PARTINORM - D.S.G. POLLOCK: TOPICS IN ECONOMETRICS...

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D.S.G. POLLOCK: TOPICS IN ECONOMETRICS FACTORISING THE THE NORMAL DISTRIBUTION The joint distribution of x and y can be factored as the product of the marginal distribution of x and the conditional distribution of y given x : (1) N ( y, x )= N ( y | x ) N ( x ) . The following notation may be adopted: (2) z = · y E ( y ) x E ( x ) ¸ ,w = · y E ( y | x ) x E ( x ) ¸ = · ε x E ( x ) ¸ . We may assume that w is a linear function of z . Then, the mapping from z to w = Qz may be represented by (3) · ε x E ( x ) ¸ = · I B 0 0 I ¸· y E ( y ) x E ( x ) ¸ , wherein (4) ε = y E ( y | x y E ( y ) B 0 { x E ( x ) } . The following dispersion matrices are deFned: (5) D ( z )=Σ zz = · Σ yy Σ yx Σ xy Σ xx ¸ ,D ( w = · Σ εε 0 xx ¸ . The o±-diaogonal blocks of D ( w ), which are C { y E ( y | x ) ,x } = 0 and its transpose, bear witness to the fact that the prediction error ε = y E ( y | x ) is uncorrelated with x , which is the instrument of the prediction. This is the consequence of the factorisation, whereby the conditional distribution is independent of the mariginal distribution.
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