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Unformatted text preview: , then b = (Var[ x i])1 Cov[ x i,yi]. This will provide a population analog to the statistics we compute with the data. ˜™™ ™ 5/26 Part 3: Least Squares Algebra U.S. Gasoline Market, 19601995 ˜™™ ™ 6/26 Part 3: Least Squares Algebra Least Squares p Example will be, Gi on x i = [1, PGi , Yi] = [1,Pgi,Yi] p Fitting criterion: Fitted equation will be yi = b1xi1 + b2xi2 + ... + bKxiK. p Criterion is based on residuals: ei = yi  b1xi1 + b2xi2 + ... + bKxiK Make ei as small as possible. Form a criterion and minimize it. ˜™™ ™ 7/26 Part 3: Least Squares Algebra Fitting Criteria p Sum of residuals: p Sum of squares: p Sum of absolute values of residuals: p Absolute value of sum of residuals p We focus on now and later ˜˜™™ ™ 8/26 1 e n i i = ∑ 2 1 e n i i = ∑ 1 e n i i = ∑ 1 e n i i = ∑ 2 1 e n i i = ∑ 1 e n i i = ∑ Part 3: Least Squares Algebra Least Squares Algebra ˜˜™™ ™ 9/26 2 1 e n i i = ′ = ∑ e e = (y  Xb)'(y  Xb) A digression on multivariate calculus. Matrix and vector derivatives. Derivative of a scalar with respect to a vector Derivative of a column vector wrt a row vector Other derivatives Part 3: Least Squares Algebra Least Squares Normal Equations ˜˜™™ ™ 10/26 2 2 1 1 e y 2 Note: Derivative of 1x1 wr n n i i i i i = = ′ ∂ ∂ = ∂ ∂ ∂ =  ∂ ∂ ∂ ∑ ∑ ( x b) b b (y  Xb)'(y  Xb) X'(y  Xb) = b (1x1) / (kx1) (2)(nxK)'(nx1) = (2)(Kxn)(nx1) = Kx1 t Kx1 is a Kx1 vector. Solution: 2 ⇒ X'(y  Xb) = 0 X'y = X'Xb Part 3: Least Squares Algebra Least Squares Solution ˜˜™™ ™ 11/26 ( 29 ( 291 1 1 Assuming it exists: = ( ) Note the analogy: = Var( ) Cov( ,y) 1 1 = Suggests something desirable about least squares n n ÷ ÷ b X'X X'y x x b X'X X'y β Part 3: Least Squares Algebra Second Order Conditions ˜˜™™ ™ 12/26 2 Necessary Condition: First derivatives = 0 2 Sufficient Condition:...
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 Fall '10
 H.Bierens
 Econometrics, Yi, Linear least squares, Σi, Stern School of Business, Squares Algebra, Professor William Greene

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