maximum likelihood method was suggested M. A. Girshick in 1947, and formalized by T. W. Anderson and H. Rubin in 1949. It is used when one is interested in estimating a single structural equation at a time (hence its name of limited information), say for observation i: The structural equations for the remaining endogenous variables Y−i are not specified, and they are given in their reduced form: Notation in this context is different than for the simple IV case. One has: : The endogenous variable(s). : The exogenous variable(s) : The instrument(s) (often denoted ) The explicit formula for the LIML is: where , and λ is the smallest characteristic root of the matrix: where, in a similar way, . In other words, λ is the smallest solution of the generalized eigenvalue problem, see : K class estimators The LIML is a special case of the K-class estimators: with: Several estimators belong to this class: κ=0: OLS κ=1: 2SLS. Note indeed that in this case, the usual projection matrix of the 2SLS κ=λ: LIML κ=λ - α (n-K): estimator. Here K represents the number of instruments, n the sample size, and α a positive constant to specify. A value of α=1 will yield an estimator that is approximately unbiased. Three-stage least squares (3SLS) The three-stage least squares estimator was introduced by . It can be seen as a special case of multi-equation GMM where the set of instrumental variables is common to all equations. If all regressors are in fact predetermined, then 3SLS reduces to seemingly unrelated regressions (SUR). Thus it may also be seen as a combination of two-stage least squares (2SLS) with SUR. Applications in social science Across fields and disciplines simultaneous equation models are applied to various observational phenomena. These equations are applied when phenomena are assumed to be reciprocally causal. The classic example is supply and demand in economics. In other disciplines there are examples such as candidate evaluations and party identification or public opinion and social policy in political science; road investment and travel demand in geography; and educational attainment and parenthood entry in sociology or demography. The simultaneous equation model requires a theory of reciprocal causality that includes special features if the causal effects are to be estimated as simultaneous feedback as opposed to one-sided 'blocks' of an equation where a researcher is interested in the causal effect of X on Y while holding