Consider the following estimators for these parameters Lemma 61 Let A B and C

Consider the following estimators for these

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Consider the following estimators for these parameters. Lemma 6.1. Let A , B and C be matrices and X , Y be vectors. Under regularity conditions (including proper dimensions): ∂X ( A prime X ) = A and ∂X ( X prime B ) = B , (6.1) ∂X ( X prime A X ) = 2 A X and (6.2) ∂X braceleftbig ( Y - C X ) prime A ( Y - C X ) bracerightbig = - 2 C prime A ( Y - C X ) . (6.3) *Proof: The regularity conditions can be found in any multivariate regres- sion textbook. Left as an exercise. Theorem 6.1. The weighted least squares regression estimators which min- imise Q j ( β ) = bracketleftbig X j - Y β j ) bracketrightbig prime V - 1 j bracketleftbig X j - Y β j ) bracketrightbig , are given by ˆ β j = ( Y prime V - 1 j Y ) - 1 Y prime V - 1 j X j = U j Y prime V - 1 j X j , (6.4) where we define U j = ( Y prime V - 1 j Y ) - 1 for notational convenience.
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62 CHAPTER 6. CREDIBILITY REGRESSION MODELS Proof: By (6.3) we have that ∂β Q j ( β ) = - 2 Y prime V - 1 j bracketleftbig X j - Y β j ) bracketrightbig , then the solution ˆ β j must satisfy the following normal equation: - 2 Y prime V - 1 j bracketleftbig X j - Y ˆ β j bracketrightbig = 0 . Hence ( Y prime V - 1 j Y ) ˆ β j = Y prime V - 1 j X j and ˆ β j = ( Y prime V - 1 j Y ) - 1 Y prime V - 1 j X j . a50 The estimator in (6.4) is expressed in terms of V j , that is the conditional covariance of X j given Θ j . The following corollary shows that it can also be expressed in terms of the unconditional covariance Cov( X j ). Corollary 6.1. Let C j = Cov( X j ) = s 2 V j + Y A Y prime then ˆ β j = ( Y prime C - 1 j Y ) - 1 Y prime C - 1 j X j . *Proof: see Lemma 6.2 below and the proof of Lemma 1.4.1, page 133 of Goovaerts et al. (1990). a50 Lemma 6.2. (i) E ( X j ) = Y E bracketleftbig β j ) bracketrightbig = Y b , (ii) E [ ˆ β j | Θ j ] = β j ) and E [ ˆ β j ] = E bracketleftbig β j ) bracketrightbig = b , (iii) Cov( X j ) = s 2 V j + Y A Y prime = C j , by definition, (iv) Cov bracketleftbig β j ) , X j bracketrightbig = A Y prime , (v) Cov bracketleftbig ˆ β j , β j ) bracketrightbig = A , (vi) Cov[ ˆ β j , X j ] = [ A + s 2 U j ] Y prime , where U j = ( Y prime V - 1 j Y ) - 1 ,
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6.2. HACHEMEISTER’S REGRESSION MODEL 63 (vii) Cov[ ˆ β j ] = A + s 2 U j , where U j = ( Y prime V - 1 j Y ) - 1 , *Proof: (i) The definition of the model E ( X j | Θ j ) = Y β j ) implies that E ( X j ) = E bracketleftbig E ( X j | Θ j ) bracketrightbig = E bracketleftbig Y β j ) bracketrightbig = Y E bracketleftbig β j ) bracketrightbig = Y b . (ii) E [ ˆ β j | Θ j ] = E bracketleftbig ( Y prime V - 1 j Y ) - 1 Y prime V - 1 j X j | Θ j bracketrightbig = ( Y prime V - 1 j Y ) - 1 Y prime V - 1 j E ( X j | Θ j ) = ( Y prime V - 1 j Y ) - 1 Y prime V - 1 j Y β j ) = β j ) , with (i) it implies that E [ ˆ β j ] = E bracketleftbig E ( ˆ β j | Θ j ) bracketrightbig = E bracketleftbig β j ) bracketrightbig . (iii) C j = Cov( X j ) = E bracketleftbig Cov( X j | Θ j ) bracketrightbig + Cov bracketleftbig E ( X j | Θ j ) bracketrightbig = E bracketleftbig σ 2 j ) V j bracketrightbig + Y Cov bracketleftbig β j ) bracketrightbig Y prime = s 2 V j + Y A Y prime .
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