W t 1 5 l u t 2 1 5 f ² f 1 1 q ² t ² 1 5 l w t u

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w t 1 " 5 L   u t 2 ± 1 " 5   F 0 ² F 1 ¡ 1 " q   ² / t ¢ ² 1 " 5 L   w t u t 2 L ARMA 1,1   AR coefficient: 5 ± " 1 ² p ² q X 1 Estimated GARCH(1,1) parameters: b 1 ± 0.619 â 1 ± 0.367 â 1 ² b 1 ± 0.986 RETURN 1987 -40 -30 -20 -10 0 10 20 30 40 GARCH-L(1,1) T 1987 -40 -30 -20 -10 0 10 20 30 40
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20 Goal: parsimonious model of regime changes y t ± ) ² C y t " 1 ² g s t u t u t ± h t v t v t L Student t with 7 degrees of freedom and unit variance h t ± a 0 ² ! j ± 1 q a j u t " j 2 ² 8 u t " 1 2 - u t " 1 t 0   P s t ± j | s t " 1 ± i   ± p ij for each j ± £ 1,2,..., N ¤ , u t j   ± y t " ) " C y t " 1   / g j g 1 q 1 for each j 1 , j 2 ,.., j q ± £ 1,2,..., N ¤ , h t j 1 , j 2 ,..., j q   ± a 0 ² a 1 ¡ u t " 1 j 1  ¢ 2 ² a 2 ¡ u t " 2 j 2  ¢ 2 ² C ² a q ¡ u t " q j q  ¢ 2 ² 8 ¡ u t " 1 j 1  ¢ 2 - u t " 1 j 1   t 0   f y t | y t " 1 , y t " 2 ,..., s t ± j 0 , s t " 1 ± j 1 ,..., s t " q ± j q   ± ´ ¡ 7 ² 1   /2 ¢ ´ 7 /2   = 7 " 2   g j 0 h t j 1 , j 2 ,..., j q   1 ² y t " ) " C y t " 1   2 7 " 2   g j 0 h t j 1 , j 2 ,..., j q  
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21 Define aggregate state s t ' as s t ' ± 1 if s t ± 1, s t " 1 ± 1,..., s t " q ± 1 s t ' ± 2 if s t ± 2, s t " 1 ± 1,..., s t " q ± 1 B s t ' ± N ' if s t ± N , s t " 1 ± N ,..., s t " q ± N Collect the densities that might be associated with each of the N ' states in an N ' 1   vector 1 t ± p y t | s t ' ± 1, ( t " 1   p y t | s t ' ± 2, ( t " 1   B p y t | s t ' ± N ' , ( t " 1   1 t © P § 8 t " 1| t " 1   1 U 1 t © P § 8 t " 1| t " 1   ± § 8 t | t
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22 RETURN 1962 1964 1966 1968 1970 1972 1974 1976 1978 1980 1982 1984 1986 -30 -20 -10 0 10 20 PROB(S=1) 1962 1964 1966 1968 1970 1972 1974 1976 1978 1980 1982 1984 1986 -0.25 0.00 0.25 0.50 0.75 1.00 1.25 PROB(S=2) 1962 1964 1966 1968 1970 1972 1974 1976 1978 1980 1982 1984 1986 -0.25 0.00 0.25 0.50 0.75 1.00 1.25 PROB(S=3) 1962 1964 1966 1968 1970 1972 1974 1976 1978 1980 1982 1984 1986 -0.25 0.00 0.25 0.50 0.75 1.00 1.25 Percent improvement of Student t SWARCH-L(4,4) in forecasting over constant variance specification: MSE ± 0.06 MAE ± 0.13 RETURN 1987 -40 -30 -20 -10 0 10 20 30 40 GARCH-L(1,1) T 1987 -40 -30 -20 -10 0 10 20 30 40 SWARCH-L(3,2) 1987 -40 -30 -20 -10 0 10 20 30 40
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23 RETURN 1962 1964 1966 1968 1970 1972 1974 1976 1978 1980 1982 1984 1986 -40 -30 -20 -10 0 10 20 30 40 GARCH-L(1,1) T 1962 1964 1966 1968 1970 1972 1974 1976 1978 1980 1982 1984 1986 -40 -30 -20 -10 0 10 20 30 40 SWARCH-L(4,2) 1962 1964 1966 1968 1970 1972 1974 1976 1978 1980 1982 1984 1986 -40 -30 -20 -10 0 10 20 30 40 Why ARCH and not GARCH? Want h t j 1 , j 2 ,..., j q   rather than h t j 1 , j 2 ,.., j t   Options for Markov-switching GARCH: (1) Gray (1996) Replace h t ± + s t ² ) s t u t " 1 2 ² * s t h t " 1 with h t ± + s t ² ) s t u t " 1 2 ² * s t h · t " 1 h · t " 1 ± ! i ± 1 N 8 i , t " 1| t " 2 + i ² ) i y t " 2 2 ² * i h · t " 2  
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24 Options for Markov-switching GARCH: (2) Haas, Mittnik, and Paolella (2004) h jt ± + j ² ) j u t " 1 2 ² * j h j , t " 1 y t ± h s t , t u t (3) your name here (2009) h t s t , s t " 1 ,..., s 1   s T , s T " 1 ,..., s 1   generated as block of Gibbs sampler problem: Even with Gaussian v t , density of s | y , 2 not known analytically solution: (?) Kim, Shephard, Chib methods might work for s | y , 2
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