TITLE: Finding conditional expectation from system of equations
QUESTION [2 upvotes]: I have three equations: 
$$X_m = \beta_0 + \beta_1 \cdot X_I + \varepsilon_{BL}$$
$$W_M = X_M + \varepsilon_{MBL}$$
$$W_I = \gamma_0 + \gamma_1 \cdot X_I + \varepsilon_{RDI}$$
The $\varepsilon$'s are normally distributed with mean 0, variance $\sigma^2$.
I need to find E[$X_I\mid W_I$] and E[$X_I\mid W_I,W_M$]. The first is straightforward because I have an equation that relates $X_I$ and $W_I$, but how do find the latter expectation?

REPLY [0 votes]: I think there's a serious error in your reasoning.  You say "The first is straightforward because I have an equation that relates $X_I$ and $W_I$".  The equation implies that $\operatorname{E}(W_I\mid X_I)=\gamma_0+\gamma_1 X_I$.
But you need $\operatorname{E}(X_I\mid W_I)$.  You haven't given enough information to find that.  It may be tempting to say that if $W_I=\gamma_0+\gamma_1 X_I$ then $X_I=(W_I-\gamma_0)/\gamma_1$ and then think that that gives you the expected value.  But that is wrong except when the correlation is $1$ or $-1$.  For example, suppose one has $\operatorname{E}(W\mid X)=X$ and the correlation between $W$ and $X$ is $0.01$.  Then $\operatorname{E}(X\mid W) = 0.0001 W$.  You have not told us the correlations.
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