Hi all,
I've been struggling a bit with the following problem:
The random variables X and Y are jointly distributed. Letand
, 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
![Cov(\epsilon, U) = E(\epsilon U) - E(\epsilon)E(U) = E(\epsilon U) = E[[Y - E(Y|X)][Y - L(Y|X)]]](http://latex.codecogs.com/png.latex?Cov(\epsilon, U) = E(\epsilon U) - E(\epsilon)E(U) = E(\epsilon U) = E[[Y - E(Y|X)][Y - L(Y|X)]])
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.
Thanks!
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?
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