Question

How is this formula for βˆ 1 related to the formula for cov(X, Y )  Express the formula for βˆ 1 as a function of the cov(X, Y )

Answer 1

To show how is this formula for related to the formula for cov(X, Y ) and to express the formula for as a function of the cov(X, Y )

Step 1:

Simple Linear Regression model is given by:

,                           (1)

where

Y is the dependent variable

X is the independent variable

is the y - intercept parameter

is the slope parameter

is the unobservable error component and is a N(0,) random variable,independent of X.

Step 2:

Taking Expectation of both sides of (1), we get:

By property of , we have E() = 0

Thus, we get:

So,

The Intercept parameter is given by:

Step 3:

By property of , we have Cov (X, ) =0

Also, we have Cov(X,1) = 0.

and

Cov(X, X) = Var(X)

Thus, we get:

Thus, the slope parameter is given by:

This is the formula for as a function of the cov(X, Y )