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174 NONLINEAR TIME SERIESPetruccelli, J., and Woolford, S. W. (1984), “A threshold AR(1) model,” Journal of AppliedProbability, 21, 270–286.Potter, S. M. (1995), “A nonlinear approach to U.S. GNP,” Journal of Applied Econometrics,10, 109–125.Priestley, M. B. (1980), “State-dependent models: a general approach to nonlinear time seriesanalysis,” Journal of Time Series Analysis,1,47–71.Priestley, M. B. (1988), Non-linear and Non-stationary Time Series Analysis, Academic Press:London.Rabemananjara, R., and Zakoian, J. M. (1993), “Threshold ARCH models and asymmetriesin volatility,” Journal of Applied Econometrics,8,31–49.Ramsey, J. B. (1969), “Tests for specification errors in classical linear least squares regressionanalysis,” Journal of the Royal Statistical Society, Series B, 31, 350–371.Ripley, B. D. (1993), “Statistical aspects of neural networks,” in Networks and Chaos—Statistical and Probablistic Aspects, eds O. E. Barndorff-Nielsen, J. L. Jensen, and W. S.Kendall, pp. 40–123. Chapman and Hall: London.Subba Rao, T., and Gabr, M. M. (1984), An Introduction to Bispectral Analysis and BilinearTime Series Models, Lecture Notes in Statistics, 24. Springer-Verlag: New York.Teräsvirta, T. (1994), “Specification, estimation, and evaluation of smooth transition autoregressivemodels,” Journal of the American Statistical Association, 89, 208–218.Tiao, G. C., and Tsay, R. S. (1994), “Some Advances in Nonlinear and Adaptive Modeling inTime Series,” Journal of Forecasting, 13, 109–131.Tong, H. (1978), “On a threshold model,” in Pattern Recognition and Signal Processing, ed.C.H. Chen, Sijhoff & Noordhoff: Amsterdam.Tong, H. (1983), Threshold Models in Nonlinear Time Series Analysis, Lecture Notes in Statistics,Springer-Verlag: New York.Tong, H. (1990), Non-Linear Time Series: A Dynamical System Approach, Oxford UniversityPress: Oxford.Tsay, R. S. (1986), “Nonlinearity tests for time series,” Biometrika, 73, 461–466.Tsay, R. S. (1989),“Testing and modeling threshold autoregressive processes,” Journal of theAmerican Statistical Association, 84, 231–240.Tsay, R. S. (1998), “Testing and modeling multivariate threshold models,” Journal of theAmerican Statistical Association, 93, 1188–1202.Venables, W. N., and Ripley, B. D. (1999), Modern Applied Statistics with S-Plus, 3rded.Springer-Verlag: New York.Watson, G. S. (1964), “Smooth regression analysis,” Sabkhya, Series A, 26, 359–372.Zakoian, J. M. (1994), “Threshold heteroscedastic models,” Journal of Economic Dynamicsand Control, 18, 931–955.
Analysis of Financial Time Series. Ruey S. TsayCopyright © 2002 John Wiley & Sons, Inc.ISBN: 0-471-41544-8CHAPTER 5High-Frequency Data Analysisand Market MicrostructureHigh-frequency data are observations taken at fine time intervals. In finance, theyoften mean observations taken daily or at a finer time scale. These data have becomeavailable primarily due to advances in data acquisition and processing techniques,and they have attracted much attention because they are important in empiricalstudy of market microstructure. The ultimate high-frequency data in finance are thetransaction-by-transaction or trade-by-trade data in security markets. Here time isoften measured in seconds. The Trades and Quotes (TAQ) database of the New YorkStock Exchange (NYSE) contains all equity transactions reported on the ConsolidatedTape from 1992 to present, which includes transactions on NYSE, AMEX,NASDAQ, and the regional exchanges. The Berkeley Options Data Base providessimilar data for options transactions from August 1976 to December 1996. Transactionsdata for many other securities and markets, both domestic and foreign,are continuously collected and processed. Wood (2000) provides some historicalperspective of high-frequency financial study.High-frequency financial data are important in studying a variety of issues relatedto trading process and market microstructure. They can be used to compare the efficiencyof different trading systems in price discovery (e.g., the open out-cry systemof NYSE and the computer trading system of NASDAQ). They can also be used tostudy the dynamics of bid and ask quotes of a particular stock (e.g., Hasbrouck, 1999;Zhang, Russell, and Tsay, 2001b). In an order-driven stock market (e.g., the TaiwanStock Exchange), high-frequency data can be used to study the order dynamic and,more interesting, to investigate the question “who provides the market liquidity.”Cho, Russell, Tiao, and Tsay (2000) use intraday 5-minute returns of more than 340stocks traded in the Taiwan Stock Exchange to study the impact of daily stock pricelimits and find significant evidence of magnet effects toward the price ceiling.However, high-frequency data have some unique characteristics that do not appearin lower frequencies. Analysis of these data thus introduces new challenges to financialeconomists and statisticians. In this chapter, we study these special characteristics,consider methods for analyzing high-frequency data, and discuss implications175
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Analysis of FinancialTime SeriesFin
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ContentsPrefacexi1. Financial Time
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CONTENTSix6.6 Black-Scholes Pricing
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PrefaceThis book grew out of an MBA
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Analysis of Financial Time Series.
