I have to solve a conditional mean where the information set is no singleton, specifically it is $I_1=\{S_1, S_2\}$ and I need to calculate $E[\bar{V}|I_1]$. My hint is that I should use the projection theorem to solve this, but my objection is that the theorem says that $$E[\bar{x}|\bar{y}]=E[\bar{x}]+\dfrac{\mathrm{cov}(\bar{x},\bar{y})}{\mathrm{Var}(\bar{y})}\cdot (\bar{y}-E[\bar{y}]),$$ but in my case instead of a singleton $\bar{y}$, the information set $I_1$ contains two elements. Is there some book in statistics which contains the projection theorem more precisely, because I've been making a search in internet for many days and I found nothing close to what I was searching for.
Thank you in advance!
The "projection theorem" you give doesn't necessarily hold for arbitrary random variables $\ \bar{x}, \bar{y}\ $. It does, however, hold if $\ \bar{x}, \bar{y}\ $ are joint-normally distributed, so presumably you can assume that the random variables $\ \bar{V}, S_1, S_2\ $ are joint-normally distributed. If that's the case, you can use multivariate analogue of your projection theorem, which, in your case, becomes $$ E\left[\bar{V}\left\vert I_1\right.\right]= E\left[\bar{V}\right] + \pmatrix{\sigma_{\bar{V}S_1},\sigma_{\bar{V}S_2}}{\large\Sigma}_{I_1}^{-1}\pmatrix{S_1 - E\left(S_1\right)\\S_2 - E\left(S_2\right)}\ ,$$ where $\ \sigma_{\bar{V}S_i}=\mathrm{cov}\left(\bar{V}, S_i\right)\ $ for $\ i=1,2\ $, and $$ {\large\Sigma}_{I_1} = \pmatrix{\mathrm{Var}\left(S_1\right)&\mathrm{cov}\left(S_1,S_2\right)\\ \mathrm{cov}\left(S_1,S_2\right)&\mathrm{Var}\left(S_1\right)} $$ is the covariance matrix of $\ I_1\ $.