Simultaneous equations model

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. One can estimate these models equation by equation; however, estimation methods that exploit the system of equations, such as generalized method of moments (GMM) and instrumental variables estimation (IV) tend to be more efficient. 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. One can estimate these models equation by equation; however, estimation methods that exploit the system of equations, such as generalized method of moments (GMM) and instrumental variables estimation (IV) tend to be more efficient. 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 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 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 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 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 = 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 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. Firstly, the rank of the matrix X of exogenous regressors must be equal to k, both in finite samples and in the limit as T → ∞ (this later requirement means that in the limit the expression 1 T X ′ X {displaystyle scriptstyle {frac {1}{T}}X'!X} should converge to a nondegenerate k×k matrix). Matrix Γ is also assumed to be non-degenerate. Secondly, error terms are assumed to be serially independent and identically distributed. That is, if the tth row of matrix U is denoted by u(t), then the sequence of vectors {u(t)} should be iid, with zero mean and some covariance matrix Σ (which is unknown). In particular, this implies that E = 0, and E = T Σ.