11 a Escola de Séries Temporais e Econometria Analysis of High Frequency Financial Time Series: Methods, Models and Software

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1 11 a Escola de Séries Temporais e Economeria Analysis of High Frequency Financial Time Series: Mehods, Models and Sofware Eric Zivo Associae Professor and Gary Waerman Disinguished Scholar, Deparmen of Economics Adjunc Associae Professor, Deparmen of Finance Universiy of Washingon Augus 1, /8/ Lecure 1 Agenda Inroducion o high frequency daa Lecure 2 Realized variance measures: heory Lecure 3 Realized variance measures: empirical analysis 8/8/2005 2

2 Daa Sources Much of he published empirical analysis of RV has been based on high frequency daa from wo sources: Olsen and Associaes proprieary FX daa se for foreign exchange The NYSE Trades and Quoaion (TAQ) daa for equiy 8/8/ Olsen FX Daa Hisorical daa made available for use in hree conferences on he saisical analysis of high frequency daa: HFDF-1993, HFDF-1996, and HF The HFDF-2000 daa is he mos commonly used daa se spo exchange raes sampled every 5 minues for he $, DM, CHF, BP, Yen over he period December 1, 1986 hrough June 30, All inerbank bid/ask indicaive quoes for he exchange raes displayed on he Reuers FXFX screen. Highly liquid marke: observaions per day per currency Oulier filered log-price a each 5-minue ick is inerpolaed from he average of bid and ask quoes for he wo closes icks, and 5-minue cc reurn is difference in he log-price. 8/8/2005 4

3 Olsen FX Daa Daa cleaning prior o compuaion of RV measures: 5-minue reurn daa is resriced o eliminae nonrading periods, weekends, holidays, and lapses of he Reuers daa feed. The slow weekend period from Friday 21:05 GMT unil Sunday 21:00 GMT is eliminaed from he sample. Holidays removed: Chrismas (December 24-26), New Year's (December 31- January 2), July 4h, Good Friday, Easer Monday, Memorial Day, Labor Day, and Thanksgiving and he day afer. Days ha conain long srings of zero or consan reurns (caused by daa feed problems) are eliminaed. 8/8/ Empirical Analysis of FX Reurns Auhor Series AB 1998 DM/$, Y/$ AB 1998 DM/$, Y/$ ABDL 2000 DM/$, Y/$ ABDL 2001 DM/$, Y/$ ABDL 2003 DM/$, Y/$ ABDM 2005 DM/$, Y/$ BNS 2001 DM/$ BNS 2002 DM/$ Sample Days, T m , , , , ,449 various , /8/2005 6

4 Disribuion of RV ABDL (2001): The Disribuion of Realized Exchange Rae Volailiy, Journal of he American Saisical Associaion. BNS (2001): Esimaing Quadraic Variaion Using Realized Variance, Journal of Applied Economerics. 8/8/ Summary Saisics for Daily RV Measures, m=228 Non-Gaussian Gaussian 8/8/2005 8

5 Uncondiional Disribuions: m=288 Source: ABDL /8/ Uncondiional Disribuions: m=288 Source: ABDL /8/

6 Correlaion Marix for Daily RV Measures 8/8/ Correlaion-in-Volailiy Effec 8/8/ Source: ABDL (2001)

7 Accuracy of RV Measures: 95% CI from BNS Asympoic heory 8/8/ Source: BNS (2002) Time Series of Daily RVOL: m=228 8/8/ Source: ABDL (2001)

8 Time Series of Daily RCOR: m=228 Source: ABDL (2001) 8/8/ SACF of Daily RV Measures: m=228 Source: ABDL (2001) 8/8/

9 Long Memory Behavior of RV Measures A saionary process y has long memory, or long range dependence, if is auocorrelaion funcion decays slowly a a hyperbolic rae: α ρk Cρ ik, as k α (0,1) 8/8/ Fracionally Differenced Processes A long memory process y can be modeled paramerically by exending an inegraed process o a fracionally inegraed process: d (1 L) ( y µ ) = u, u ~ I(0) 0 < d < 0.5: saionary long memory 0.5 d < 1: nonsaionary long memory 8/8/

