(presence of bid-ask spread) (discreteness of price) (price adjustment) Ro (1984) 1) Harris (1990) Amihud and Mendeson (1987) 1 (partia
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1 39, 1, : Dependence of Microstructure Noise: High-Frequency Data Anaysis Masato Ubukata ( ) 30 (1) ( / ) 1 (2) (3) In the iterature of market microstructure, an impact of market trading system on asset pricing has been studied. On the transaction price mode in high-frequency financia econometrics, the impact is caed as microstructure noise or microstructure effect. The statistica anaysis of the microstructure noise sheds more ight on the impact of market reguarity and trading mechanism on asset pricing in financia markets. In this paper, we test for dependence of the microstructure noise in Japanese stocks traded on the Osaka Securities Exchange. Main resuts for the thirty individua stocks are: (1) noise-to-signa ratio, which is the ratio between a variance estimate of the noise and an integrated variance estimate, has positive and negative reationship with tick size and number of transactions per day, respectivey, (2) the noise in thicky traded stock has more significanty negative autocovariance than thiny traded stocks, and (3) the noise among some assets is associated with asymmetric and different patterns of cross-sectiona dependence. : ( ) (E-mai: ubukata@mai.econ.osakau.ac.jp)
2 (presence of bid-ask spread) (discreteness of price) (price adjustment) Ro (1984) 1) Harris (1990) Amihud and Mendeson (1987) 1 (partia adjustment) (overshooting) 2) Hansen and Lunde (2006) (noise-to-signa ratio : NSR) Medahhi (2002) Ubukata and Oya (2009a) 2 1) Hasbrouck (2007)
3 3 Ubukata and Oya (2009a) 30 1 NSR ( / ) NSR NSR NSR (ead-ag) 2) 2 Reaized Voatiity (RV) RV Zhou (1996) Zhang et a. (2005) Zhang (2006) Zhou (1996) Reaized Voatiity Zhang et a. (2005) RV two scaes RV Zhang (2006) two scaes V mutipe scae Bandi and Russe (2008) 2 (Mean Squared Error: MSE) Aït-Sahaia et a. (2005) Zhang et a. (2005) Zhang (2006) Hansen and Lunde (2006) Barndorff-Niesen et a. (2008) RV (2007) (2007) McAer and Medeiros (2008) Voev and Lunde (2007) Hayashi and Yoshida (2005)
4 Ubukata and Oya (2009a) 2 {P 1 (t)} {P 2 (t)} 3) dp (t) = σ (t)dw (t), = 1, 2, d W 1, W 2 t = ρ (t)dt, ρ (t) ( 1, 1) t [0, T ] (2.1) W (t) P (0) > 0 σ (t) 4) 1 i r 1,i := P 1 (t i ) P 1 (t i 1 ) t i r 1,i i 2 j r 2,j := P 2 (s j ) P 2 (s j 1 ) s j r 2,j j P P 1 (t i ) = P 1 (t i ) + η(t i ), P 2 (s j ) = P 2 (s j ) + δ(s j ), (2.3) 3) Ubukata and Oya (2009a) Ubukata and Oya (2009b) 4) [0, T ] (Integrated voatiity: IV) (Integrated covariance: IC) Z T Z T IV,T = σ 2 (u)du, IC T = σ 1 (u)σ 2 (u)ρ (u)du, = 1, 2. (2.2) 0 0 Reaized Voatiity Reaized Covariance Cumuative Covariance IV IC Reaized Covariance Reaized Voatiity Cumuative Covariance Hayashi and Yoshida (2005)
