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By Eckmann J.-P., Guenin M.

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100) To establish this result, it is convenient to use the following lemma in which the relation between the OLS residual vector e = (e1 , . . 83) and the equation-wise residual vectors εˆ j = yj − Xj βˆj = Nj yj (j = 1, . . 101) are clarified. 102) i=1 Proof. The proof is the calculation: Let Mj = Xj (Xj Xj )+ Xj and so Inj = Mj + Nj . Then we have ej = yj − Xj (X X)−1 X y p = yj − Xj (X X)−1 Xi yi i=1 GENERALIZED LEAST SQUARES ESTIMATORS 47 p = (Mj + Nj )yj − Xj (X X)−1 Xi (Mi + Ni )yi i=1 p = (Xj βˆj + εˆ j ) − Xj (X X)−1 Xi Xi βˆi i=1 p Xi Xi (βˆi − βˆj ) , = εˆ j − Xj (X X)−1 i=1 where the last equality is due to p Xj βˆj = Xj (X X)−1 Xi Xi βˆj .

23) with E(εj ) = 0 and Cov(εj ) = σj2 Inj (j = 1, . . , p), where the coefficient vectors are restricted as β1 = · · · = βp ≡ β. 25) where σ 2 = σ12 , θ = σ12 /σ22 , λ(θ ) = θ − 1 and D = Here θ ∈ ≡ (0, ∞), and (θ ) is positive definite on 0 0 0 In 2 . 1) is of the form     X1 y1 0     .. y =  ...  : n × 1, X =   : n × k, .  yp  β1  ..  β =  .  : k × 1, βp  0  Xp ε1  ..  ε = . 27) it is called a p-equation SUR model, which was originally formulated by Zellner (1962), where p yj : m × 1, Xj : m × kj , n = pm, k = kj j =1 and ⊗ denotes the Kronecker product.

In this case, a natural estimator of β is a GME with unknown in b( ) replaced by an estimator ˆ , which we shall call a GLSE. 42) is called a GLSE if ˆ is almost surely positive definite and is a function of the OLS residual vector e, where e = Ny with N = In − X(X X)−1 X . 44) that is, the set of (n − k) columns of the matrix Z forms an orthonormal basis of the orthogonally complementary subspace of the column space L(X) of X. 44) and fix it throughout. 42) is in general highly nonlinear in y and hence it is generally difficult to investigate its finite sample properties.

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### Methodes Aigebriques en Mecanique Statistique by Eckmann J.-P., Guenin M.

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