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2 5 Lo and MacKinlay (1990a) de Jong and Nijman (1997) Cohen et al. (1983) Lo and MacKinlay (1990a b) Cohen et al. (1983) de Jong and Nijman (1997) de Jong and Nijman (1997) Parzen (1963) Cohen et al. (1983) Robinson (1985) Lo and Mackinlay (1990a b) Conley et al. (1995) Ghysels et al. (1995)
3 ( 1a-1c, 1a-1c, 2a-2c)
4 3 3 1a-1c 2a-2c ( 3a-3c) 3 6 3a-3c ( 2a-2c, 4a-4c, 5a-5c) 2a-2c 4a-4c a-5c
5 ( 6a-6c) 6a-6c de Jong and Nijman (1997) de Jong and Nijman (1997) 2 2 p t q t p t i q t j p t 2 p ti+1 p ti = t i+1 t=t i +1 Δp t (1) 5
6 Δp t = p t p t 1 (1) t i i 2 y ij (p ti+1 p ti )(q tj+1 q tj )= t i+1 t=t i +1 t j+1 Δp t s=t j +1 Δq s (2) (deterministic) 2 E[y ij ]=E[ t i+1 t=t i +1 t j+1 Δp t t=t j +1 Δq s ]= t i+1 t j+1 t=t i +1 s=t j +1 γ t s (3) γ k = Cov[Δp t, Δq t k ]=E[Δp t Δq t k ] Δq t k = q t k q t k 1 3 γ k x ij (k) x ij (k) =max(0,min(t i+1,t j+1 + k) max(t i,t j + k)) (4) x ij (k) x ij (k) y ij E[y ij x ij ]= K k= K x ij (k)γ k (5) K -K 5 2K+1 γ k γ x ij (2K+1 x ij (k) y ij x ijγ + e ij (6) γ y ij x ij 2 y ij 1 γ γ k = γ k x ij Δp t Δq t γ k γ k ρ k = ( γ p (7) 0 γq 0) 1/2 γ p 0 γ q 0 Δp t Δq t γ p 0 γ q 0 t 3.1 6
7 7a-7c 3 K) 5 10 K=6,1 K=15,de Jong and Nijman (1997) S & P 500 K 5 10 K=8,10, 1 K=20,25 ( 7a) k= k=1 5 de Jong and Nijman (1997) S & P 500 k=1 bid-ask bounce) k= ( 7b) k=1 1 k= t k=14 t 10 ( 7c) 7
8 k= k= k= a-8c 3 K) 5 10 K=6,1 K= K=8,10, 1 K=20,25 ρ k = Cov[Δp t, Δq t k ]/(γ0γ p 0) q 1/2 p q k k k k 8b 8c 2 2 ( 8a) 5 k= k=0 t 10 k= t k=
9 5 10 k k= k=0 10% ( 8b) 5 k= 1 0 k= 1 t t k=0 10 k=, 1 0, 6 k= k= k= k= ( 8c) 5 k= 4, 1,0,1 t k= 0 10 k= 4,0,1 k=0 k=1 10 k= k=1 5 1 k= 9, 8, 2, 8 k= 9, 2 k= 8,
10 k=0 1 k= 2 5 k= 1 1 k= 9 1 k= 8 1 k=8 5 k=1 1 k= (2008), ρ k = γ k ρ (γ p k = γ k 0 γq 0 )1/2 ( γ 0 γq p 0 )1/2 γ 0, p γ q 0 γ k γ k γ k γ p 0 γ p 0 White (1980) γ q 0 γ q 0 y ij x ij (k) y X 6 y = Xγ + e (8) 2 1 y p ij (p ti+1 p ti )(p tj+1 p tj ) (9) 10
11 y q ij (q ti+1 q ti )(q tj+1 q tj ) (10) y p ij y q ij 8 y p = X p γ p + e p (11) y q = X q γ q + e q (12) y p X p y p ij y p ij x p ij(k) y q X q y q ij y q ij x q ij(k) ( γ γ) AV [( γ γ)] = (X X) 1 X E[ee ]X(X X) 1 (13) AV [ ] ( γ p γ p ) ( γ q γ q ) AV [( γ p γ p )] = (X p X p ) 1 X p E[e p e p ]X p (X p X p ) 1 (14) AV [( γ q γ q )] = (X q X q ) 1 X q E[e q e q ]X q (X q X q ) 1 (15) ( γ γ) ( γ p γ p ) ACOV [( γ γ), ( γ p γ p )] = (X X) 1 X E[ee p ]X p (X p X p ) 1, (16) ( γ γ) ( γ q γ q ) ACOV [( γ γ), ( γ q γ q )] = (X X) 1 X E[ee q ]X q (X q X q ) 1, (17) ( γ p γ p ) ( γ q γ q ) ACOV [( γ p γ p ), ( γ q γ q )] =(X p X p ) 1 X p E[e p e q ]X q (X q X q ) 1, (18) 16 γ γ γ p γ p ( γ γ) ( γ p γ p ) γ k γ k γ p 0 γ p Ω γ q 0 γ q 0 ( ρ k ρ k ) delta method ( Hayashi (2000) ) ρ k γ k,γ0,γ p q 0 ρ k f(γ k,γ0,γ p 0) q ρ k γ k,γ0,γ p q 0 11
