I'm trying to work out the following conditional expectation:
$E[\epsilon_t z_t|\epsilon_t + z_t = k, \epsilon_{t-1}, z_{t-1}, \epsilon_{t-2}, z_{t-2},...]$
where $k$ is known and $\epsilon_{t}$ and $z_t$ are independent and normal distributed random variables with zero means and variances equal to $\sigma^2$ and $\tau^2$ respectively.
Any hints?
Regards, Paul
Assume without loss of generality that the random variables $x$ and $y$ are independent and centered normal with variances $\sigma^2$ and $\tau^2$ respectively, and that one looks for $$ E[xy\mid x+y]. $$ Let $z=x+y$, then $t=\tau^2x-\sigma^2y$ is independent of $z$ since $(z,t)$ is centered normal and $E[zt]=0$, and $$ (\sigma^2+\tau^2)\cdot(x,y)=(\sigma^2z+t,\tau^2z-t), $$ hence $$ (\sigma^2+\tau^2)^2xy=\sigma^2\tau^2z^2+(\tau^2-\sigma^2)zt-t^2. $$ Note that $E[z^2\mid z]=z^2$, $E[t^2\mid z]=E[t^2]=\tau^2\sigma^2(\sigma^2+\tau^2)$, $E[zt\mid z]=zE[t\mid z]$ and $E[t\mid z]=E[t]=0$. Hence, $$ (\sigma^2+\tau^2)^2E[xy\mid z]=\sigma^2\tau^2z^2-\tau^2\sigma^2(\sigma^2+\tau^2), $$ that is, $$ E[xy\mid x+y]=\frac{\sigma^2\tau^2}{(\sigma^2+\tau^2)^2}\left((x+y)^2-(\sigma^2+\tau^2)\right). $$