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hwk2 - Financial Data Analysis Professor S Kou Department...

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Financial Data Analysis Professor S. Kou, Department of IEOR, Columbia University HWK 2 Solution 1. (a) Since R i ° N ( ° i ; ± 2 ) ; X i j R i ° N ( R i ; 1) ; we have the posterior density f ( R i j X i ) / f ( X i j R i ) f ( R i ) = 1 p 2 ² exp ° ± 1 2 ( X i ± R i ) 2 ± ² 1 ± p 2 ² exp ( ± 1 2 ² R i ± ° i ± ³ 2 ) / exp ° ± 1 2 ² ± 2 R i X i + R 2 i + R 2 i ± 2 R i ° i ± 2 ³± = exp ( ± 1 2 (1 + ± 2 ) R 2 i ± 2 R i ´ ° i + ± 2 X i µ ± 2 !) = exp ( ± 1 2 1 ± 2 = (1 + ± 2 ) ( R 2 i ± 2 R i ´ ° i + ± 2 X i µ (1 + ± 2 ) )) / exp 8 < : ± 1 2 1 ± 2 = (1 + ± 2 ) ( R i ± ´ ° i + ± 2 X i µ (1 + ± 2 ) ) 2 9 = ; : Therefore, R i j X i ° N ² ° i + ± 2 X i (1 + ± 2 ) ; ± 2 = (1 + ± 2 ) ³ : In other words, the posterior distribution of R i j X i is normal with the mean given by E [ R i j X i ] = ° i + ± 2 X i (1 + ± 2 ) = ° i + ± 2 X i ± ± 2 ° i (1 + ± 2 ) = ° i + ² 1 ± 1 1 + ± 2 ³ ( X i ± ° i ) : (b) Since R i ° N ( ° i ; ± 2 ) ; X i j R i ° N ( R i ; 1) ; we have X i = R i + " i ; " i ° N (0 ; 1) = ° i + ³ i + " i ; ³ i ° N (0 ; ± 2 ) ; where ³ i and " i are independent. Thus, R i ° N ( ° i ; 1 + ± 2 ) : 1
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Next, S : = N X i =1 ( X i ± ° i ) 2 = N X i =1 ( ³ i + " i ) 2 ° (1 + ± 2 ) ´ 2 N ; because ³ i + " i ° N (0 ; 1 + ± 2 ) ; where ´ 2 k is the chi-square distribution with d.f. being N . 2. First of all, we have to download the adjusted closing prices on Jan 2, 2001, and Dec 31, 2001 for Dow 30 stocks.
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