10 Esimaing d Nonparameric esimaion Geweke-Porer-Hudak (GPH) logperiodogram regression Local While esimaor Phillips-Kim modified GPH esimaor Andrews-Guggenberger biased correced GPH esimaor Parameric esimaion ARFIMA(p,d,q) model wih normal errors 8/8/ GPH Esimaed of d Noe: Mulivariae esimaion of common d using (RLVOL D, RLVOL Y, RLVOL DY ) is 0.4 8/8/

11 Temporal Aggregaion and Scaling Laws The fracional differencing parameer d is invarian under emporal aggregaion If x is fracionally inegraed wih parameer d hen var([ x ] ) [ x ] = h = c h h x 2d + 1 h j = 1 h( 1) + j ( ) ( ) ln var([ x ] ) 2d + 1 ln( h) h 8/8/ Temporal Aggregaion and Esimaed of d GPH Esimaes of d 8/8/

12 Temporal Aggregaion and Scaling Laws RV RLVOL Source: ABDL (2001) 8/8/ Disribuion of Reurns Sandardized by RV ABDL (2000): Exchange Rae Reurns Sandardized by Realized Volailiy Are (Nearly) Gaussian, Mulinaional Finance Journal 8/8/

13 Sochasic Volailiy Model Assume daily reurns r may be decomposed following a sandard condiional volailiy model r = σ ε σ = laen volailiy ε ~ iid (0,1) E[ r ] = σ 2 2 8/8/ Sandardized Reurns Compue reurns sandardized by esimaes of condiional volailiy r ˆ ε = ˆ σ ˆ σ = RVOL, m = 288 GARCH (1,1) ˆ σ = ˆ σ GARCH(1,1): σ = w+ αr + βσ /8/

14 Mulivariae Sandardized Reurns Sandardized reurns based RCOV ˆ ε D, r 1/2 D, RCOV ˆ ε = Y, r Y, RCOV 1/2 = Cholesky facor of RCOV 8/8/ Squared reurns Forecass of daily σ GARCH(1,1) RV, m=48 8/8/

15 Comparison of Volailiy Forecass Squared reurns are unbiased bu very noisy GARCH(1,1) forecass are smooher han RV forecass; do no uilize informaion beween ime -1 and (exponenially weighed average of pas reurns) RV forecass make exclusive use of informaion beween ime -1 and ; beer forecas of ime volailiy 8/8/ Summary Saisics 8/8/

16 Disribuion of Daily Reurns Source: ABDL (2000) 8/8/ Disribuion of Sandardized Reurns RV RCOV 8/8/ Source: ABDL (2000)

17 Scaerplo of Daily Reurns Source: ABDL (2000) 8/8/ Scaerplo or Sandardized Reurns RV RCOV Source: ABDL (2000) 8/8/

18 SACF of Squared Reurns RAW RV RCOV DM/$ Yen/$ DM/$, Yen/$ 8/8/ Conclusions Daily reurns sandardized by RV measures are nearly Gaussian Suppors diffusion model for reurns Alernaive o copula mehods for characerizing mulivariae disribuions Advanages for value-a-risk compuaion RV provides superior volailiy forecass 8/8/

19 Modeling and Forecasing RV ABDL (2003): Modeling and Forecasing Realized Volailiy, Economerica 8/8/ Tradiional Condiional Volailiy Models Normal GARCH(1,1) Log-Normal SV model r = σε, ε ~ iid N(0,1) σ = w+ αr + βσ r = σ ε, ε ~ iid N(0,1) lnσ = δ + φln σ + σ u, u ~ iid N(0,1) u E[ ε u] = 0 8/8/

20 Advanages of Using RV RV provides an observable esimae of laen volailiy Sandard ime series models (e.g. ARIMA) may be used o model and forecas RV Mulivariae ime series models may be used model and forecas RCOV, RCOR 8/8/ Trivariae Sysem of Exchange Raes RLVOLD/$, y = RLVOLY /$,, m= 48 RLVOL Y / D, 1 RCOV = RV + RV RV 2 ( ) D /$, Y /$ D/$, Y /$, Y / D, Fi models for y in sample: 12/1/86-12/1/96 Forecas y ou-of-sample: 12/2/96 6/30/99 8/8/

21 SACF of Daily DM/$ RLVOL: m= (1 L) ( y i µ i) Source: ABDL (2003) 8/8/ SACF of Daily Yen/$ RLVOL: m= (1 L) ( y i µ i) 8/8/2005 Source: ABDL (2003) 42

