Econometric take home APPS_Part_46

# Econometric take home APPS_Part_46 - Appendix D Large...

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Appendix D Large Sample Distribution Theory There are no exercises for Appendix D. 183

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Appendix E Computation and Optimization 1. Show how to maximize the function f ( β ) = () 1 2 2 2 π e c −− β / with respect to β for a constant, c , using Newton's method. Show that maximizing log f ( β ) leads to the same solution. Plot f ( β ) and log f ( β ). The necessary condition for maximizing f ( β ) is d f ( β )/ d β = 1 2 2 2 e c β / [-( β - c )] = 0 = -( β - c )f( β ). The exponential function can never be zero, so the only solution to the necessary condition is β = c . The second derivative is d 2 f ( β )/d β 2 = -( β - c )d f ( β )/d β - f( β ) = [( β - c ) 2 - 1] f ( β ). At the stationary value b = c , the second derivative is negative, so this is a maximum. Consider instead the function g ( β ) = log f ( β ) = -(1/2)ln(2 π ) - (1/2)( β - c ) 2 . The leading constant is obviously irrelevant to the solution, and the quadratic is a negative number everywhere except the point β = c . Therefore, it is obvious that this function has the same maximizing value as f ( β ). Formally, dg ( β )/ d β = -( β - c ) = 0 at β = c , and d 2 g ( β )/d β 2 = -1, so this is indeed the maximum. A sketch of the two functions appears below. Note that the transformed function is concave everywhere while the original function has inflection points. 184
2. Prove that Newton’s method for minimizing the sum of squared residuals in the linear regression model will converge to the minimum in one iteration. The function to be maximized is f ( β ) = ( y - X β ) ( y - X β ). The required derivatives are f ( β )/ ∂β = - X ( y - X β ) and 2 f ( β )/ ∂β∂β∂ X X . Now, consider beginning a Newton iteration at an arbitrary point, β 0 . The iteration is defined in (12-17), β 1 = β 0 - ( X X ) -1 {- X ( y - X β 0 )} = β 0 + ( X X ) -1 X y - ( X X ) -1 X X β 0 = ( X X ) -1 X y = b .

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## This note was uploaded on 11/13/2011 for the course ECE 4105 taught by Professor Dr.fang during the Spring '10 term at University of Florida.

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Econometric take home APPS_Part_46 - Appendix D Large...

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