In: Statistics and Probability
Please explain what the assumption of Cov(x,u)=0 means and why it is so important for regression to return unbiased estimates of the relationship of interest.
Let us first understand the assumption of Cov(x,u)=0
Assumptions :
Let us do by making a simple assumption about the error
The average value of u, the error term, in the population is 0 that is E(u)=0
This is not a restrictive assumption, since we can always use to normalize E(u) to 0
e.g. use to normalize average ability to zero
We have to assume that the average value of u does not depend on the value of x
x and u are independent, that is
We need some way of estimating the true relationship using the sample data
We Can do this using some of our assumptions First we need to realize that our main assumption of E(u|x) = E(u) = 0
also implies Cov(x,u) = E(xu) = 0
because From basic probability we know that: Cov(x,u) = E(xu) – E(x)E(u)
given E(u)=0 (by assumption) and Cov(x,u)=0 (by same assumption plus independence) then E(xu)=0