Ct=α+β*yt + Ut (1)
{
yt=Ct+It (2)
Ct=α/(1-β) + β*It/(1-β) + Ut/(1-β)
α׳= α/(1-β) (3)
β׳=β/(1-β) (4) Ct=α׳+β׳*It+U׳t (5)
(5) α׳ β׳
(3) (4)
α= α׳/(1+ β׳) β= β׳/(1+ β׳)
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y=α+β*x+U x, z.
β
β= cov(y;z) / cov(x;z)
β=cov(y;z) / cov(x;z) = ov[(α+β*x+u);z] / cov(x;z) =
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= ( cov(α;z) + cov(β-x;z) + cov(u;z) ) / cov(x;z) = β + ( cov(u;z) / cov(x;z) )
cov(u;z) =>0 cov (x;z), , x z , .
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t=α+β*yt+Ut
{
yt=Ct+It
, Yt 2 Yt , .
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Ct=α+β*It+Ut
b=cov(Ct,It) / cov(yt,It)
b=b׳/(1+b׳)= ( cov(Ct;It)/var(It) ) / ( cov(Ct;It)/var(It) ) = cov(Ct;It) / ( var(It)+cov(Ct;It) )
yt=Ct+It => cov(y;I)=cov[(C+I);I]=cov(Ct;It)+var(I) => b=b
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y1=α+β1*x1+β2*x2+β3*y2+U1
{
y2=j+β4*x4+β5*y1+U2
y1=α+β1*x1+β2*x2+β3*j+β3*β4*x4+β3β5*y1
y1=(α+β3*j)/(1-β3*β5) + β1*x1/(1-β3*β5) + β2*x2/(1-β3*β5) + β3*β4*x4/(1-β3*β5)
y2=j+β4*x4+β5*α+β5*β1*x1+β5*β2*x2+β5*β3*y2
y2=(j+β5*α)/(1-β5*β3) + β4*x4/(1-β5*β3) + β5*β1*x1/(1-β5*β3) +β5*β2*x2/(1-β5*β3)
y1=α׳+β׳1*x1+β׳2*x2+β׳3*x4+U׳4
{ (2)
y2=j׳+β׳4*x4+β׳5*x1+β׳6*x2+U׳2
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