Joanna Fijałkowska – Przewidywalność stóp zwrotu z akcji. Wyniki dla USA i Polski

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(

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β

β

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μ

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)

(

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(

2

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k i t k r =1 +

∑

– α α β β β β β β β β ε ε ε ε ε ŘFHQDZXMÚFLXORJDU\WPLF]Q\PZRNUHVLH! 1 1 + + = + t + t t p p μ ε ε ε ε ε σ σ ε ) ln( 1 1 + + = t t P p 1 1 + + = + t t z μ 1 1 + + = t t p z 0 ) , cov( ₁ = ≠ + t t 0 ) , cov( 2 1 2 + t t (r ,t 1,... + –1 – + t n t r r ) 2 1 2 1 var var var varrt + rt+ + rt+ +…+ rt+n =n / n . ) ( ) ( ) ( t n t r nVar r Var n VR = ) ( n t r Var ) (rt Var ) ( ^ _n VR )) 1 ( 2 , 0 ( ~ ) 1 ) ( (VR^ n N n nT (n = 10, T = 77) (n = 10, T= 109) t t t t t t _P D P R R R 1 1 1 1 1 1 1 + 1 + + + + + + = = 1 + t R t + 1, 1 + t P t+ 1, 1 + t D t + 1 P/D = exp(p d) ) ( 1 1 j t j t j j t t t d const E d r p + + = + 0 ) ( limj j pt+j dt+j = = + + = = 1 1 _{) cov( , )} , cov( ) var( j t j t t t j j t t t t d p d d p d r p B t t t t A t t t t d p d p d p d p ) var( ) , cov( ) var( , cov( 1 k t t k i= rt+k dp + + + + + + + + + + + = = = = = = = ) ~ ( 1 t t t u dp (dp 1) 2002 ,... 1990 1989 ,... 1926 ~ ~ 2 1 = = t t p d dp p d dp p d p d t t t t 1 p d , d p₂ t t t dp r ₁ t t t dp u dp 1

(

2

)

2 1/ 3 1 ) ˆ ( O T T E u u u 2 u 1 1 + + = + t + t t x r μ 0 ) ( _t = E 1 1 ~ + + = + t + t t dp r μ T p d~ ˆ ˆ ˆμ+ 1 ~ ˆ+ dpT+ μ ) ( +₁ – T T r E ) ( 2 1 + + T T r E μ = ≥ + | ) (r_{t 1} D_t E } 0 , { = r i D_{t t i} Δ – – – – – – – – – – – – – – – – – – – – – – ≈ _ρ ρ ρ ρ ρ σ σ σ σ θ θ ρ ∞ ∞ ∞ Δ Δ →

∑

∑

=1 1 j j– ρ ∞

∑

+j t r =1 1 j j– ρ ∞

∑

= + 1 1 j t j j– _d – – – ρ ∞ Δ

∑

∑

k i t k r =1 +

∑

– α α β β β β β β β β ε ε ε ε ε ŘFHQDZRNUHVLH!"#"! 1 1 + + = + t + t t p p μ ε ε ε ε ε σ σ ε ) ln( 1 1 + + = t t P p 1 1 + + = + t t z μ 1 1 + + = t t p z 0 ) , cov( 1 = ≠ + t t 0 ) , cov( 2 1 2 + t t (r ,_{t 1},... + –₁ – + t n t r r ) 2 1 2 1 var var var varr_t + r_t+ + r_t+ +…+ r_t+n =n / n . ) ( ) ( ) ( t n t r nVar r Var n VR = ) ( n t r Var ) (rt Var ) ( ^ _n VR )) 1 ( 2 , 0 ( ~ ) 1 ) ( (VR^ n N n nT (n = 10, T = 77) (n = 10, T= 109) t t t t t t _P D P R R R 1 1 1 1 1 1 1 + 1 + + + + + + = = 1 + t R t + 1, 1 + t P t+ 1, 1 + t D t + 1 P/D = exp(p d) ) ( 1 1 j t j t j j t t t d const E d r p + + = + 0 ) ( limj j pt+j dt+j = = + + = = 1 1 _{) cov( , )} , cov( ) var( j t j t t t j j t t t t d p d d p d r p B t t t t A t t t t d p d p d p d p ) var( ) , cov( ) var( , cov( 1 k t t k i= rt+k dp + + + + + + + + + + + = = = = = = = ) ~ ( 1 t t t u dp (dp 1) 2002 ,... 1990 1989 ,... 1926 ~ ~ 2 1 = = t t p d dp p d dp p d p d t t t t 1 p d , d p₂ t t t dp r 1 t t t dp u dp ₁

(

2

)

