Deriving the variance of the OLS estimator

Problem

This problem is at a higher level. You may want to skip it for now (or try to understand as much as you can) and return to it later in the course.

In this problem we will prove that

\[Var\left( b_2 \mid x \right)= \frac{σ^2}{\sum_{i=1}^{n}{ {\left( x_i-\bar{x} \right)}^2 }}\]

We start from the statistical formula for \(b_2\) :

\[b_2=β_2+ \frac{\sum_{i=1}^{n}{ \left( x_i-\bar{x} \right)ε_i }}{\sum_{i=1}^{n}{ {\left( x_i-\bar{x} \right)}^2 }}\]

Which we write as (see Proving that the slope OLS estimator is unbiased)

\[b_2=β_2+\sum_{i=1}^{n}{ a_iε_i }\]

where

\[a_i= \frac{\left( x_i-\bar{x} \right)}{\sum_{i=1}^{n}{ {\left( x_i-\bar{x} \right)}^2 }}\]

a) Show that

\[Var\left( b_2|x \right)=Var\left( \sum_{i=1}^{n}{ a_iε_i } \mid x \right)\]

b) Remember that for two independent random variables \(X\) and \(Y\) we have \(Var\left( X+Y \right)=Var\left( X \right)+Var\left( Y \right)\) . Show that

\[Var\left( \sum_{i=1}^{n}{ a_iε_i } \mid x \right)=\sum_{i=1}^{n}{ Var\left( a_iε_i \mid x \right) }\]

c) Show that

\[\sum_{i=1}^{n}{ Var\left( a_iε_i \mid x \right) }=\sum_{i=1}^{n}{ a_i^2Var\left( ε_i \mid x \right) }\]

d) Show that

\[\sum_{i=1}^{n}{ a_i^2Var\left( ε_i|x \right) }=σ^2\sum_{i=1}^{n}{ a_i^2 }\]

e) Show that

\[\sum_{i=1}^{n}{ a_i^2 }= \frac{1}{\sum_{i=1}^{n}{ {\left( x_i-\bar{x} \right)}^2 }}\]

We are finished and we have

\[Var\left( b_2|x \right)= \frac{σ^2}{\sum_{i=1}^{n}{ {\left( x_i-\bar{x} \right)}^2 }}\]