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Math Help - Regression models that ignore regressors

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    Regression models that ignore regressors

    Hi,

    Suppose X, Y are independent discrete random variables with jont distribution P(X, Y). My regression model is of the form

    Z = E[Z | X, Y] + \epsilon = f(X, Y) + \epsilon,

    where the random variable \epsilon is the noise-term. Next, assume that we ignore our knowledge about Y. The we have another regression model

    Z' = E[Z' | X] + \eta = g(X) + \eta,

    where \eta is the noise term. We can express g(X) as

    g(X) = \sum_y P(Y = y) f(X, y).

    My first question is: How can we interpret the right hand side of the last equation? Since X and Y are independent, we have for each x

    g(x) = \sum_y P(y) f(x, y) = \sum_y P_{Y | X}(y | x) f(x, y) = E[f(X, Y) | X = x].

    So the function g at x that has no knowledge about Y can be regarded as the conditional expectation of f given X = x. Is this argumentation correct? If independence does not hold, how do you call the expression

     \sum_y P(y) f(x, y),

    which looks a bit like an expectation?

    The last question is concerned with the error term \eta. From the above, we have

    Z' = \sum_y P(y) Z - \sum_y P(Y)\epsilon - \eta.

    If I assume that Z' = \sum_y P(y) Z, then I may conclude

    \eta = \sum_y P(Y)\epsilon.

    But under which conditions is my assumption about Z' valid?

    Thanks und best wishes,

    samosa
    Last edited by samosa; November 14th 2012 at 12:45 AM.
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