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Econometrics-I-3

Sufficient condition second derivatives = āˆ‚ = āˆ‚

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Unformatted text preview: Sufficient Condition: Second derivatives ... = āˆ‚ = - āˆ‚ āˆ‚ āˆ‚ Ć· āˆ‚ āˆ‚ ā€² ā€² āˆ‚ āˆ‚ āˆ‚ (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 āˆ‚ Ɨ āˆ‚ Ɨ X'X Part 3: Least Squares Algebra Does b Minimize eā€™e ? ˜˜ā„¢ā„¢ ā„¢ 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 = = = = = = = = = Ī£ Ī£ Ī£ Ī£ Ī£ Ī£ āˆ‚ = āˆ‚ āˆ‚ Ī£ Ī£ Ī£ 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 = āˆ‘ positive definite m x x atrix ' Part 3: Least Squares Algebra Sample Moments - Algebra ˜˜ā„¢ ā„¢ 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 ... ... ... ... = ... ... ... ... ... ... ... ... ... = = = = = = = = = = Ī£ Ī£ Ī£ Ī£ Ī£ Ī£ Ī£ Ī£ Ī£ Ī£ 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 ... = ... ... = = = Ī£ ā€² Ī£ iK i i n i i i iK ik n i i i x x x x x x x x x Part 3: Least Squares Algebra Positive Definite Matrix ˜˜ā„¢ ā„¢ 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 āˆ‘-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 v = 0 Xa = 0 b X'X X'y e'e Part 3: Least Squares Algebra Algebraic Results - 1 ˜˜ā„¢ ā„¢ 16/26 1 n i i = = āˆ‘ In the population: E[ ' ] = 1 In the sample: e n i X x Īµ Part 3: Least Squares Algebra Residuals vs. Disturbances ˜˜ā„¢ ā„¢ 17/26 ā€²- = Īµ ā€²- = i i i i i i Disturbances (population) y Partitioning : = E[ | ] +...
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Sufficient Condition Second derivatives = āˆ‚ = āˆ‚ āˆ‚ āˆ‚...

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