Download e-book for kindle: Generalized Least Squares by Takeaki Kariya

By Takeaki Kariya

ISBN-10: 0470866977

ISBN-13: 9780470866979

ISBN-10: 0470866985

ISBN-13: 9780470866986

Generalised Least Squares adopts a concise and mathematically rigorous procedure. it is going to supply an up to date self-contained creation to the unified thought of generalized least squares estimations, adopting a concise and mathematically rigorous procedure. The publication covers intensive the 'lower and top bounds approach', pioneered via the 1st writer, that is extensively considered as the most important and great tool for generalized least squares estimation, supporting the reader advance their realizing of the idea. The ebook additionally includes routines on the finish of every bankruptcy and purposes to statistical data, econometrics, and biometrics, allowing use for self-study or as a path textual content.

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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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Generalized Least Squares by Takeaki Kariya

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