Covariance of BLP error and CEF error
I've been struggling a bit with the following problem:
The random variables X and Y are jointly distributed. Let and , where is the CEF and is the BLP. Determine whether the following is true or false: .
My working out is as follows:
We are given that
Hence, , assuming .
Is that okay? The zero covariance between espilon & U seems like a strange result to me, but there are no clues in my textbook.
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 , where .
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?