5 5 η δ u(t) = (η(t) δ(t)) (a) E[ u(t)] = 0 (b) 2 Γ() = E[ u(t) u (t )] = γ η() γ ηδ () = 0, for a > m. γ δη () γ δ () (c) E u(t) u (s) 4β < for a t, s β > 1 (d) P u(t), = 1, 2 5) (b) γ ηδ () = E[η(t)δ(t )] γ δη () = E[η(t )δ(t)] = E[η(t)δ(t+)] γ ηδ () γ() γ δη () γ() γ() ρ η () ρ δ () ρ() Hayashi and Yoshida (2005) Cumuative Covariance (2.2) Integrated Covariance 1 2 Ubukata and Oya (2009a) 1 2 (I i := (t i 1, t i ], J j := (s j 1, s j ]) E IJ [ ] V IJ [ ] (d) E IJ [ r 1,i r 1,j ] = E IJ [ (η(ti ) η(t i 1 ) )( η(t j ) η(t j 1 ) )], for a i j. (2.4) 5) Ubukata and Oya (2009a) Hansen and Lunde (2006) Ubukata and Oya (2009b) Hansen and Lunde (2006)
6 γ η () X 1,,ij = r 1,i r 1,j, for a i, j such that t j 1 t i = 0. (2.5) 1 t j 1 t i = 0 X 1,,ij t j := t j t j 1 t i := t i t i 1 E IJ [X 1,,ij ] = E IJ [η(t i )η(t j )] E IJ [η(t i 1 )η(t j )] E IJ [η(t i )η(t j 1 )] + E IJ [η(t i 1 )η(t j 1 )] = γ η (t j t i ) γ η (t j t i 1 ) γ η (t j 1 t i ) + γ η (t j 1 t i 1 ) = γ η ( + t j ) γ η ( + t j + t i ) γ η () + γ η ( + t i ) (2.6) γ η () γ η ( + t j ) γ η ( + t j + t i ) γ η ( + t i ) γ η () 1 (b) 1 γ η () = 0 for a > m 1 > 0 m 1 ( )( ) X (±) 1,,ij = r( ) 1,i r(+) 1,j = P (t i ) P (t ( ) i 1 ) P (t (+) j ) P (t j 1 ), (2.7) for a i, j such that t j 1 t i = 0. t (+) j t (+) j t i > m 1 t ( ) i 1 t j 1 t ( ) i 1 > m 1 r ( ) 1,i r(+) 1,j (t( ) i 1, t i] (t j 1, t (+) j ] 2 (2.7) (2.7) t j 1 t i = γ η () m 1 E IJ [X (±) 1,,ij ] = γ η() X (±) (±) 1, 1,,ij i {X(±) 1,,k }N k=1 N (±) 1, X (±) 1,,k γ η() ˆγ η () N (±) 1,
7 7 2 (2.7) ˆγ η () = 1 N (±) 1, N (±) 1, k=1 X (±) 1,,k, N (±) 1, (ˆγ η() γ η ()) a N(0, ω 2 η,), (2.8) ωη, 2 ˆγ η() ωη, 2 = im N (±) 1, E IJ[N (±) 1, (ˆγ η() γ η ()) 2 ] Ubukata and Oya (2009a) {X (±) 1,,k }N 1, k=1 ω 2 η, {X (±) (±) 1, 1,,k }N k=1 M 1, {X (±),h 1, } := ( X (±) 1,,hM 1, +1, X(±) 1,,hM 1, +2,, X(±) 1,,(h+1)M 1, ), 0 h K1, 1, K 1, = [N (±) 1, /M 1,], s.t. M 1, and M 1, /N (±) 1, 0 as N (±) 1, (2.9) K 1, M 1, {X (±),h 1, } K 1, {X (±),h 1, } ([ ] ) ω 2 η, N (±) 1, ˆω 2 η, = M 1, K 1, K 1, 1 h=0 ( X(±),h 1,,M 1, 1 K 1, 1 K 1, h=0 X (±),h 1,,M 1, ) 2 (2.10) X (±),h 1,,M 1, {X (±),h 1, } M 1, 3 ω 2 η, = 0 ( ) γ η (0) = 0 γ η (0) > 0 N (±) 1, τ η (0) 1 τ η (0) = ( N (±) 1,0 ˆγ η(0) ˆω η,0 ) 2 a χ 2 (1) (2.11)
