Econometric take home APPS_Part_42

Econometric take home APPS_Part_42 - characteristic roots...

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characteristic roots and vectors of AA . The inverse square root defined in Section B.7.12 would also provide a method of transforming x to obtain the desired covariance matrix. 18. The density of the standard normal distribution, denoted φ ( x ), is given in (C-28). The function based on the i th derivative of the density given by H i = [(-1) i d i φ ( x )/ dx i ]/ φ ( x ), i = 0,1,2,. .. is called a Hermite polynomial . By definition, H 0 = 1. (a) Find the next three Hermite polynomials. (b) A useful device in this context is the differential equation d r φ ( x )/ dx r + xd r -1 φ ( x )/ dx r -1 + ( r -1) d r -2 φ ( x )/ dx r -2 = 0. Use this result and the results of part a. to find H 4 and H 5 . The crucial result to be used in the derivations is d φ ( x )/d x = - x φ ( x ). Therefore, d 2 φ ( x )/ dx 2 = ( x 2 - 1) φ ( x ) and d 3 φ ( x )/ dx 3 = (3 x - x 3 ) φ ( x ). The polynomials are H 1 = x , H 2 = x 2 - 1, and H 3 = x 3 - 3 x . For part (b), we solve for d r φ ( x )/ dx r = - xd r -1 φ ( x )/ dx r -1 - ( r -1) d r -2 φ ( x )/ dx r -2 Therefore, d 4 φ ( x )/ dx 4 = - x (3 x - x 3 ) φ ( x ) - 3( x 2 - 1) φ ( x ) = ( x 4 - 6 x 2 + 3) φ ( x ) and d 5 φ ( x )/ dx 5 = (- x 5 + 10 x 3 - 15 x ) φ ( x ). Thus, H 4 = x 4 - 6 x 2 + 3 and H 5 = x 5 - 10 x 3 + 15 x . ± 19. Continuation: orthogonal polynomials : The Hermite polynomials are orthogonal if x has a standard normal distribution. That is, E [ H i H j ] = 0 if i j. Prove this for the H 1 , H 2 , and H 3 which you obtained above. E [ H 1 ( x ) H 2 ( x )] = E [ x ( x 2 - 1)] = E [ x 3 - x ] = 0 since the normal distribution is symmetric. Then, E [ H 1 ( x ) H 3 ( x )] = E [ x ( x 3 - 3 x )] = E [ x 4 - 3 x 2 ] = 0. The fourth moment of the standard normal distribution is 3 times the variance. Finally, E [ H 2 ( x ) H 3 ( x )] = E [( x 2 - 1)( x 3 - 3 x )] = E [ x 5 - 4 x 3 + 3 x ] = 0 because all odd order moments of the normal distribution are zero. (The general result for extending the preceding is that in a product of Hermite polynomials, if the sum of the subscripts is odd, the product will be a sum of odd powers of x , and if even, a sum of even powers. This provides a method of determining the higher moments of the normal distribution if they are needed. (For example, E [ H 1 H 3 ] = 0 implies that E [ x 4 ] = 3 E [ x 2 ].) 20. If x and y have means μ x and μ y and variances and and covariance σ xy , what is the approximation of the covariance matrix of the two random variables f 1 = x / y and f 2 = xy ? σ x 2 y 2 The elements of J Σ J N are (1,1) = σ μ σμ μ μ x y y x y xy x y 2 2 2 2 43 2 +− ( 1 , 2 ) = σ - σ / x 2 y 2 μ x 2 μ y 4 ( 2 , 2 ) = σ μ + + 2 σ xy μ x μ y . x 2 y 4 σ y 2 μ x 2 21. Factorial Moments . For finding the moments of a distribution such as the Poisson, a useful device is the factorial moment. (The Poisson distribution is given in Example 3.1.) The density is f ( x ) = e - λ λ x / x !, x = 0,1,2,. ..
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Econometric take home APPS_Part_42 - characteristic roots...

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