Question

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.

Answer 1

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$\beta _{0}$ to normalize E(u) to 0

e.g. use $\beta _{0}$  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

$E(y|x) = \beta _{0}+\beta _{1}x$

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