8 > 0 γ() = 0 γ() 0 N (±) 1, τ η () τ η () := N (±) 1, ˆγ η() ˆω η, a N(0, 1) (2.12) m 1 m 1 Ubukata and Oya (2009a) m 1 (2.5) X 1,,ij i {X 1,,k } N 1, k=1 N 1, N (±) 1, X 1,,N1, E IJ [X 1,,k ] = 0 for a k E IJ [X 1,,k ] 0 for a k X 1,,N1, σ 2 1, = im N 1, E IJ [N 1, ( X 1,,N1, E IJ [ X 1,,N1, ]) 2 ] ˆσ 2 1, N 1, m 1 τ η () τ η () := N1, X1,,N1, ˆσ 1, a N(0, 1), (2.13) L E IJ [X 1,L,k ] = 0 E IJ [X 1,L 1,k ] = 0 = max { τη () > } m 1 6) γ() 1 2 Y,ij = r 1,i r 2,j, for a i, j, such that t i 1 s j = if > 0 t i 1 s j = 0 or s j 1 t i = 0 if = 0 s j 1 t i = if < 0 (2.14) 3 > 0 < 0 > 0 1 (b) γ() = γ ηδ () 2 1 6) Ubukata and Oya (2009a)
9 9 3 (2.14) (a) > 0 (b) < 0 < = 0 ( t i 1 s j = 0 or s j 1 t i = 0 ) > 0 Y,ij E IJ [ Y,ij ] = γ( + t i ) γ( + t i + s j ) γ() + γ( + s j ) (2.15) t i := t i t i 1 s j := s j s j 1 1 (b) γ η () = 0 for a > m + c > 0 and > m c > 0 γ() (2.14) r 1,i r(+) 2,j if > 0 Y (±),ij = r (+) 1,i r(+) 2,j + r( ) 1,i r ( ) 2,j if = 0 r ( ) 1,i r ( ) 2,j if < 0 (2.16) 4 (a) (b) (c) > 0 = 0 < 0 r (+) 1,i r(+) 2,j r( ) 1,i r( ) 2,j (t i 1, t (+) i ] (s (+) j 1, s j] (t ( ) i 1, t i] (s j 1, s ( ) j ] t (+) i 1 t (+) i s j > m + c s (+) j 1 2 t i 1 s (+) j 1 > m+ c t ( ) i 1 1 s j 1 t ( ) i 1 > m c s( ) j 2 s ( ) j t i > m c > 0 t i 1 s j γ() m + c
10 (2.16) (a) > 0 (b) = 0 (c) < 0 E IJ [Y (±) (±),ij ] = γ() Y,ij i {Y (±),k (±) }N k=1 ˆγ() = 1 N (±) N (±) k=1 N (±) Y (±),k, N (±) (ˆγ() γ()) a N(0, ω 2 ). (2.17)
11 11 ω 2 ˆγ() ω2 = im N (±) E IJ[N (±) (ˆγ() γ()) 2 ] {Y (±),k k=1 ω 2 ˆω2 γ() = 0 γ() 0 N (±) }N (±) N (±) τ ˆγ() a () := N(0, 1) (2.18) ˆω (2.16) m + c m c (2.13) (2.14) Y,ij i {Y,k } N k=1 N N (±) E IJ [Y,k ] = 0 for a k E IJ [Y,k ] 0 for a k N m + c m c τ() N Ȳ,N a τ() := N(0, 1) (2.19) ˆσ [( ˆσ 2 σ2 = im ]) ) 2 ] 1/2 N E IJ (Ȳ,N E IJ [Ȳ,N N m + c m c X (±) (±) 1,,k Y,k N (±) (±) 1, N
12 (9:00 11:00) (12:30 15:10) 4 40 (16800 ) ( 1 ) 1 1 (16800 ) 1 7) ( ) ) ˆγ η (0) Reaized Voatiity (RV) (noise-to-signa ratio: NSR) = 0 (2.8) (2.11) 5% γ η (0) = 0 5% γ η (0) = 0 5% 7) 1 8)
13 13 1 ( ) ( ) (1911) (1944) (4078) (4527) (4528) (4536) (5449) (5702) (6141) (6222) (6457) (6594) (6645) (6804) (6839) (6929) (6963) (6981) (7309) (7744) (7947) (7974) (8219) (8367) (8545) (8806) (9045) (9665) (9783) (9832) ) (16800 ) 1 NSR NSR ˆγ η (0) RV
14 ( ) RV (1911) (1944) (4078) (4527) (4528) (4536) (5449) (5702) (6141) (6222) (6457) (6594) (6645) (6804) (6839) (6929) (6963) (6981) (7309) (7744) (7947) (7974) (8219) (8367) (8545) (8806) (9045) (9665) (9783) (9832) ) = 0 (2.8) (2.11) γ η (0) = 0 γ η (0) > 0 5% 3.84 RV previous tick interpoation RV 5 Andersen and Boersev (1998) RV RV
