Q) Consider the one-variable regression model Yi = ß0 + ß1Xi + Ui and suppose that it satisfies the least squares assumptions. Suppose that Yi is measured with error, so that data are Ybari= Yi + Wi, where Wi is the measurement error, which is i.i.d. and independent of Yi and Xi. Consider the population regression Ybari = ß0 + ß1Xi + Vi, where Vi is the regression error, using the mis-measured dependent variable, Ybari.
*Note that 0 and 1 n the equations are subscripts

A) Show that Vi = Ui + Wi
My solution:
Yi = ß0 +ß1Xi +Ui
Ybari = Yi + Wi
Ybari = ß0 + ß1Xi + Vi

n -> ∞
Ybari = Yi + Wi
= Yi - Ybari + Wi
= [ß0 + ß1Xi + Ui] - {ß0 + ß1Xbari + Vi} +Wi
= ß0 + ß1Xi +Ui - ß0 - ß1Xbari - Vi + Wi
= ß0 - ß0 + ß1(Xi - Xbari) + Ui - Vi + Wi
= ß0 - ß0 + ß1(Xi-Xi) + Ui - Vi + Wi
= Ui - Vi + Wi
Vi = Ui + Wi

Since n -> ∞ , Xbari can be rewritten as Xi
My Reasoning: Since it's asking me to prove that Vi = Ui + Wi
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I'm not sure if did this question right, is it right?
If this is right, could someone to explain to me why it is right?
if it is wrong, could someone who me where i went wrong with out giving the answer away ?

This post was edited by zyQuzA0e5esy2y on Jan 21 2016 04:58pm