ABSTRACT
A simultaneous-equations model can be represented as https://www.w3.org/1998/Math/MathML"> Z Α + Ε = [ Y , X ] [ Γ B ] + Ε = Y Γ + X B + E = 0 , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203180754/695bd69b-d1f9-4baf-8f3a-2907edb27a88/content/math_1043_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> where Y is an n × m matrix of n observations y_{ti} on m jointly dependent (current endogenous) variables, X is an n × k matrix of n observations x_{ti} on k predetermined (exogenous and lagged-endogenous) variables, ^{1} Γ is a nonsingular m × m matrix of unknown parameters γ_{ij} (which may be normalized so that its diagonal elements are equal to – 1), B is a k × m matrix of unknown parameters β_{ij} , and E is an n × m matrix of random errors ε_{ti} (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 ⊗ I n , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203180754/695bd69b-d1f9-4baf-8f3a-2907edb27a88/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 = [a_{ij}B]. Thus, it is assumed that E{ε_{ti} ε_{t′j}} = σ_{ij} δ_{tt′} where δ_{tt′} 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 = [σ_{ ij } ] 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 = – I_{m} , (8.1.1) reduces to the multivariate multiple-regression model (1.1.1).