design knowledge and appropriate diagnostics.

Bias Reduction Is Not Instrument Validation

The distinction between estimator behavior and instrument validity is essential. Suppose the true structural relationship is $y=\beta x+\gamma z+u$, but $z$ is treated as an excluded instrument even though it has a direct effect $\gamma$ on $y$. If the first-stage relationship is $x=\pi z+v$, then, in a simplified just-identified case with the relevant exogeneity conditions for $u$ and $v$, the IV ratio has probability limit

\frac{\operatorname{Cov}(z,y)}{\operatorname{Cov}(z,x)} = \frac{\beta\operatorname{Cov}(z,x)+\gamma\operatorname{Var}(z)} {\pi\operatorname{Var}(z)} = \beta+\frac{\gamma}{\pi}.

For example, if $\beta=2$, $\gamma=0.10$, and $\pi=0.25$, the ratio is $2+0.10/0.25=2.4$, not 2. In this hypothetical example the discrepancy is due to exclusion failure. Replacing 2SLS with LIML or Fuller does not remove the direct effect and cannot, by itself, recover the intended causal parameter.

Likewise, using LIML or Fuller does not ensure strong identification. With weak instruments, estimates can be unstable and ordinary confidence intervals or tests can be misleading. Estimation and inference are separate decisions: a point estimator that is less biased under a many-instrument approximation does not automatically supply valid weak-instrument-robust inference.

Worked Example: Distinguishing Bias Adjustment from Identification

Suppose a hypothetical analyst has a large instrument set, and the first-stage evidence suggests that the instruments jointly predict treatment. The analyst calculates both 2SLS and LIML, obtaining estimates of 0.30 and 0.36. A Fuller estimate is 0.35. The values are in units of outcome per unit of treatment.

The disagreement is a reason to examine sensitivity to the estimator and the many-instrument setting; it is not proof that 0.35 is the correct causal effect. To support a causal interpretation, the analyst still needs a defensible argument that the instruments affect the outcome only through treatment and are uncorrelated with the structural error. The analyst also needs evidence adequate for the identification and inference claims being made.

If the instruments are invalid, none of the three estimates is rescued by this comparison. If identification is weak, a point estimate alone does not reveal the reliability of a conventional confidence interval. The responsible conclusion is therefore conditional: Fuller and LIML may reduce a particular source of estimator bias, but the design assumptions and inferential procedure remain decisive.

Common Mistakes and Practical Interpretation

“LIML is always better than 2SLS.” The methods have different finite-sample behavior; neither is guaranteed to be closer to the causal effect in every sample or model.
“Fuller fixes weak instruments.” The Fuller adjustment modifies $\kappa$; it does not create first-stage information or guarantee reliable weak-IV inference.
“An estimator with less many-instrument bias validates the instruments.” Bias reduction does not establish relevance, exclusion, or exogeneity. Those are substantive assumptions about the design.
Rounding the adjustment too early. Calculate $\alpha/(n-q)$ from the stated values and carry adequate precision through the numerator and denominator. If a rounded value is used, the displayed arithmetic must use that same value consistently.

When reporting results, state the estimator, instrument count or rank convention, the Fuller $\alpha$ if applicable, and the coefficient with its units. Explain why the instruments are credible and report inference suited to the identification setting. Comparing 2SLS, LIML, and Fuller can reveal sensitivity to many-instrument bias, but it cannot replace those steps.

Key takeaway: LIML chooses a data-dependent k-class parameter, and Fuller($\alpha$) subtracts $\alpha/(n-q)$ from it. These estimators can reduce many-instrument bias under suitable conditions, but they do not repair invalid instruments, guarantee strong identification, or automatically make conventional inference reliable.

Check Your Understanding

Answer each question using the model and conventions stated above.

In the k-class equation, what value of $\kappa$ gives 2SLS, and what value gives OLS after partialling out included covariates?
For $n=150$, $q=30$, and $\alpha=1$, calculate the Fuller adjustment and write $\kappa_F$ in terms of $\kappa_{\text{LIML}}$.
Why does a Fuller estimate closer to 2SLS than LIML in one sample not establish that Fuller has less bias in every sample?
If an instrument directly affects the outcome, which causal assumption fails, and can switching from 2SLS to LIML repair that failure?
Why does a point estimate from LIML or Fuller not, by itself, justify using conventional confidence intervals with weak instruments?