2 1/ 3 1 ) ˆ ( O T T E u u u 2 u 1 1 + + = + t + t t x r μ 0 ) ( _t = E 1 1 ~ + + = + t + t t dp r μ T p d~ ˆ ˆ ˆμ+ 1 ~ ˆ+ dpT+ μ ) ( +₁ – T T r E ) ( 2 1 + + T T r E μ = ≥ + | ) (rt 1 Dt E } 0 , { = r i D_{t t i} Δ – – – – – – – – – – – – – – – – – – – – – – ≈ _ρ ρ ρ ρ ρ σ σ σ σ θ θ ρ ∞ ∞ ∞ Δ Δ →

∑

∑

=1 1 j j– ρ ∞

∑

+j t r =1 1 j j– ρ ∞

∑

= + 1 1 j t j j– _d – – – ρ ∞ Δ

∑

∑

k i=1rt+k

∑

– α α β β β β β β β β ε ε ε ε ε ŘVNïDGQLNORVRZ\ZRNUHVLH!"#"! 1 1 + + = + t + t t p p μ ε ε ε ε ε σ σ ε ) ln( 1 1 + + = t t P p 1 1 + + = + t t z μ 1 1 + + = t t p z 0 ) , cov( ₁ = ≠ + t t 0 ) , cov( 2 1 2 + t t (r ,t 1,... + –1 – + t n t r r ) 2 1 2 1 var var var varrt + rt+ + rt+ +…+ rt+n =n / n . ) ( ) ( ) ( t n t r nVar r Var n VR = ) ( n t r Var ) (rt Var ) ( ^ _n VR )) 1 ( 2 , 0 ( ~ ) 1 ) ( (VR^ n N n nT (n = 10, T = 77) (n = 10, T= 109) t t t t t t _P D P R R R 1 1 1 1 1 1 1 + 1 + + + + + + = = 1 + t R t + 1, 1 + t P t+ 1, 1 + t D t + 1 P/D = exp(p d) ) ( 1 1 j t j t j j t t t d const E d r p + + = + 0 ) ( limj j pt+j dt+j = = + + = = 1 1 _{) cov( , )} , cov( ) var( j t j t t t j j t t t t d p d d p d r p B t t t t A t t t t d p d p d p d p ) var( ) , cov( ) var( , cov( 1 k t t k i= rt+k dp + + + + + + + + + + + = = = = = = = ) ~ ( 1 t t t u dp (dp 1) 2002 ,... 1990 1989 ,... 1926 ~ ~ 2 1 = = t t p d dp p d dp p d p d t t t t 1 p d , d p₂ t t t dp r ₁ t t t dp u dp 1

(

2

)

2 1/ 3 1 ) ˆ ( O T T E u u u 2 u 1 1 + + = + t + t t x r μ 0 ) ( _t = E 1 1 ~ + + = + t + t t dp r μ T p d~ ˆ ˆ ˆμ+ 1 ~ ˆ+ dpT+ μ ) ( +₁ – T T r E ) ( 2 1 + + T T r E μ = ≥ + | ) (r_{t 1} D_t E } 0 , { = r i D_{t t i} Δ – – – – – – – – – – – – – – – – – – – – – – ≈ _ρ ρ ρ ρ ρ σ σ σ σ θ θ ρ ∞ ∞ ∞ Δ Δ →

∑

∑

=1 1 j j– ρ ∞

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+j t r =1 1 j j– ρ ∞

∑

= + 1 1 j t j j– _d – – – ρ ∞ Δ

∑

∑

k i t k r =1 +

∑

– α α β β β β β β β β ε ε ε ε ε ŘSDUDPHWU _{ +LSRWH]D'$%()*#+$,-ZSRVWDFLSRGDQHMSU]H]/RL0DF.LQOD\DMHVWEïÈG]HQLHPORVRZ\P]bGU\IHP'ODWHJR} ZUöZQDQLXZ\VWÚSXMHGU\I+

