[2] Simultaneity poses challenges for the estimation of the statistical parameters of interest, because the Gauss–Markov assumption of strict exogeneity of the regressors is violated.
[3] This situation prompted the development, spearheaded by the Cowles Commission in the 1940s and 1950s,[4] of various techniques that estimate each equation in the model seriatim, most notably limited information maximum likelihood and two-stage least squares.
Postmultiplying the structural equation by Γ −1, the system can be written in the reduced form as This is already a simple general linear model, and it can be estimated for example by ordinary least squares.
into the individual factors Β and Γ −1 is quite complicated, and therefore the reduced form is more suitable for prediction but not inference.
The identification conditions require that the system of linear equations be solvable for the unknown parameters.
The rank condition, a stronger condition which is necessary and sufficient, is that the rank of Πi0 equals ni, where Πi0 is a (k − ki)×ni matrix which is obtained from Π by crossing out those columns which correspond to the excluded endogenous variables, and those rows which correspond to the included exogenous variables.
In simultaneous equations models, the most common method to achieve identification is by imposing within-equation parameter restrictions.
To illustrate how cross equation restrictions can be used for identification, consider the following example from Wooldridge[6] where z's are uncorrelated with u's and y's are endogenous variables.
The method is called “two-stage” because it conducts estimation in two steps:[7] If the ith equation in the model is written as where Zi is a T×(ni + ki) matrix of both endogenous and exogenous regressors in the ith equation, and δi is an (ni + ki)-dimensional vector of regression coefficients, then the 2SLS estimator of δi will be given by[7] where P = X (X ′X)−1X ′ is the projection matrix onto the linear space spanned by the exogenous regressors X.
The “limited information” maximum likelihood method was suggested by M. A. Girshick in 1947,[13] and formalized by T. W. Anderson and H. Rubin in 1949.
One has: The explicit formula for the LIML is:[15] where M = I − X (X ′X)−1X ′, and λ is the smallest characteristic root of the matrix: where, in a similar way, Mi = I − Xi (Xi′Xi)−1Xi′.
[18][19] It can be seen as a special case of multi-equation GMM where the set of instrumental variables is common to all equations.
In other disciplines there are examples such as candidate evaluations and party identification[21] or public opinion and social policy in political science;[22][23] road investment and travel demand in geography;[24] and educational attainment and parenthood entry in sociology or demography.