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In: Statistics and Probability

How is this formula for βˆ 1 related to the formula for cov(X, Y ) Express the...

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 )

Solutions

Expert Solution

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 )

orchestra answered 1 year ago

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