12 F (γ k,γ0,γ p 0) q ( F (γ k,γ0,γ p 0) q ρk, ρ k γ k γ p, ρ ) k 0 γ q 0 ( ) 1 = (γ0γ p 0) q, 1 γ k 1/2 2 (γ0) p 3/2 (γ0) q, 1 γ k 1/2 2 (γ0) p 1/2 (γ0) q 3/2 (19) ( ρ k ρ k ) AV [( ρ k ρ k )] = F (γ k,γ0,γ p 0)ΩF q (γ k,γ0,γ p 0) q (20) ( ρ k ρ k ) ÂV [( ρ k ρ k )] = F ( γ k, γ 0, p γ 0) q ΩF ( γ k, γ 0, p γ 0) q (21) ρ k , Cohen, K., Hawawimi, G., Maier, S., Schwartz, R., Whitcomb, D., Friction in the trading process and the estimation of systematic risk. Joumal of Financial Economics 12, Conley, T., Hansen, L.P., Luttmer, E., Scheinkman, J., Short term interest rates as subordinated diffusions. Unpublished working paper. de Jong, F., Nijman, T., High frequency analysis of lead-lag relationships between financial markets, Journal of Empirical Finance 4, Ghysels, E., Gourieroux. C., Jasiak, J., Market time and asset price movements: Theory and estimation. Discussion paper CIRANO and CREST. Hayashi, F., Econometrics, Princeton University Press, Princeton. 12
13 Lo, A., MacKinlay, A.C., 1990a. An econometric analysis of infrequent trading. Journal of Econometrics 45, Lo, A., MacKinlay, A.C., 1990b. When are contrarian profits due to stock market overreaction?. Review of Financial Studies 3, Parzen, E., On spectral analysis with missing observations and amplitude modulation. Shankya, series A 25, Robinson, P.M., Testing for serial correlation in regression with missing observations. Joumal of the Royal Statistical Society B 47, White, H., 1980, A hetoroskedasticity-consistent covariance matrix estimator and a direct test for heteroskedasticity, Econometrica 48, a
14 1b c
15 2a :00 9:30 10:00 10:30 12:30 13:00 13:30 14:00 14:30 15: No. 1 No. 2 No. 3 (a) :00 9:30 10:00 10:30 12:30 13:00 13:30 14:00 14:30 15: No. 1 No. 2 No. 3 No. 4 No. 5 No. 6 (b)
16 2b :00 9:30 10:00 10:30 12:30 13:00 13:30 14:00 14:30 15: No. 1 No. 2 No. 3 (a) :00 9:30 10:00 10:30 12:30 13:00 13:30 14:00 14:30 15: No. 1 No. 2 No. 3 No. 4 No. 5 No. 6 (b)
17 2c :00 9:30 10:00 10:30 12:30 13:00 13:30 14:00 14:30 15: No. 1 No. 2 No. 3 (a) :00 9:30 10:00 10:30 12:30 13:00 13:30 14:00 14:30 15: No. 1 No. 2 No. 3 No. 4 No. 5 No. 6 (b)
18 1a b c
19 2a :% No No No No No No b :% No No No No No No c :% No No No No No No
20 3a (h:m:s) No. 1 9 : : : : No. 2 9 : : : : No. 3 9 : : : : No. 4 9 : : : : No. 5 9 : : : : No. 6 9 : : : : b (h:m:s) No. 1 9 : : : : No. 2 9 : : : : No. 3 9 : : : : No. 4 9 : : : : No. 5 9 : : : : No. 6 9 : : : : c (h:m:s) No. 1 9 : : : : No. 2 9 : : : : No. 3 9 : : : : No. 4 9 : : : : No. 5 9 : : : : No. 6 9 : : : :
21 4a 30 9:00 9:30 10:00 10:30 12:30 13:00 13:30 14:00 14:30 15:00 No No No No No No b 30 9:00 9:30 10:00 10:30 12:30 13:00 13:30 14:00 14:30 15:00 No No No No No No c 30 9:00 9:30 10:00 10:30 12:30 13:00 13:30 14:00 14:30 15:00 No No No No No No
22 5a 30 9:00 9:30 10:00 10:30 12:30 13:00 13:30 14:00 14:30 15:00 No No No No No No b 30 9:00 9:30 10:00 10:30 12:30 13:00 13:30 14:00 14:30 15:00 No No No No No No c 30 9:00 9:30 10:00 10:30 12:30 13:00 13:30 14:00 14:30 15:00 No No No No No No
23 6a 30 x1,000 9:00 9:30 10:00 10:30 12:30 13:00 13:30 14:00 14:30 15:00 No No No No No No :00-9: b 30 x1,000 9:00 9:30 10:00 10:30 12:30 13:00 13:30 14:00 14:30 15:00 No No No No No No :00-9: c 30 x1,000 9:00 9:30 10:00 10:30 12:30 13:00 13:30 14:00 14:30 15:00 No No No No No No :00-9:
