$1.$ Consider the following ARCH$(1)$ model:

yt $=\backslash$mu $+$ut

ut $=$ vt$\backslash$sigma t$,$ vt is i$.$i$.$d$.$ with mean $0$ and variance $1$

$\backslash$sigma t$2=\backslash$alpha $0+\backslash$alpha $1$ u $2$t $1.$ Please compute the following:

$($a$).$ E$[\backslash$sigma $2$ t$+1$

$($b$).$ E$[\backslash$sigma $2$ t$+2$

$|\backslash$Omega $]$ t

$|\backslash$Omega $]$ t

$($c$).$ E$[\backslash$sigma $2|\backslash$Omega $]$ t$+3$ t

$($d$).$ Derive a general formula for E$[\backslash$sigma $2$ t$+$s

$|\backslash$Omega $]$ for any s $>=2.$ t