426 1 1 n n x e x y xb part 3 least squares algebra

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™    4/26 = = 1 1   n n X e 0 X (y - Xb)
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Part 3: Least Squares Algebra Population and Sample Moments    We showed that E[e i| x i] = 0 and Cov[ x i , e i] =  0 .  If it is,  and if E[ y | X ] =  Xb , 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
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Part 3: Least Squares Algebra U.S. Gasoline Market, 1960-1995 ™    6/26
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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
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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 =
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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
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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) = 0 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
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Part 3: Least Squares Algebra Least Squares Solution ™    11/26 ( 29 ( 29 -1 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
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Part 3: Least Squares Algebra Second Order Conditions ™    12/26 2 Necessary Condition: First derivatives = 0 2 Sufficient Condition: Second derivatives ...
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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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