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 $\displaystyle \epsilon = Y - E(Y|X)$ and $\displaystyle U = Y - L(Y|X)$, where $\displaystyle E(Y|X)$ is the CEF and $\displaystyle L(Y|X)$ is the BLP. Determine whether the following is true or false: $\displaystyle Cov(\epsilon, U) = Var(\epsilon)$.

My working out is as follows:

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

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

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

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

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

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

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

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

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

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!

Re: Covariance of BLP error and CEF error

Would it help if I explained the question a little more?

The CEF is the conditional expectation function, i.e., the expectation of the conditional distribution of Y given X, and the BLP is the best linear prediction of the CEF, i.e., the function that minimizes $\displaystyle E(W^2)$, where $\displaystyle W = E(Y|X) - (a + b X)$.

In my workings, I've taken advantage of the law of iterated expectations (i.e., the marginal expectation of Y is the expectation of its conditional expectation), and the iterated product law (i.e., the expected product of X and Y is equal to the expected product of and the conditional expectation of Y given X).

I suppose the thing that is confusing me, is that the parameters of the BLP (alpha and beta) are functions of random variables, so are random variables (right?), so should I really be pulling them outside the expectations operator and treating them as constants?