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ASSET RETURNS 3Multiperiod Simple R
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ASSET RETURNS 5r t [k] =ln(1 + R t
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DISTRIBUTIONAL PROPERTIES OF RETURN
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10 FINANCIAL TIME SERIES AND THEIR
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REFERENCES 21Federal Reserve Bank o
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CORRELATION AND AUTOCORRELATION FUN
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CORRELATION AND AUTOCORRELATION FUN
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WHITE NOISE AND LINEAR TIME SERIES
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SIMPLE AUTOREGRESSIVE MODELS 29whic
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SIMPLE AUTOREGRESSIVE MODELS 31wher
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SIMPLE AUTOREGRESSIVE MODELS 33wher
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SIMPLE AUTOREGRESSIVE MODELS 35expa
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SIMPLE AUTOREGRESSIVE MODELS 37Tabl
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SIMPLE AUTOREGRESSIVE MODELS 39Mode
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SIMPLE AUTOREGRESSIVE MODELS 41The
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SIMPLE MOVING-AVERAGE MODELS 43wher
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SIMPLE MOVING-AVERAGE MODELS 45s-rt
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SIMPLE MOVING-AVERAGE MODELS 47The
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SIMPLE ARMA MODELS 49A time series
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SIMPLE ARMA MODELS 51operator, the
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SIMPLE ARMA MODELS 53Table 2.5. Sam
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SIMPLE ARMA MODELS 55This represent
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UNIT-ROOT NONSTATIONARITY 57ˆp h (
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UNIT-ROOT NONSTATIONARITY 59log-pri
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SEASONAL MODELS 61which is commonly
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SEASONAL MODELS 63Series : xSeries
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SEASONAL MODELS 65models use the sa
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• ••• •••••••
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REGRESSION MODELS WITH TIME SERIES
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REGRESSION MODELS WITH TIME SERIES
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LONG-MEMORY MODELS 73=(k − d −
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APPENDIX A: SOME SCA COMMANDS 75est
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EXERCISES 77relation. Draw your con
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STRUCTURE OF A MODEL 813.2 STRUCTUR
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THE ARCH MODEL 83corrected asset re
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THE ARCH MODEL 85Series : xACF0.0 0
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THE ARCH MODEL 87sample mean from t
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THE ARCH MODEL 89where Ɣ(x) is the
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THE ARCH MODEL 91Series : resiSerie
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THE GARCH MODEL 93other softwares a
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THE GARCH MODEL 95ahead forecast ca
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THE GARCH MODEL 97The fitted AR(3)
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THE GARCH MODEL 99Series : stdatACF
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THE GARCH-M MODEL 101two models. Th
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THE EXPONENTIAL GARCH MODEL 103To b
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THE EXPONENTIAL GARCH MODEL 105eros
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THE CHARMA MODEL 107ˆσ 2 864 (3)
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RANDOM COEFFICIENT AUTOREGRESSIVE M
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THE LONG-MEMORY STOCHASTIC VOLATILI
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AN ALTERNATIVE APPROACH 113where Va
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APPLICATION 115(a) IBMlog-rtn-30 -1
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APPLICATION 117equationThe fitted m
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KURTOSIS OF GARCH MODELS 119where K
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SOME RATS PROGRAMS FOR ESTIMATING V
Page 131 and 132: EXERCISES 123• Is there evidence
Page 133 and 134: REFERENCES 125Melino, A., and Turnb
Page 135 and 136: NONLINEAR TIME SERIES 127To put non
Page 137 and 138: NONLINEAR MODELS 129Example 4.1. Co
Page 139 and 140: NONLINEAR MODELS 131jumps when it b
Page 141 and 142: NONLINEAR MODELS 133the series and
Page 143 and 144: NONLINEAR MODELS 135els to the mont
Page 145 and 146: NONLINEAR MODELS 137growth-2 -1 0 1
Page 147 and 148: NONLINEAR MODELS 139average of y t
Page 149 and 150: NONLINEAR MODELS 141indicating that
Page 151 and 152: NONLINEAR MODELS 143s T,2t=1T∑K h
Page 153 and 154: NONLINEAR MODELS 145f i (.) of Eq.