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μ

1 1 + + = t t p z 0 ) , cov( ₁ = ≠ + t t 0 ) , cov( 2 1 2 + t t (r ,t 1,... + –1 – + t n t r r ) 2 1 2 1 var var var varrt+ rt+ + rt+ +…+ rt+n =n / n . ) ( ) ( ) ( t n t r nVar r Var n VR = ) ( n t r Var ) (rt Var ) ( ^ _n VR )) 1 ( 2 , 0 ( ~ ) 1 ) ( (VR^ n N n nT (n = 10, T = 77) (n = 10, T= 109) t t t t t t R R P _PD R 1 1 1 1 1 1 1 + 1 + + + + + + = = 1 + t R t + 1, 1 + t P t+ 1, 1 + t D t + 1 P/D = exp(p d) ) ( 1 1 j t j t j j t t t d const E d r p + + = + 0 ) ( limj j pt+j dt+j = = + + = = 1 1 _{) cov( , )} , cov( ) var( j t j t t t j j t t t t d p d d p d r p B t t t t A t t t t d p d p d p d p ) var( ) , cov( ) var( , cov( 1 k t t k i= rt+k dp + + + + + + + + + + + = = = = = = = ) ~ ( 1 t t t u dp (dp ₁) 2002 ,... 1990 1989 ,... 1926 ~ ~ 2 1 = = t t p d dp p d dp p d p d t t t t 1 p d , d p2 t t t dp r 1 t t t dp u dp 1

(

2

)

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∑

∑

=1 1 j j– ρ ∞

∑

+j t r =1 1 j j– ρ ∞

∑

= + 1 1 j t j j– _d – – – ρ ∞ Δ

∑

∑

k i=1rt+k

∑

– α α β β β β β β β β ε ε ε ε ε , 1 1 + + = + t+ t t p p μ ε ε ε ε ε σ σ ε ) ln( ₁ 1 + + = t t P p 1 1 + + = + t t z

μ

1 1 + + = t t p z 0 ) , cov( ₁ = ≠ + t t 0 ) , cov( 2 1 2 + t t (r ,_{t 1},... + –₁ – + t n t r r ) 2 1 2 1 var var var varrt + rt+ + rt+ +…+ rt+n =n / n . ) ( ) ( ) ( t n t r nVar r Var n VR = ) ( n t r Var ) (rt Var ) ( ^ _n VR )) 1 ( 2 , 0 ( ~ ) 1 ) ( (VR^ n N n nT (n = 10, T = 77) (n = 10, T= 109) t t t t t t _P D P R R R 1 1 1 1 1 1 1 + 1 + + + + + + = = 1 + t R t + 1, 1 + t P t+ 1, 1 + t D t + 1 P/D = exp(p d) ) ( 1 1 j t j t j j t t t d const E d r p + + = + 0 ) ( limj j pt+j dt+j = = + + = = 1 1 _{) cov( , )} , cov( ) var( j t j t t t j j t t t t d p d d p d r p B t t t t A t t t t d p d p d p d p ) var( ) , cov( ) var( , cov( 1 k t t k i= rt+k dp + + + + + + + + + + + = = = = = = = ) ~ ( 1 t t t u dp (dp 1) 2002 ,... 1990 1989 ,... 1926 ~ ~ 2 1 = = t t p d dp p d dp p d p d t t t t 1 p d , d p₂ t t t dp r 1 t t t dp u dp 1

(

2

)

2 1/ 3 1 ) ˆ ( O T T E u u u 2 u 1 1 + + = + t+ t t x r μ 0 ) ( _t = E 1 1 ~ + + = + t + t t dp r μ T p d~ ˆ ˆ ˆμ+ 1 ~ ˆ+ dpT+ μ ) ( +1 – T T r E ) ( 2 1 + + T T r E μ = ≥ + | ) (rt1 Dt E } 0 , { = r i Dt t i Δ – – – – – – – – – – – – – – – – – – – – – – ≈ ρ ρ ρ ρ ρ σ σ σ σ θ θ ρ ∞ ∞ ∞ Δ Δ →

∑

∑

=1 1 j j– ρ ∞

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∑

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– α α β β β β β β β β ε ε ε ε ε 

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μ

1 1 + + = t t p z 0 ) , cov( ₁ = ≠ + t t 0 ) , cov( 2 1 2 + t t (r ,_{t 1},... + –₁ – + t n t r r ) 2 1 2 1 var var var varrt+ rt+ + rt+ +…+ rt+n =n / n . ) ( ) ( ) ( t n t r nVar r Var n VR = ) ( n t r Var ) (rt Var ) ( ^ _n VR )) 1 ( 2 , 0 ( ~ ) 1 ) ( (VR^ n N n nT (n = 10, T = 77) (n = 10, T= 109) t t t t t t _P D P R R R 1 1 1 1 1 1 1 + 1 + + + + + + = = 1 + t R t + 1, 1 + t P t+ 1, 1 + t D t + 1 P/D = exp(p d) ) ( 1 1 j t j t j j t t t d const E d r p + + = + 0 ) ( limj j pt+j dt+j = = + + = = 1 1 _{) cov( , )} , cov( ) var( j t j t t t j j t t t t d p d d p d r p B t t t t A t t t t d p d p d p d p ) var( ) , cov( ) var( , cov( 1 k t t k i= rt+k dp + + + + + + + + + + + = = = = = = = ) ~ ( 1 t t t u dp (dp 1) 2002 ,... 1990 1989 ,... 1926 ~ ~ 2 1 = = t t p d dp p d dp p d p d t t t t 1 p d , d p₂ t t t dp r 1 t t t dp u dp 1

