Consider an observation-driven model in which Y t given X t is binomial with parameters n and X t ,…

Consider an observation-driven model in which Ytgiven Xtis binomial with parameters and Xt, i.e.,

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a. Show that the observation equation with state variable transformed by the logit transformation Wt = ln(Xt /(1 − Xt )) follows an exponential family

p(yt|wt ) = exp{ytwt− b(wt ) c(yt )}.

Determine the functions b(·and c(·).

b. Suppose that the state Xthas the beta density

p(xt+1|y(t)f (xt+1αt+1|t, λ t+1|t ),

where

f (xα, λ) = [B(α, λ)]−1xα−1(1 − x)λ−1< x < 1,

.

c. Under the assumptions made in (b), show that E(Xt|Y(t)) = E(Xt+1|Y(t)) and Var(Xt|Y(t))<Var(Xt+1|Y(t))

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