Page 155 and 156: NONLINEAR MODELS 1474.1.9.1 Feed-Fo
Page 157 and 158: NONLINEAR MODELS 1494.1.9.2 Trainin
Page 159 and 160: NONLINEAR MODELS 151prob0.0 0.2 0.4
Page 161 and 162: NONLINEARITY TESTS 153idea behind v
Page 163 and 164: NONLINEARITY TESTS 155C l (δ, T )
Page 165 and 166: NONLINEARITY TESTS 157and obtain th
Page 167 and 168: NONLINEARITY TESTS 159• Step 2: C
Page 169 and 170: FORECASTING 161returns. In summary,
Page 171 and 172: FORECASTING 163in m 11 and m 22 ind
Page 173 and 174: APPLICATION 165unem. rate4 6 8 10
Page 175 and 176: APPLICATION 167Table 4.4. Out-of-Sa
Page 177 and 178: APPENDIX A 169compute a0 = 0.1, a1
Page 179 and 180: REFERENCES 171P(s t = 2 | s t−1 =
Page 181: REFERENCES 173Fan, J. (1993), “Lo
Page 185 and 186: NONSYNCHRONOUS TRADING 177t − k t
Page 187 and 188: BID-ASK SPREAD 179The lag-1 autocov
Page 189 and 190: EMPIRICAL CHARACTERISTICS 181The ma
Page 191 and 192: n(trades)0 20406080 1200 1000 2000
Page 193 and 194: EMPIRICAL CHARACTERISTICS 185after-
Page 195 and 196: MODELS FOR PRICE CHANGES 187deep un
Page 197 and 198: MODELS FOR PRICE CHANGES 189σ 2i (
Page 199 and 200: MODELS FOR PRICE CHANGES 191A i = 1
Page 201 and 202: MODELS FOR PRICE CHANGES 193( )piln
Page 203 and 204: DURATION MODELS 195For the IBM data
Page 205 and 206: DURATION MODELS 197(a) is the avera
Page 207 and 208: DURATION MODELS 199actions might no
Page 209 and 210: DURATION MODELS 201Series : xACF0.0
Page 211 and 212: DURATION MODELS 203reduces to that
Page 213 and 214: DURATION MODELS 205Series : xACF0.0
Page 215 and 216: THE PCD MODEL 207where w(α) denote
Page 217 and 218: THE PCD MODEL 209(a) All transactio
Page 219 and 220: THE PCD MODEL 2110 10 20 30 40 500
Page 221 and 222: THE PCD MODEL 213⎧⎪⎨1f (x |
Page 223 and 224: THE PCD MODEL 215Generalized Gamma
Page 225 and 226: THE PCD MODEL 217smpl 2 3534compute
Page 227 and 228: REFERENCES 219(a) Use the data to c
Page 229 and 230: Analysis of Financial Time Series.
Page 231 and 232: STOCHASTIC PROCESSES 223write a con
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STOCHASTIC PROCESSES 2253. stationa
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ITO’S LEMMA 2276.3.2 Stochastic D
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ITO’S LEMMA 229This result shows
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DISTRIBUTIONS OF PRICE AND RETURN 2
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BLACK-SCHOLES EQUATION 233and G t =
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BLACK-SCHOLES FORMULA 235risk prefe
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BLACK-SCHOLES FORMULA 237The stock
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BLACK-SCHOLES FORMULA 239(a) Call o
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AN EXTENSION OF ITO’S LEMMA 2410
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STOCHASTIC INTEGRAL 243using the It
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JUMP DIFFUSION MODELS 245where w t
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JUMP DIFFUSION MODELS 247where it i
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JUMP DIFFUSION MODELS 249sudden jum
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APPENDIX A 251Pricing formulas for
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EXERCISES 253which involves the CDF
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REFERENCES 255Bakshi, G., Cao, C.,
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VALUEATRISK 257log return-0.2 -0.1
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RISKMETRICS 259we use log returns r
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RISKMETRICS 261investor is$10,000,0
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ECONOMETRIC APPROACH TO VAR 263Cons
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ECONOMETRIC APPROACH TO VAR 265The
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QUANTILE ESTIMATION 267Using the fo
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QUANTILE ESTIMATION 269ˆx 0.01 = p
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EXTREME VALUE THEORY 271= 1 −= 1
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EXTREME VALUE THEORY 273shows that
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EXTREME VALUE THEORY 275{ [] }r n(i
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EXTREME VALUE THEORY 2777.5.3 Appli
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EXTREME VALUE TO VAR 2797.6 AN EXTR
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EXTREME VALUE TO VAR 281VaR =−2.5
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EXTREME VALUE TO VAR 283stock. The
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A NEW APPROACH TO VAR 285series (e.