(

2

)

2 1/ 3 1 ) ˆ ( O T T E u u u 2 u 1 1 + + = + t+ t t x r μ 0 ) ( _t = E 1 1 ~ + + = + t + t t dp r μ T p d~ ˆ ˆ ˆμ+ 1 ~ ˆ+ dpT+ μ ) ( +1 – T T r E ) ( 2 1 + + T T r E μ = ≥ + | ) (rt1 Dt E } 0 , { = r i Dt t i Δ – – – – – – – – – – – – – – – – – – – – – – ≈ ρ ρ ρ ρ ρ σ σ σ σ θ θ ρ ∞ ∞ ∞ Δ Δ →

∑

∑

=1 1 j j– ρ ∞

∑

+j t r =1 1 j j– ρ ∞

∑

= + 1 1 j t j j– _d – – – ρ ∞ Δ

∑

∑

k i=1rt+k

∑

– α α β β β β β β β β ε ε ε ε ε PLDï\MHGQDNRZHUR]NïDG\LE\ï\QLH]DOHĝQHDbLFKZDULDQFMDƱ_{E\ïDVWDïDWR} 1 1 + + = + t + t t p p μ ε ε ε ε ε σ σ ε ) ln( ₁ 1 + + = t t P p 1 1 + + = + t t z

μ

1 1 + + = t t p z 0 ) , cov( 1 = ≠ + t t 0 ) , cov( 2 1 2 + t t (r ,t 1,... + –1 – + t n t r r ) 2 1 2 1 var var var varrt+ rt+ + rt+ +…+ rt+n =n / n . ) ( ) ( ) ( t n t r nVar r Var n VR = ) ( n t r Var ) (rt Var ) ( ^ _n VR )) 1 ( 2 , 0 ( ~ ) 1 ) ( (VR^ n N n nT (n = 10, T = 77) (n = 10, T= 109) t t t t t t R R P _PD R 1 1 1 1 1 1 1 + 1 + + + + + + = = 1 + t R t + 1, 1 + t P t+ 1, 1 + t D t + 1 P/D = exp(p d) ) ( 1 1 j t j t j j t t t d const E d r p + + = + 0

)

(

limj j pt+j dt+j = = + + = = 1 1 _{) cov( , )} , cov( ) var( j t j t t t j j t t t t d p d d p d r p B t t t t A t t t t d p d p d p d p ) var( ) , cov( ) var( , cov( 1 k t t k i= rt+k dp + + + + + + + + + + + = = = = = = = ) ~ ( 1 t t t u dp (dp 1) 2002 ,... 1990 1989 ,... 1926 ~ ~ 2 1 = = t t p d dp p d dp p d p d t t t t 1 p d , d p2 t t t dp r 1 t t t dp u dp 1

(

2

)

2 1/ 3 1 ) ˆ ( O T T E u u u 2 u 1 1 + + = + t+ t t x r μ 0 ) ( t = E 1 1 ~ + + = + t + t t dp r μ T p d~ ˆ ˆ ˆμ+ 1 ~ ˆ+ dpT₊ μ

)

(

+1 – T T r E

)

(

2 1 + + T T r E μ = ≥ + | ) (rt 1 Dt E } 0 , { = r i Dt t i Δ – – – – – – – – – – – – – – – – – – – – – – ≈ _ρ ρ ρ ρ ρ σ σ σ σ θ θ ρ ∞ ∞ ∞ Δ Δ →

∑

∑

=1 1 j j– ρ ∞

∑

+j t r =1 1 j j– ρ ∞

∑

=1 + 1 j t j j– _d – – – ρ ∞ Δ

∑

∑

k i=1rt+k

∑

– α α β β β β β

β

β

β

ε ε ε ε ε  2GFK\OHQLHVWDQGDUGRZHVWRS\]ZURWXVNïDGDMÈFHMVLÚ]%VWöSZbXMÚFLXMHGQRRNUHVRZ\PE\ïRE\ QDWRPLDVWUöZQH 1 1 + + = + t+ t t p p μ ε ε ε ε ε σ σ ε ) ln( 1 1 + + = t t P p 1 1 + + = + t t z