24 7a Lag 10 min 5 min 1min ( 2.724) ( 9.200) (17.871) (-1.051) ( 1.134) (-7.081) (-0.337) (-1.397) ( 1.077) (-1.825) ( 1.328) ( 1.264) (-0.356) (-0.584) ( 0.871) ( 0.221) (-0.753) (-1.301) ( 0.936) ( 0.590) (-0.276) ( 1.408) (-1.844) (-0.095) ( 1.853) (-1.128) (-0.343) (-0.303) ( 1.000) (-0.799) heteroskedasticity and autocorrelation consistent 24
25 7b Lag 10 min 5 min 1min ( 7.274) ( 7.801) (20.705) (-0.893) (-3.662) (-8.358) (-0.682) (-0.816) ( 2.724) (-0.491) ( 1.334) (-3.147) ( 0.384) (-1.200) (-0.475) ( 0.248) (-0.230) (-1.401) ( 0.313) ( 1.487) (-0.349) (-0.082) (-1.230) (-0.258) ( 1.070) (-0.106) (-0.765) (-0.774) ( 1.936) ( 0.217) heteroskedasticity and autocorrelation consistent 25
26 7c Lag 10 min 5 min 1min ( 9.911) ( 9.731) (18.457) ( 1.664) ( 1.994) (-2.467) (-0.533) ( 1.150) (-2.483) (-0.509) (-0.606) (-0.417) (-0.087) ( 0.110) ( 0.487) ( 0.063) (-0.505) ( 1.009) (-0.154) ( 0.047) (-0.598) (-0.989) (-1.053) (-2.193) ( 0.090) ( 0.052) ( 1.513) (-0.713) (-0.490) ( 0.430) heteroskedasticity and autocorrelation consistent 26
27 8a Lag 10 min 5 min 1min ( 0.216) (-0.353) (-0.721) ( 1.160) (-1.038) ( 0.886) ( 1.201) ( 1.019) (-0.292) ( 1.001) (-0.913) ( 2.654) ( 0.175) ( 0.146) ( 2.898) (-0.246) ( 1.082) ( 2.297) (-1.453) ( 0.715) ( 0.900) ( 1.546) (-0.168) ( 5.293) (-0.517) ( 6.839) ( 0.255) ( 4.317) ( 7.770) ( 0.945) ( 2.192) ( 4.939) (-0.876) (-0.893) ( 1.761) (-0.417) (-0.039) (-0.541) (-0.306) (-0.642) (-0.982) (-1.383) (-0.865) ( 0.696) (-1.661) ( 0.446) ( 1.107) ( 0.998) ( 0.422) (-1.733) (-1.044) ( 0.985) ( 1.951) (-1.652) ( 1.878) (-1.905) ( 0.310) heteroskedasticity and autocorrelation consistent 27
28 8b Lag 10 min 5 min 1min (-1.535) (-0.047) ( 1.268) ( 0.330) ( 1.482) (-0.032) (-1.589) ( 0.267) (-0.742) ( 0.771) ( 0.421) (-1.411) ( 0.055) (-1.680) (-0.144) ( 0.151) ( 1.592) (-3.614) (-1.972) ( 1.046) ( 0.558) (-0.309) (-1.539) (-3.211) ( 2.850) (-2.205) (-0.907) (-4.074) (-8.007) (-3.641) ( 1.044) (-0.502) ( 2.009) (-1.540) (-0.126) (-0.635) ( 1.045) (-0.886) (-1.503) ( 0.276) (-0.446) ( 1.259) (-0.648) (-0.053) (-1.678) ( 2.139) ( 0.597) ( 0.922) ( 0.254) ( 1.706) (-1.224) ( 0.402) ( 1.192) (-0.946) (-0.940) ( 0.911) (-1.012) heteroskedasticity and autocorrelation consistent 28
29 8c Lag 10 min 5 min 1min ( 0.591) ( 0.975) ( 0.084) (-1.057) ( 0.914) ( 0.434) (-2.175) ( 2.696) (-0.977) (-0.250) ( 1.527) ( 0.172) ( 0.399) ( 1.196) (-0.417) (-2.035) (-1.974) (-0.019) ( 0.880) ( 1.379) (-0.377) (-0.286) (-1.699) (-2.630) (-0.628) (-4.590) ( 0.665) (-3.569) (-6.523) (-0.468) ( 4.397) (-4.084) (-0.222) ( 1.005) ( 0.509) ( 0.214) ( 0.623) (-0.632) ( 1.667) (-0.259) ( 1.392) ( 1.013) (-1.004) ( 0.327) ( 1.487) ( 0.800) ( 1.404) ( 1.509) (-0.039) ( 2.935) ( 0.728) (-0.527) (-0.474) (-1.252) ( 1.203) ( 1.832) (-1.059) heteroskedasticity and autocorrelation consistent 29
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