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
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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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Econometric take home APPS_Part_46 - Appendix D Large...

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