Covariance of BLP error and CEF error

Hi all,

I've been struggling a bit with the following problem:

The random variables X and Y are jointly distributed. Let $\epsilon = Y - E(Y|X)$ and $U = Y - L(Y|X)$ , where $E(Y|X)$ is the CEF and $L(Y|X)$ is the BLP. Determine whether the following is true or false: $Cov(\epsilon, U) = Var(\epsilon)$ .

My working out is as follows:

We are given that $Var(\epsilon) = E[Var(\epsilon|X)] = E(\sigma^2_{Y|X})$

$Cov(\epsilon, U) = E(\epsilon U) - E(\epsilon)E(U) = E(\epsilon U) = E[[Y - E(Y|X)][Y - L(Y|X)]]$

$= E[[Y - E(Y|X)][Y - (\alpha + \beta X)]]$

$= E[Y^2 - YE(Y|X) - YL(Y|X) + E(Y|X)L(Y|X)]$

$= E(Y^2) - E[E(Y|X)] - E[Y(\alpha + \beta X)] + E[E(Y|X)(\alpha + \beta X)]$

$= E(Y^2) - E(Y^2) - [\alpha E(Y) + \beta E(YX)] + E[\alpha E(Y|X) + \beta XE(Y|X)]$

$= - [\alpha E(Y) + \beta E(YX)] + [\alpha E[E(Y|X)] + \beta E[XE(Y|X)]]$

$= - [\alpha E(Y) + \beta E(YX)] + [\alpha E[Y] + \beta E[XY]] = 0$

Hence, $Cov(\epsilon, U) \not= Var(\epsilon)$ , assuming $E(\sigma^2_{Y|X}) \not= 0$ .

Is that okay? The zero covariance between espilon & U seems like a strange result to me, but there are no clues in my textbook.

Thanks!