Lecture+1+Review+SLR

Thesearetheordinaryleastsquaresolsestimatorsfortheinte

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Unformatted text preview: 2∑ yi − α − β xi = 0 ˆ ∂α i =1 ( ) 6 (1) −2 n ˆˆ ∂f (α , β ) ˆˆ = −2∑ xi yi − α − β xi = 0 ˆ ∂β i =1 ( ) (2) These are two linear equations in two unknowns; they are called the first order conditions (normal equations). • ˆ FOC for α : ∑( n i =1 • ˆ FOC for β : ) n ˆ ∑u ˆˆ yi − α − β xi = 0 , same as ( n ) ˆ ∑ xi yi − α − βˆ xi = 0 , same as i =1 i i =1 = 0 , i.e., residuals sum to zero n ˆ ∑xu i =1 ii = 0 , i.e., the sample variance or correlation between the residual and X is zero ˆ Solve the first equation for α ˆ α= 1n 1n ∑ yi − βˆ n ∑ xi = y − βˆ x n i =1 i =1 (3) y and x = sample means of Y and X. ˆ Plug in the second equation (3), and solve for β , we get n ˆ β= ∑ ( x − x )( y − y ) i i =1 i n ∑ (x − x ) i =1 2 i ˆˆ Therefore, the values of α , β that minimize the sum of squared residuals (SSR) are n ˆ β= ∑ ( x − x )( y − y ) i i =1 i n ∑ (x − x ) i =1 2 i ˆ ˆ α = y −βx n n n n i =1 i =1 i =1 i =1 Note that ∑ ( xi − x )( yi − y ) = ∑ ( xi − x ) yi = ∑ xi ( yi − y ) = ∑ xi yi −...
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This document was uploaded on 03/11/2014.

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