Econometrics-I-3

Y xby xb xy xb b y xby xb y xby xb b b b b k 1 column

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= = - ÷ ∂ ∂ (y - Xb)'(y - Xb) X'(y - Xb) b (y - Xb)'(y - Xb) (y - Xb)'(y - Xb) b b b b K 1 column vector = 1 K row vector = 2 × × X'X
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Part 3: Least Squares Algebra Does b Minimize e’e ? ™    13/26 2 1 1 1 1 2 1 1 2 2 1 2 1 1 2 1 2 2 1 1 1 2 1 ... ... 2 ... ... ... ... ... If there were a single b, we would require this to be po n n n i i i i i i i iK n n n i i i i i i i iK n n n i iK i i iK i i iK x x x x x x x x x x x x x x x = = = = = = = = = Σ Σ Σ Σ Σ Σ = ∂ ∂ Σ Σ Σ e'e X'X = 2 b b' 2 1 sitive, which it would be; 2 = 2 0. The matrix counterpart of a positive number is a . n i i x = positive definite m x x atrix '
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Part 3: Least Squares Algebra Sample Moments - Algebra ™    14/26 2 2 1 1 1 1 2 1 1 1 1 2 1 2 2 1 2 1 1 2 1 2 2 1 2 2 1 2 1 1 1 2 1 1 2 ... ... ... ... = ... ... ... ... ... ... ... ... ... = = = = = = = = = = Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ n n n i i i i i i i iK i i i i iK n n n n i i i i i i i iK i i i i iK i n n n i iK i i iK i i iK iK i iK i x x x x x x x x x x x x x x x x x x x x x x x x x x x x x X'X = [ ] 2 1 2 1 1 2 1 ... = ... ... = = = Σ Σ iK i i n i i i iK ik n i i i x x x x x x x x x
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Part 3: Least Squares Algebra Positive Definite Matrix ™    15/26 Matrix   is positive definite if   is > 0 for any  .     Generally hard to check.  Requires a look at     characteristic roots (later in the course).     For some matrices, it is easy to verify.    i C a'Ca a X'X K 2 k k=1 = v 0 -1 s     one of these.   =  ( )( ) = ( ) ( ) =  Could   =  ?    means  .  Is this possible? Conclusion:   = ( )  does indeed minimize  . a'X'Xa a'X' Xa Xa ' Xa v'v v 0 v = 0 Xa = 0 b X'X X'y e'e
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Part 3: Least Squares Algebra Algebraic Results - 1 ™    16/26 1 n i i = = In the population: E[ ' ] =  1 In the sample:       e n i X 0 x 0 ε
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Part 3: Least Squares Algebra Residuals vs. Disturbances ™    17/26 - = ε - = i i i i i i Disturbances (population)  y Partitioning  :                      =   E[ | ]  +   Residuals (sample)            y e Partitio x y y y X x b β ε = conditional mean + disturbance ning  :                      =     +   y y Xb X' e = projection + residual ( Note : Projection 'into the column space of )
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Part 3: Least Squares Algebra Algebraic Results - 2 p A “residual maker”   M   =  ( I  -  X ( X’X )-1 X’ ) p e  =  y  -  Xb y  -  X ( X’X )-1 X’y  =  My   p My  =   The residuals that result when  y
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