15 15 5 RV 5 5 (previous tick interpoation) 9) 22 NSR ) 8 NSR 22 NSR 8 NSR NSR 1 NSR NSR NSR i = β 0 + β 1 og(t ICKSIZE i ) + β 2 og(t RADESIZE i ) + ɛ i (3.20) NSR i T ICKSIZE i T RADESIZE i i NSR ) 3 NSR NSR 1 NSR 9) Barucci and Renó (2002) RV 10) Hansen and Lunde (2006) 30 NSR NSR Hansen and Lunde (2006) 11)
16 Constant og(t ICKSIZE) og(t RADESIZE) t ) NSR i = β 0 +β 1 og(t ICKSIZE i )+β 2 og(t RADESIZE i )+ɛ i NSR i T ICKSIZE i T RADESIZE i i 1 White (1980) 22 (2.12) γ η () = 0 5% (70 ) i.i.d. (2.5) (2.6) γ η (0) if = 0 E[X 1,,ij ] = 0 if 1 (3.21) 1 Ro (1984)
17 17 (2.3) ( ) (2.5) (2.6) E[X 1,0,ij ] = γ η ( t j ) γ η ( t j + t i ) γ η (0) + γ η ( t i ) (3.22) γ η (0) E[X 1,0,ij ]/γ η (0) = ( 1 ρ η ( t j ) ρ η ( t i ) + ρ η ( t j + t i ) ) (3.23) ρ η ( t j ) ρ η ( t j + t i ) ρ η ( t i ) 1 ρ η ( t j ) ρ η ( t i ) + ρ η ( t j + t i ) > 0 (3.23) 1 (2.6) (2.6) Ro (1984) Ro (1984) P (t) = V (t) + q(t)c, V (t) = V (t 1) + u(t) (3.24) P (t) V (t) q(t)c q(t) +1-1
18 (4078) (4528) (4536) (5449) (5702) (6141) (6457) (6594) (6645) (6929) (6981)
19 ) (2.8) (2.12) γη() = 0 γη() 0 5%
20 (7309) (7744) (7947) (7974) (8219) (8367) (8545) (9045) (9665) (9783) (9832)
21 ) (2.8) (2.12) γη() = 0 γη() 0 5%
22 (4528) 0.5 (6141) (6645) (6981) (7974) (8219) t ( Pr ( q(t) = +1 ) = Pr ( q(t) = 1 ) = 0.5) u(t) u(t) q(t) Ro (1984) ( k Corr[q(t), q(t k)] = φ(k) = 0) φ(k) φ(k) 1 c 2 (1 2φ(1) + φ(2)) if k = 1 Cov[ P (t), P (t k)] = c 2 (φ(k 1) 2φ(k) + φ(k + 1)) if k > 1 (3.25) φ(1) 0 1 k φ(k 1) φ(k) φ(k + 1) φ(k) 1 Ro (1984)
23 23 5 ˆρ η () = ˆγ η ()/ˆγ η (0) Hansen and Lunde (2006) 1 ( ) ( ) (9:05 10:55 12:35 15:05) m + c m c (2.19) (2.17) γ() (2.18) ( 1 2) : ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 8 γ() 2 δ(t ) 1 η(t) 1 η(t ) 2 δ(t) 6 7
24 ( 0 ),,,,,,,, (4528, 6594) (4528, 7974) (5449, 7974) (5702, 6981) (6141, 9783) (6981, 7974) (6981, 8545) (7947, 9832)
25 ) (2.18) γ() = 0 γ() 0 5% 2 δ(t ) 1 η(t)
26 ( 0 ),,,,,,,, (4528, 6594) (4528, 7974) (5449, 7974) (5702, 6981) (6141, 9783) (6981, 7974) (6981, 8545) (7947, 9832)
27 ) 1 η(t ) 2 δ(t)
28 (4528, 6594) (5449, 7974) (6141, 9783) (6981, 8545) (4528, 7974) (5702, 6981) (6981, 7974) (7947, 9832) ( ) : ( ) : ( ) : ( ) : ( ) : ( ) : ( ) : ( ) : ˆρ() = ˆγ()/ ˆγ η (0)ˆγ δ (0) ( ) ( ) (2.15) (ead-ag)
29 29 Chordia et a. (2005) 5 1 (order imbaance) Chordia et a. (2005) (noise-to-signa ratio: NSR) ( / ) NSR NSR NSR 22 22
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