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Consider the simple linear regression model Yi = 1 + 2Xi + ui .
Consider the simple linear regression model Yi = β1 + β2Xi + ui . Show that the estimate of β2 can be written as βˆ 2 = σˆxy σˆ 2 x , where ˆσxy is the estimate of the covariance between Y and X and ˆσ 2 x is the estimate of the variance of X. (Hint: Write out σˆxy σˆ 2 x in its summation form to get an idea of what your end goal is. Start with the normal equations (the first order conditions for minimizing RSS) and manipulate until you obtain the desired result.)