Simultaneous Equations Model - Structural and Reduced Form

Structural and Reduced Form

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 ith 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 ith 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 ith 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 = 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 = is a T×m matrix of the error terms.

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. Unfortunately, the task of decomposing the estimated matrix into the individual factors Β and Γ −1 is quite complicated, and therefore the reduced form is more suitable for prediction but not inference.