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Math Help - Covariance of BLP error and CEF error

  1. #1
    tlm
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    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!
    Last edited by tlm; November 28th 2012 at 06:58 AM.
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  2. #2
    tlm
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    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 E(W^2), where 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?
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