μ

1 1 + + = t t p z 0 ) , cov( 1 = ≠ + t t 0 ) , cov( 2 1 2 + t t (r ,t 1,... + –1 – + t n t r r ) 2 1 2 1 var var var varrt+ rt+ + rt+ +…+ rt+n =n / n . ) ( ) ( ) ( t n t r nVar r Var n VR = ) ( n t r Var ) (rt Var ) ( ^ _n VR )) 1 ( 2 , 0 ( ~ ) 1 ) ( (VR^ n N n nT (n = 10, T = 77) (n = 10, T= 109) t t t t t t R R P _PD R 1 1 1 1 1 1 1 + 1 + + + + + + = = 1 + t R t + 1, 1 + t P t+ 1, 1 + t D t + 1 P/D = exp(p d) ) ( 1 1 j t j t j j t t t d const E d r p + + = + 0 ) ( limj j pt+j dt+j = = + + = = 1 1 _{) cov( , )} , cov( ) var( j t j t t t j j t t t t d p d d p d r p B t t t t A t t t t d p d p d p d p ) var( ) , cov( ) var( , cov( 1 k t t k i= rt+k dp + + + + + + + + + + + = = = = = = = ) ~ ( 1 t t t u dp (dp 1) 2002 ,... 1990 1989 ,... 1926 ~ ~ 2 1 = = t t p d dp p d dp p d p d t t t t 1 p d , d p₂ t t t dp r 1 t t t dp u dp 1

(

2

)

2 1/ 3 1 ) ˆ ( O T T E u u u 2 u 1 1 + + = + t+ t t x r μ 0 ) ( t = E 1 1 ~ + + = + t + t t dp r μ T p d~ ˆ ˆ ˆμ+ 1 ~ ˆ+ dpT+ μ ) ( +1 – T T r E ) ( 2 1 + + T T r E μ = ≥ + | ) (rt1 Dt E } 0 , { = r i Dt t i Δ – – – – – – – – – – – – – – – – – – – – – – ≈ ρ ρ ρ ρ ρ σ σ σ σ θ θ ρ ∞ ∞ ∞ Δ Δ →

∑

∑

=1 1 j j– ρ ∞

∑

+j t r =1 1 j j– ρ ∞

∑

= + 1 1 j t j j– _d – – – ρ ∞ Δ

∑

∑

k i=1rt+k

∑

– α α β β β β β β β β ε ε ε ε ε  :DULDQFMHVWöS]ZURWXGRNWöU\FKZ\]QDF]HQLDXĝ\WRDQDOL]RZDQ\FKV]HUHJöZQLHSR]ZDODMÈ QDSRWZLHUG]HQLHSRZ\ĝV]\FK]DOHĝQRĂFL7DEHODSRND]XMHĝHZDULDQFMDORJDU\WPLF]Q\FKSUHPLL ]DU\]\NRZ\]QDF]RQ\FKQDED]LHLQGHNVX1<6(GODKRU\]RQWöZLQZHVW\F\MQ\FKGïXĝV]\FKQLĝ MHGQRRNUHVRZHFKDUDNWHU\]XMHVLÚWHQGHQFMÈPDOHMÈFÈ2GFK\OHQLHVWDQGDUGRZH]PQLHMV]DVLÚQD WRPLDVWV]\EFLHMQLĝZbWHPSLHRGZURWQRĂFLSLHUZLDVWNDNZDGUDWRZHJR]bOLF]E\RNUHVöZXZ]JOÚG QLRQ\FKSU]\Z\OLF]DQLXVWRS\]ZURWX:SU]\SDGNXSROVNLFKGDQ\FKZDULDQFMDZXMÚFLXMHGQR RNUHVRZ\PZ]UDVWDZUD]]Z\GïXĝDQLHPKRU\]RQWXLQZHVW\F\MQHJRMHĂOLMHGQDNSUDZG]LZDMHVW KLSRWH]DEïÈG]HQLDORVRZHJRWRSRZLQQDSR]RVWDÊVWDïDWDEHOD:XMÚFLXPDWHPDW\F]Q\PUöĝ QLFHPLÚG]\U\]\NLHPZ\UDĝRQ\PZbSRVWDFLZDULDQFMLGODRNUHVöZNUöWNLFKLbGïXJLFKVÈPRĝOLZH MHG\QLHZWHG\JG\RF]HNLZDQHVWRS\]ZURWX]PLHQLDMÈVLÚZbF]DVLH =DSRPRFÈWHVWXLORUD]XZDULDQFMLPRĝQD]ZHU\ILNRZDÊLVWRWQRĂÊUöĝQLFPLÚG]\ZDULDQFMD PL GOD UöĝQ\FK KRU\]RQWöZ F]DVRZ\FK 7HVW WHQ ]RVWDï ]DSUH]HQWRZDQ\ PLQ Zb DUW\NXïDFK /R! 0DF.LQOD\D  RUD] &RFKUDQHijD  ,ORUD] ZDULDQFML SU]\ VSHïQLHQLX KLSRWH]\ ]HURZHM! PöZLÈFHMĝHVWRS\]ZURWXVÈQLH]DOHĝQHLPDMÈLGHQW\F]QHUR]NïDG\SRZLQLHQZ\QRVLÊ,ORUD] ZDULDQFMLGOD%RNUHVRZHMVWRS\]ZURWXZ\UDĝDVLÚQDVWÚSXMÈF\PZ]RUHP !! ! ! ! !!!_!!! !!! 1 1 + + = + t+ t t p p μ ε ε ε ε ε σ σ ε ) ln( 1 1 + + = t t P p 1 1 + + = + t t z

