Normal Equations Proof

# Normal Equations Proof - AGEC 317 Dr Capps Fall 2008...

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Unformatted text preview: AGEC 317 Dr. Capps Fall 2008 Derivation of Ordinary Least Squares Estimators u x y i i i + + = β β 1 where y i is a dependent variable, x i is an independent right-hand side (RHS) variable, i u is the error term (unobservable), β β 1 and are coefficients. The ordinary least squares procedure minimizes the error sum of squares (SSE). The minimization problem is given as follows: b)- (1 ) ( 2 a)- (1 ) 1 ( 2 . . . ˆ ) , ( 1 ^ 1 ^ ^ 1 1 ^ 1 ^ ^ 2 1 ^ 1 ^ 1 2 ^ 1 ^ =- -- = ∂ ∂ =- -- = ∂ ∂ -- = = ∑ ∑ ∑ ∑ = = = = i n i i i n i i i n i i i n i i x x y SSE x y SSE C O F x y u SSE Minimize β β β β β β β β β β Step 1: Derive the OLS estimate of ) ˆ ( β β 1 ^ 1 ^ = -- ∑ = n i i i x y β β divide equation (1-a) by -2 1 1 ^ 1 1 ^ =-- ∑ ∑ ∑ = = = n i i n i n i i x y β β re-arrange ∑ ∑ = = =-- n i i n i i x n y 1 1 ^ 1 ^ β β divide both sides by n ....
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Normal Equations Proof - AGEC 317 Dr Capps Fall 2008...

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