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Consider the simple linear regression model Yi = ÃŽÂ²1 + ÃŽÂ²2Xi + ui . Show that the estimate of ÃŽÂ²2 can be written as ÃŽÂ²Ã‹â€ 2 = ÃÆ’Ã‹â€ xy ÃÆ’Ã‹â€ 2 x , where Ã‹â€ ÃÆ’xy is the estimate of the covariance between Y and X and Ã‹â€ ÃÆ’ 2 x is the estimate of the variance of X. (Hint: Write out ÃÆ’Ã‹â€ xy ÃÆ’Ã‹â€ 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.)
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