μ

1 1 + + = t t p z 0 ) , cov( 1 = ≠ + t t 0 ) , cov( 2 1 2 + t t (r ,_t 1,... + –1 – + t n t r r ) 2 1 2 1 var var var varr_t + r_t+ + r_t+ +…+ r_t+n =n / n . ) ( ) ( ) ( t n t r nVar r Var n VR = ) ( n t r Var ) (r_t Var ) ( ^ _n VR )) 1 ( 2 , 0 ( ~ ) 1 ) ( (VR^ n N n nT (n = 10, T = 77) (n = 10, T= 109) t t t t t t _P D P R R R 1 1 1 1 1 1 1 + 1 + + + + + + = = 1 + t R t + 1, 1 + t P t+ 1, 1 + t D t + 1 P/D = exp(p d) ) ( 1 1 j t j t j j t t t d const E d r p + + = + 0

)

(

limj j pt+j dt+j = = + + = = 1 1 _{) cov( , )} , cov( ) var( j t j t t t j j t t t t d p d d p d r p B t t t t A t t t t d p d p d p d p ) var( ) , cov( ) var( , cov( 1 k t t k i= rt+k dp + + + + + + + + + + + = = = = = = = ) ~ ( 1 t t t u dp (dp 1) 2002 ,... 1990 1989 ,... 1926 ~ ~ 2 1 = = t t p d dp p d dp p d p d t t t t 1 p d , d p2 t t t dp r 1 t t t dp u dp 1

(

2

)

2 1/ 3 1 ) ˆ ( O T T E u u u 2 u 1 1 + + = + t + t t x r μ 0 ) ( _t = E 1 1 ~ + + = + t + t t dp r μ T p d~ ˆ ˆ ˆμ+ 1 ~ ˆ+ dpT+ μ

)

(

+1 – T T r E

)

(

2 1 + + T T r E μ = ≥ + | ) (rt1 Dt E } 0 , { = r i Dt t i Δ – – – – – – – – – – – – – – – – – – – – – – ≈ ρ ρ ρ ρ ρ σ σ σ σ θ θ ρ ∞ ∞ ∞ Δ Δ →

∑

∑

=1 1 j j– ρ ∞

∑

+j t r =1 1 j j– ρ ∞

∑

= + 1 1 j t j j– _d – – – ρ ∞ Δ

∑

∑

k i=1rt+k

∑

– α α β β β β β

β

β

β

ε ε ε ε ε !     JG]LH 1 1 + + = + t+ t t p p μ ε ε ε ε ε σ σ ε ) ln( 1 1 + + = t t P p 1 1 + + = + t t z

μ

1 1 + + = t t p z 0 ) , cov( ₁ = ≠ + t t 0 ) , cov( 2 1 2 + t t (r ,t 1,... + –1 – + t n t r r ) 2 1 2 1 var var var varrt+ rt+ + rt+ +…+ rt+n =n / n . ) ( ) ( ) ( t n t r nVar r Var n VR = ) ( n t r Var ) (rt Var ) ( ^ _n VR )) 1 ( 2 , 0 ( ~ ) 1 ) ( (VR^ n N n nT (n = 10, T = 77) (n = 10, T= 109) t t t t t t R R P _PD R 1 1 1 1 1 1 1 + 1 + + + + + + = = 1 + t R t + 1, 1 + t P t+ 1, 1 + t D t + 1 P/D = exp(p d) ) ( 1 1 j t j t j j t t t d const E d r p + + = + 0 ) ( limj j pt+j dt+j = = + + = = 1 1 _{) cov( , )} , cov( ) var( j t j t t t j j t t t t d p d d p d r p B t t t t A t t t t d p d p d p d p ) var( ) , cov( ) var( , cov( 1 k t t k i= rt+k dp + + + + + + + + + + + = = = = = = = ) ~ ( 1 t t t u dp (dp ₁) 2002 ,... 1990 1989 ,... 1926 ~ ~ 2 1 = = t t p d dp p d dp p d p d t t t t 1 p d , d p2 t t t dp r 1 t t t dp u dp 1

