Simultaneous equations models are a type of statistical model in which the dependent variables are functions of other dependent variables, rather than just independent variables. This means some of the explanatory variables are jointly determined with the dependent variable, which in economics usually is the consequence of some underlying equilibrium mechanism. Take the typical supply and demand model: whilst typically one would determine the quantity supplied and demanded to be a function of the price set by the market, it is also possible for the reverse to be true, where producers observe the quantity that consumers demand and then set the price. 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. And while it would be natural to estimate all simultaneous equations at once, this often leads to a computationally costly non-linear optimization problem even for the simplest system of linear equations. This situation prompted the development, spearheaded by the Cowles Commission in the 1940s and 1950s, of various techniques that estimate each equation in the model seriatim, most notably limited information maximum likelihood and two-stage least squares.
Structural and reduced form Suppose there are m regression equations of the form
y i t = y − i , t ′ γ i + x i t ′ β i + u i t , i = 1 , … , m , {\displaystyle y_{it}=y_{-i,t}'\gamma _{i}+x_{it}'\;\!\beta _{i}+u_{it},\quad i=1,\ldots ,m,}
where i is the equation number, and t = 1, ..., T is the observation index. In these equations xit is the ki×1 vector of exogenous variables, yit is the dependent variable, y−i,t is the ni×1 vector of all other endogenous variables which enter the ith equation on the right-hand side, and uit are the error terms. The “−i” notation indicates that the vector y−i,t may contain any of the y’s except for yit (since it is already present on the left-hand side). The regression coefficients βi and γi are of dimensions ki×1 and ni×1 correspondingly. Vertically stacking the T observations corresponding to the ith equation, we can write each equation in vector form as
y i = Y − i γ i + X i β i + u i , i = 1 , … , m , {\displaystyle y_{i}=Y_{-i}\gamma _{i}+X_{i}\beta _{i}+u_{i},\quad i=1,\ldots ,m,}
where yi and ui are T×1 vectors, Xi is a T×ki matrix of exogenous regressors, and Y−i is a T×ni 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
Y Γ = X B + U . {\displaystyle Y\Gamma =X\mathrm {B} +U.\,}
This representation is known as the structural form. In this equation Y = [y1 y2 ... ym] is the T×m matrix of dependent variables. Each of the matrices Y−i is in fact an ni-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 −γi 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 Xi is a ki-columned submatrix of X. Matrix Β has size k×m, and each of its columns consists of the components of vectors βi and zeros, depending on which of the regressors from X were included or excluded from Xi. Finally, U = [u1 u2 ... um] 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
Y = X B Γ − 1 + U Γ − 1 = X Π + V . {\displaystyle Y=X\mathrm {B} \Gamma ^{-1}+U\Gamma ^{-1}=X\Pi +V.\,}
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 Π ^ {\displaystyle \scriptstyle {\hat {\Pi }}} into the individual factors Β and Γ −1 is quite complicated, and therefore the reduced form is more suitable for prediction but not inference.
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