Limited information maximum likelihood

Simultaneous equation models are a type of statistical model in the form of a set of linear simultaneous equations. They are often used in econometrics.

Suppose there are m regression equations of the form

where i is the equation number, and t = 1, ..., T is the observation index. In these equations x_it is the k_i×1 vector of exogenous variables, y_it is the dependent variable, y_−i,t is the n_i×1 vector of all other endogenous variables which enter the i^th equation on the right-hand side, and u_it are the error terms. The “−i” notation indicates that the vector y_−i,t may contain any of the y’s except for y_it (since it is already present on the left-hand side). The regression coefficients β_i and γ_i are of dimensions k_i×1 and n_i×1 correspondingly. Vertically stacking the T observations corresponding to the i^th equation, we can write each equation in vector form as

where y_i and u_i are T×1 vectors, X_i is a T×k_i matrix of exogenous regressors, and Y_−i is a T×n_i matrix of endogenous regressors on the right-hand side of the i^th equation. Finally, we can move all endogenous variables to the left-hand side and write the m equations jointly in vector form as

This representation is known as the structural form. In this equation Y = [y₁y₂ ... y_m] is the T×m matrix of dependent variables. Each of the matrices Y_−i is in fact an n_i-columned submatrix of this Y. The m×m matrix Γ, which describes the relation between the dependent variables, has a complicated structure. It has ones on the diagonal, and all other elements of each column i are either the components of the vector −γ_i or zeros, depending on which columns of Y were included in the matrix Y_−i. The T×k matrix X contains all exogenous regressors from all equations, but without repetitions (that is, matrix X should be of full rank). Thus, each X_i is a k_i-columned submatrix of X. Matrix Β has size k×m, and each of its columns consists of the components of vectors β_i and zeros, depending on which of the regressors from X were included or excluded from X_i. Finally, U = [u₁u₂ ... u_m] is a T×m matrix of the error terms.

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