(

2

)

2 1/ 3 1 ) ˆ ( O T T E u u u 2 u 1 1 + + = + t+ t t x r μ 0 ) ( t = E 1 1 ~ + + = + t + t t dp r μ T p d~ ˆ ˆ ˆμ+ 1 ~ ˆ+ dpT+ μ ) ( +1 – T T r E ) ( 2 1 + + T T r E μ = ≥ + | ) (r_t1 D_t E } 0 , { = r i Dt t i Δ – – – – – – – – – – – – – – – – – – – – – – ≈ ρ ρ ρ ρ ρ σ σ σ σ θ θ ρ ∞ ∞ ∞ Δ Δ →

∑

∑

=1 1 j j– ρ ∞

∑

+j t r =1 1 j j– ρ ∞

∑

= + 1 1 j t j j– _d – – – ρ ∞ Δ

∑

∑

k i=1rt+k

∑

– α α β β β β β β β β ε ε ε ε ε ļZDULDQFMD%RNUHVRZHMVWRS\]ZURWX 1 1 + + = + t+ t t p p μ ε ε ε ε ε σ σ ε ) ln( 1 1 + + = t t P p 1 1 + + = + t t z

μ

1 1 + + = t t p z 0 ) , cov( 1 = ≠ + t t 0 ) , cov( 2 1 2 + t t (r ,t 1,... + –1 – + t n t r r ) 2 1 2 1 var var var varrt+ rt+ + rt+ +…+ rt+n =n / n . ) ( ) ( ) ( t n t r nVar r Var n VR = ) ( n t r Var ) (rt Var ) ( ^ _n VR )) 1 ( 2 , 0 ( ~ ) 1 ) ( (VR^ n N n nT (n = 10, T = 77) (n = 10, T= 109) t t t t t t R R P _PD R 1 1 1 1 1 1 1 + 1 + + + + + + = = 1 + t R t + 1, 1 + t P t+ 1, 1 + t D t + 1 P/D = exp(p d) ) ( 1 1 j t j t j j t t t d const E d r p + + = + 0 ) ( limj j pt+j dt+j = = + + = = 1 1 _{) cov( , )} , cov( ) var( j t j t t t j j t t t t d p d d p d r p B t t t t A t t t t d p d p d p d p ) var( ) , cov( ) var( , cov( 1 k t t k i= rt+k dp + + + + + + + + + + + = = = = = = = ) ~ ( 1 t t t u dp (dp 1) 2002 ,... 1990 1989 ,... 1926 ~ ~ 2 1 = = t t p d dp p d dp p d p d t t t t 1 p d , d p₂ t t t dp r 1 t t t dp u dp 1

(

2

)

2 1/ 3 1 ) ˆ ( O T T E u u u 2 u 1 1 + + = + t+ t t x r μ 0 ) ( t = E 1 1 ~ + + = + t + t t dp r μ T p d~ ˆ ˆ ˆμ+ 1 ~ ˆ+ dpT+ μ ) ( +1 – T T r E ) ( 2 1 + + T T r E μ = ≥ + | ) (rt1 Dt E } 0 , { = r i Dt t i Δ – – – – – – – – – – – – – – – – – – – – – – ≈ ρ ρ ρ ρ ρ σ σ σ σ θ θ ρ ∞ ∞ ∞ Δ Δ →

∑

∑

=1 1 j j– ρ ∞

∑

+j t r =1 1 j j– ρ ∞

∑

= + 1 1 j t j j– _d – – – ρ ∞ Δ

∑

∑

k i=1rt+k

∑

– α α β β β β β β β β ε ε ε ε ε ļZDULDQFMDMHGQRRNUHVRZHMVWRS\]ZURWX $V\PSWRW\F]Q\UR]NïDG 1 1 + + = + t+ t t p p μ ε ε ε ε ε σ σ ε ) ln( 1 1 + + = t t P p 1 1 + + = + t t z

