By Pedroza C.
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Extra info for A Bayesian forecasting model: predicting U.S. male mortality (2006)(en)(21s)
23) on the basis of observations (y, X) and assumptions already stated. This will be done by choosing a ˆ which then will be used to calculate the conditional suitable estimator β, expectation E(y|X) = Xβ and an estimate for the error variance σ 2 . 37) where C : K × T and d : K × 1 are nonstochastic matrices to be determined by minimizing a suitably chosen risk function. First we have to introduce some deﬁnitions. 7 βˆ is called a homogeneous estimator of β if d = 0; otherwise βˆ is called heterogeneous.
25) which is the same as the least squares estimator of β, and the minimum dispersion matrix is σ 2 (X ′ X)−1 . 26) Proof: Let a + By be an unbiased estimator of β. Then E(a + By) = a + BXβ = β ∀β ⇒ a = 0 , BX = I . 22), it is suﬃcient that 0 = = cov(By, c′ Z ′ y) ∀c σ 2 BZc ∀c ⇒ BZ = 0 ⇒ B = AX ′ for some A . 28): BX = I , B = AX ′ . 5. Case 2: Rank(X) = r < K (deﬁciency in rank) and rank(Z) = T − r, in which case X ′ X is singular. We denote any g-inverse of X ′ X by (X ′ X)− . The consequences of deﬁciency in the rank of X, which arises in many practical applications, are as follows.
Y, X) = ⎝ ... ⎝ ⎠ (1) (K) . ⎠ . 2) 34 3. The Multiple Linear Regression Model and Its Extensions where y = (y1 , . . , yT )′ is a T -vector, xi = (x1i , . . , xKi )′ is a K-vector and x(j) = (xj1 , . . , xjT )′ is a T -vector. 1): yt = x′t β + et , t = 1, . . 3) where β ′ = (β1 , . . 4) where X is a T × K design matrix of T observations on each of the K explanatory variables and e = (e1 , . . , eT )′ . If x1 = (1, . . 4). We consider the problems of estimation and testing of hypotheses on β under some assumptions.