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CIRJE Discussion Papers can be downloaded without charge from:

http://www.e.u-tokyo.ac.jp/cirje/research/03research02dp.html

Discussion Papers are a series of manuscripts in their draft form.

They are not intended for

circulation or distribution except as indicated by the author.

For that reason Discussion Papers may

not be reproduced or distributed without the written consent of the author.

CIRJE-F-515

Estimating Stochastic Volatility Models Using

Daily Returns and Realized Volatility

Simultaneously

Makoto Takahashi

Graduate School of Economics, University of Tokyo

Yasuhiro Omori

University of Tokyo

Toshiaki Watanabe

Hitotsubashi University

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(27)

) , , , % , , %, ,% ) " ' & C +3 C CD * Æ 3 " 3 $% Æ $ Æ " ".>"+,,'# * +B% 3 3 &D%" 3  ½ 3 ¼ ' C%&D% ( C, ( C ( (   +> & ( ( *+B%  ½  ½ 3  ½ 3 " * (   C ( C D, ' " C   "H !   3 3 ! 33   ! $% D, !   3   !33 ! 3   ! $+ D,C ! 33 ! 3 $%$+ " C 3 + + 3 + C   3 33 + , C C % 3 3 C $%$+   +D C 3 3 + D, D,C /

(28)

C ¼ C 23 CD%D, 4C 5 2 2323 ¼ C 6 7 , , 7 6 7 , , 7 6 7 , , 7 6 6 C 5 2 23 CD%D, $B 7 C 5 2 2323   CD+D, 7 C, $1 , %$B +$13C3 3 ¼ 3 " C % + D + D 2 23 D       2   23 D 3 CD%D, % CD,C C % + D + D 2 23 D       2   23 D 3 3 D 3 CD, 2 23 C   D3 3 +3+ C% % , C 2   23 C , C%   3   + C+ " % 5 "6 575 H 6 C % + D   2 23 D     2   23 D   CD%D, % CD,C % + D   2 23 D     2   23 D   D   CD, 7 C     2   23   2   23 C+ # $% ! £   3 3 .>"+,,' H

(29)

  % +   D I D 2 2 ¼ ¼ I D % + I ¼ 5 2 22 ¼ I C % +   D I D I ¼ 3 3I % + 3 3 I ¼ I 43 3 I CD ! £   3 3 I I I 4 4 3C3I! CI 5 35* " $+ J 8 &9 . CD+D, 8 C I 6 8     I 7 8 C I 6 & C 8 9 C I 7&     9 C, 9 C, . C I 9&     .   . C I * " I CI: D8   . I : C3I D&   9 I 3 CD%D, J " I C; 3 D<= CD%D, 3 C3 D> = CD%D, = ) , ; C%D&   9 < C&   % 9 > C , #*+B%   "$%$+ " #83 * 5 ! 35 "" K " $+ ! 4* T"* C C TC "CT / " 1 "   > 3 " 8:8: 2 % * 3 "* J& ( ( ( C,( C " * 3  ½ 3 (   ?( "* @C%&D%

(30)

+ * 3 "   * B * T3 "   * 1 * 3 "   ¾ * 2 * 3 "   * - 3 " " " " - # "" " #D % + % 3 % +   3 3   3+ +% >83 " %% "A   C   33   3+ 3 D   3 C % 3 D   3 J A   "" # ! % # % % " # $ - " T   > " " T   B T T 3 % + DB + T % + T   T   + 3 C D % T   CT   D   1 1 ¼ 1 C 3+ 3 3 ¼ > T83 T   B T J T   T   B T ""     3 % + + 3     3 % + + 3 % " # $ - *+B+* 5 35 HI#J##9# ((>K777;#7*70 HI#J## # L#9# (('M 7;1 #51!!)&(#" 06 #%@# #%#!88& 1J#7 B01)8# &&-N8(-#%@# #%#:2#.O#02#P# ((!%27* 7.0'!#,)N>'

(31)

7 8'#, #.##*# ((-###5@7 #M76 G"#K0##"# ((!"@KM227F *7 ')#!'>N ,! # .# !8>' 7 B !8>' 7 # 0#!>>!&! 2##!88-7@22*7 8(#!)!)N!) ! 2# # ((! 7 B P# * *# 9# 0 506# P270?-"P# #)-'8N)',8 2##@2#0#!88-MP ,8#) >N))-2# #*# B# ((! 7 P *7 8'# >(N &! # #!8& %27#G*7 .0!(#,(>N,) *###"#!88-%7 & #))8N)-( :2#.O#!8&&070/:?#"1J :# * # (( K24K / MK/# ) %2 K/7# 9 :2#*#$#*# (( Æ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

(32)

L#9# (('0Æ7 2 ! # !(!8N!(,)

L#9## # HI#J# ((- 71:1

References

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