Mean Zero Measurement Error and Two Variable Regression

Question: What do you think mean zero measurement error in the measure of class attendance will do to your estimate of the returns to class attendance and why (please derive this)?


Gist 

Mean zero measurement error in  leads to attenuation bias in the regression coefficient estimate on the x1 parameter - As the as the variance of the error term increases (as people are more incorrect as to their estimate of x1) the coefficient estimate gets biased down.  



"Proof"

Where ,  where  is the observed value (observed with error,  is the true value, and  is the mean zero error term, with 



Now,  appears in four places. We now need to consider x_1 given its error. Let's take these four occurrences of x_1 one by one. 


 

1) First occurrence of x_1:

   as usual.


2)  Second:



Going term by term of the five terms above, in the probability limit













Third Location:




Through a similar process to 







Fourth Occurrence of x_1:






Also unchanged:







So after all that, let's plug in what we found for these probability limits







Okay! So the bias in our estimate for the  depends on  


Note that if  the term on the right goes to one, and we have no bias.


If   (and since it's a squared term, it'll be positive if it  ain't  zero), then will decrease our estimate of Just like in the  inivariate case.




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