1. Consider the normal measurement error model for (y, X, x|Z) with yi |Xi , Zi ∼ N

α + βXi + γ Zi, σ2

ε



, xi |Xi ∼ NXi, σ2

δ



, Xi |Zi ∼ N

μX + κ Zi, σ2

η



, where Z is error free. Show how with transformed X and γ this model can be converted to a specification for (y, X, x) involving a regression of x on Z, namely yi |X∗

i , Zi ∼ N

α + βX∗

i

+ γ

∗Zi, σ2

ε



, xi |X∗

i

∼ NX∗

i

+ κ Zi, σ2

δ



, Xi ∼ N

μX, σ2

η



.

Obtain the joint marginal density of the observations y and x given the parameters {α, β,

γ *, X∗i , κ, μX , σ2

ε , σ2

δ , σ2

η

}.