## ABSTRACT

A simultaneous-equations model can be represented as https://www.w3.org/1998/Math/MathML">ZΑ+Ε=[Y,X][ ΓB ]+Ε=YΓ+XB+E=0,https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203180754/c73c8eaa-6e2d-45b3-bd8b-866a06ca944e/content/math_1043_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> where Y is an n × m matrix of n observations yti on m jointly dependent (current endogenous) variables, X is an n × k matrix of n observations xti on k predetermined (exogenous and lagged-endogenous) variables,1 Γ is a nonsingular m × m matrix of unknown parameters γij (which may be normalized so that its diagonal elements are equal to – 1), B is a k × m matrix of unknown parameters βij, and E is an n × m matrix of random errors εti(t = 1, 2, …, n; i = 1,2, …, m) with conditional means and variances https://www.w3.org/1998/Math/MathML">E{E| X}=0,   E{(col E)(col E)′|X}=Σ⊗In,https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203180754/c73c8eaa-6e2d-45b3-bd8b-866a06ca944e/content/math_1044_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> where col E denotes the nm × 1 column vector of columns of E,2 Σ is an m × m symmetric matrix which is assumed to be positive-definite, and ⊗ denotes the Kronecker product A ⊗ B = [aijB]. Thus, it is assumed that E{εti εt′j} = σij δtt′ where δtt′ is the Kronecker delta, which implies absence of serial correlation. (This assumption can certainly be relaxed, but in order to concentrate on the simultaneous-equations issue it will be convenient to leave this problem aside.) The m × m matrix Σ = [σij ] is usually called the “contemporaneous” or “simultaneous” variance matrix in contrast to the n × n sample variance matrix V which in this case is assumed to be In. Note that in the special case Γ = – Im, (8.1.1) reduces to the multivariate multiple-regression model (1.1.1).