# 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 *Y _{t}*given

*X*is binomial with parameters

_{t}*n*and

*X*, i.e.,

_{t}a. Show that the observation equation with state variable transformed by the logit transformation *Wt *= ln*(X _{t} /(*1 −

*X*follows an exponential family

_{t}))*p(y _{t}*|

*w*= exp{

_{t})*y*−

_{t}w_{t}*b(w*+

_{t})*c(y*}

_{t})*.*

Determine the functions *b(*·*) *and *c(*·*)*.

b. Suppose that the state *X _{t}*has the beta density

*p(x _{t}*

_{+1}|y

^{(t)}*)*=

*f (x*

_{t}_{+1};

*α*

_{t}_{+1|}

_{t}*, λ*

_{t}_{+1|}

_{t}*),*

where

*f (x*; *α, λ) *= [*B(α, λ)*]^{−1}*x ^{α}*

^{−1}

*(*1 −

*x)*

^{λ}^{−1}

*,*0

*< x <*1

*,*

.

c. Under the assumptions made in (b), show that *E(X _{t}*|Y

*) =*

^{(t)}*E(X*

_{t}_{+1}|Y

*) and Var(*

^{(t)}*X*|Y

_{t}*)*

^{(t)}*<*Var(

*X*

_{t}_{+1}|Y

*)*

^{(t)}