22 SACF of Daily Yen/DM RLVOL: m= (1 L) ( y i µ i ) Source: ABDL (2003) 8/8/ FI-VAR(5) Model (VAR-RV) Φ µ = ε 0.4 ( L)(1 L) ( y ) ε ~ iid N (0, Ω) Φ ( L) = I Φ L Φ L /8/

23 Alernaive Models VAR-ABS: VAR(5) fi o r AR-RV: univariae AR(5) fi o (1-L) 0.4 RLVOL i, Daily GARCH(1,1): normal-garch(1,1) fi o daily reurns r i, Daily RiskMerics: exponenially weighed moving average model for r i, ² wih λ=0.94 Daily FIEGARCH(1,1): univariae fracionally inegraed exponenial GARCH(1,1) fi o r i, Inra-day FIEGARCH deseason/filer: univariae fracionally inegraed exponenial GARCH(1,1) fi o 30- minue filered and deseasonalized reurns r i,+. 8/8/ Forecas Evaluaion RVOL = b + b RVOL ˆ + b RVOL ˆ + error VAR RV model i, 0 1 i, 2 i, RVOL ˆ = 1-day ahead forecas from RV-VAR VAR RV i, RVOL ˆ = 1-day ahead forecas from alernaive model model i, H : b = 0, b = 1, b = /8/

24 Findings RV-VAR is consisenly bes forecasing model in-sample and ou-of-sample: highes R 2 from forecas evaluaion regressions. Rarely rejec H 0 : b 0 =0, b 1 =1, b 2 =0 for RV- VAR model RV-AR is close o RV-VAR 8/8/ Forecass of Daily RVOL: VAR-RV vs. GARCH(1,1) 8/8/

25 NYSE TAQ Daa Inra-day rade and quoaion informaion for all securiies lised on NYSE, AMEX, and NASDAQ. The mos acive period for equiy markes is during he rading hours of he NYSE beween 9:30 a.m. EST unil 4:00 p.m. EST. No as liquid as FX markes 8/8/ NYSE TAQ Daa Equiy reurns are generally subjec o more pronounced marke microsrucure effecs (e.g., negaive firs order serial correlaion caused by bid-ask bounce effecs) han FX daa. As a resul, equiy reurns are ofen filered o remove hese microsrucure effecs prior o he consrucion of RV measures. A common filering mehod involves esimaing an MA(1) or AR(1) model o he reurns, and hen consrucing he filered reurns as he residuals from he esimaed model. 8/8/

26 Empirical Analysis of TAQ Daa Andersen, Bollerslev, Diebold, Ebens (2001): The Disribuion of Realized Sock Reurn Volailiy, Journal of Financial Economics Analyze 30 Dow Jones Indusrial Average Socks over he period 1/2/93 5/29/98 Resric analysis o NYSE exchange hours T=1,336; m=79 5-minue reurns 8/8/ Summary of Findings Resuls for equiy reurns are similar o hose for FX reurns RLVOL, RCOR are approximaely Gaussian RV measures exhibi long memory Daily reurns sandardized by RVOL are nearly Gaussian Lile evidence of leverage effec Evidence of facor srucure in mulivariae sysem of RV measures 8/8/

27 Disribuion of Daily RLVOL: Alcoa Solid line: RLVOL Dashed line: normal densiy Source: ABDE (2001) 8/8/ Disribuion of Daily RCOR: Alcoa,Exxon Solid line: RCOR Dashed line: normal densiy Source: ABDE (2001) 8/8/

28 Time Series of Daily RLVOL: Alcoa Source: ABDE (2001) 8/8/ Time Series of Daily RCOR: Alcoa, Exxon Source: ABDE (2001) 8/8/

29 Disribuion of Daily Sandardized Reurns for Alcoa Solid line: reurns/rvol Dashed line: normal densiy 8/8/ Evidence for Facor Srucure RLVOL Alcoa 8/8/2005 RLVOL Exxon 58

30 Evidence of Facor Srucure RCOR Alcoa,i RLVOL 8/8/2005 Alcoa 59 Evidence of Facor Srucure Average RCOR Alcoa,I i Alcoa, Exxon Average RCOR Exxon,I i Alcoa, Exxon 8/8/

31 Direcions for Fuure Research Coninued developmen of mehods for exploiing he volailiy informaion in highfrequency daa Volailiy modeling and forecasing in he high-dimensional mulivariae environmens of pracical financial economic relevance 8/8/