LIML

LIML, or Limited-Information Maximum Likelihood, is an estimation method used in econometrics for linear instrumental variables models that include endogenous regressors. It is designed to extract information from the equation of interest using the available instruments, while not requiring a full specification of the entire system.

Conceptually, LIML treats the model as a likelihood problem limited to the equation being estimated. It can be viewed as a maximum likelihood estimator under a normal-error assumption, but only the information relevant to the instrumented relationship is exploited. This makes LIML a middle ground between two common approaches: 2SLS, which uses a purely stage-wise procedure, and full information maximum likelihood, which requires specifying the entire system of equations.

Properties and use cases. LIML tends to perform more robustly than 2SLS when instruments are weak or numerous, showing reduced finite-sample bias in many situations. When instruments are strong, LIML is asymptotically equivalent to other efficient IV estimators. It is not universally superior; in some contexts, FIML or alternative methods may offer advantages, especially in small samples or in nonstandard settings. Computationally, LIML is more demanding than 2SLS but is readily available in standard econometric software as an option for IV estimation.

Implementation and interpretation. Practitioners estimate the model with instrumental variables, selecting LIML to obtain parameter estimates and standard errors that are often more robust to weak instruments. Inference can be complemented with tests designed for weak instruments and overidentification, such as related statistics used in LIML contexts.

Related topics include 2SLS, FIML, and methods for weak instruments.