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Simultaneous equation models are a form of statistical model in the form of a set of linear simultaneous equations. They are often used in econometrics. == Structural and reduced form == Suppose there are ''m'' regression equations of the form : where ''i'' is the equation number, and 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 ''i''th 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 ''i''th 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 ''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 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, is a ''T×m'' matrix of the error terms. Postmultiplying the structural equation by , 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 is quite complicated, and therefore the reduced form is more suitable for prediction but not inference. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「simultaneous equations model」の詳細全文を読む スポンサード リンク
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