μ

1 1 + + = t t p z 0 ) , cov( 1 = ≠ + t t 0 ) , cov( 2 1 2 + t t (r ,t 1,... + –1 – + t n t r r ) 2 1 2 1 var var var varrt+ rt+ + rt+ +…+ rt+n =n / n . ) ( ) ( ) ( t n t r nVar r Var n VR = ) ( n t r Var ) (rt Var ) ( ^ _n VR )) 1 ( 2 , 0 ( ~ ) 1 ) ( (VR^ n N n nT (n = 10, T = 77) (n = 10, T= 109) t t t t t t R R P _PD R 1 1 1 1 1 1 1 + 1 + + + + + + = = 1 + t R t + 1, 1 + t P t+ 1, 1 + t D t + 1 P/D = exp(p d) ) ( 1 1 j t j t j j t t t d const E d r p + + = + 0 ) ( limj j pt+j dt+j = = + + = = 1 1 _{) cov( , )} , cov( ) var( j t j t t t j j t t t t d p d d p d r p B t t t t A t t t t d p d p d p d p ) var( ) , cov( ) var( , cov( 1 k t t k i= rt+k dp + + + + + + + + + + + = = = = = = = ) ~ ( 1 t t t u dp (dp 1) 2002 ,... 1990 1989 ,... 1926 ~ ~ 2 1 = = t t p d dp p d dp p d p d t t t t 1 p d , d p₂ t t t dp r 1 t t t dp u dp 1

(

2

)

2 1/ 3 1 ) ˆ ( O T T E u u u 2 u 1 1 + + = + t+ t t x r μ 0 ) ( t = E 1 1 ~ + + = + t + t t dp r μ T p d~ ˆ ˆ ˆμ+ 1 ~ ˆ+ dpT+ μ ) ( +1 – T T r E ) ( 2 1 + + T T r E μ = ≥ + | ) (rt1 Dt E } 0 , { = r i Dt t i Δ – – – – – – – – – – – – – – – – – – – – – – ≈ ρ ρ ρ ρ ρ σ σ σ σ θ θ ρ ∞ ∞ ∞ Δ Δ →

∑

∑

=1 1 j j– ρ ∞

∑

+j t r =1 1 j j– ρ ∞

∑

= + 1 1 j t j j– _d – – – ρ ∞ Δ

∑

∑

k i=1rt+k

∑

– α α β β β β β β β β ε ε ε ε ε SU]\VSHïQLHQLXKLSRWH]\]HURZHMPDSRVWDÊ !! ! ! ! _! 1 1 + + = + t + t t p p μ ε ε ε ε ε σ σ ε ) ln( ₁ 1 + + = t t P p 1 1 + + = + t t z

μ

1 1 + + = t t p z 0 ) , cov( ₁ = ≠ + t t 0 ) , cov( 2 1 2 + t t (r ,_t 1,... + –1 – + t n t r r ) 2 1 2 1 var var var varrt + rt+ + rt+ +…+ rt+n =n / n . ) ( ) ( ) ( t n t r nVar r Var n VR = ) ( n t r Var ) (r_t Var ) ( ^ _n VR )) 1 ( 2 , 0 ( ~ ) 1 ) ( (VR^ n N n nT (n = 10, T = 77) (n = 10, T= 109) t t t t t t R R P _PD R 1 1 1 1 1 1 1 + 1 + + + + + + = = 1 + t R t + 1, 1 + t P t+ 1, 1 + t D t + 1 P/D = exp(p d) ) ( 1 1 j t j t j j t t t d const E d r p + + = + 0

)

(

limj j pt+j dt+j = = + + = = 1 1 _{) cov( , )} , cov( ) var( j t j t t t j j t t t t d p d d p d r p B t t t t A t t t t d p d p d p d p ) var( ) , cov( ) var( , cov( 1 k t t k i= rt+k dp + + + + + + + + + + + = = = = = = = ) ~ ( 1 t t t u dp (dp 1) 2002 ,... 1990 1989 ,... 1926 ~ ~ 2 1 = = t t p d dp p d dp p d p d t t t t 1 p d , d p2 t t t dp r 1 t t t dp u dp 1

(

2

)

2 1/ 3 1 ) ˆ ( O T T E u u u 2 u 1 1 + + = + t+ t t x r μ 0 ) ( _t = E 1 1 ~ + + = + t + t t dp r μ T p d~ ˆ ˆ ˆμ+ 1 ~ ˆ+ dpT₊ μ

)

(

+1 – T T r E

)

(

2 1 + + T T r E μ = ≥ + | ) (rt 1 Dt E } 0 , { = r i D_{t t i} Δ – – – – – – – – – – – – – – – – – – – – – – ≈ _ρ ρ ρ ρ ρ σ σ σ σ θ θ ρ ∞ ∞ ∞ Δ Δ →

∑

∑

=1 1 j j– ρ ∞

∑

+j t r =1 1 j j– ρ ∞

∑

= + 1 1 j t j j– _d – – – ρ ∞ Δ

∑

∑

k i=1rt+k

∑

– α α β β β β β

β

β

β

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