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A NEW APPROACH TO VAR 287In Eq. (7.
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A NEW APPROACH TO VAR 289Example 7.
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A NEW APPROACH TO VAR 291feature is
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A NEW APPROACH TO VAR 293(a) Homoge
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A NEW APPROACH TO VAR 295Table 7.4.
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REFERENCES 297a Gaussian GARCH(1, 1
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CROSS-CORRELATION 301correlation co
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CROSS-CORRELATION 303where ̂D is t
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CROSS-CORRELATION 305Table 8.1. Sum
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30yr20yr10yr5yr1yr-10 0 510-10 0 51
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VAR MODELS 3098.2 VECTOR AUTOREGRES
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VAR MODELS 311where G = Cov(b t ).
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VAR MODELS 313where φ 0 and a t ar
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VAR MODELS 315To specify the order
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VAR MODELS 317the S&P 500 index. Th
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VECTOR MA MODELS 319[ ] [ ] [ ] [r1
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VECTOR MA MODELS 321turn used to co
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VECTOR ARMA MODELS 323For the VAR(1
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VECTOR ARMA MODELS 325log-rate0.0 0
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VECTOR ARMA MODELS 327interest-rate
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CO-INTEGRATION 329Series : x1ACF0.0
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CO-INTEGRATION 331Stock Exchange; o
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THRESHOLD CO-INTEGRATION 333ously b
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PRINCIPAL COMPONENT ANALYSIS 3358.6
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PRINCIPAL COMPONENT ANALYSIS 337(c
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PRINCIPAL COMPONENT ANALYSIS 339(a)
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FACTOR ANALYSIS 341(a) 5 stock retu
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FACTOR ANALYSIS 343andCov(r, F) = E
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FACTOR ANALYSIS 345matrix P to tran
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FACTOR ANALYSIS 347Example 8.9. In
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APPENDIX A. REVIEW OF VECTORS AND M
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APPENDIX A. REVIEW OF VECTORS AND M
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APPENDIX B. MULTIVARIATE NORMAL DIS
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REFERENCES 355(b) Build a bivariate
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REPARAMETERIZATION 359To illustrate
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REPARAMETERIZATION 361where ⊥ den
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GARCH MODELS FOR BIVARIATE RETURNS
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VECTOR VOLATILITY MODELS 377using t
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VECTOR VOLATILITY MODELS 379large-c
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VECTOR VOLATILITY MODELS 381(a) S&P
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FACTOR-VOLATILITY MODELS 383conside
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APPLICATION 385[ ]σ11,t=σ 22,t⎡
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MULTIVARIATE t DISTRIBUTION 387σ 1
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APPENDIX A. SOME REMARKS ON ESTIMAT
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APPENDIX A. SOME REMARKS ON ESTIMAT
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REFERENCES 393(a) Compute the sampl
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GIBBS SAMPLING 397mean adding auxil
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BAYESIAN INFERENCE 399distributions
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BAYESIAN INFERENCE 401normal with m
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ALTERNATIVE ALGORITHMS 403distribut
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ALTERNATIVE ALGORITHMS 40510.4.2 Me
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REGRESSION WITH SERIAL CORRELATIONS
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REGRESSION WITH SERIAL CORRELATIONS
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MISSING VALUES AND OUTLIERS 411y h
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MISSING VALUES AND OUTLIERS 413Cons
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MISSING VALUES AND OUTLIERS 415we h
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MISSING VALUES AND OUTLIERS 417c3t-
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STOCHASTIC VOLATILITY MODELS 419tha
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STOCHASTIC VOLATILITY MODELS 421whe
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STOCHASTIC VOLATILITY MODELS 423den
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STOCHASTIC VOLATILITY MODELS 425The
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STOCHASTIC VOLATILITY MODELS 427(a)
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•••MARKOV SWITCHING MODELS 42
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MARKOV SWITCHING MODELS 431(a) GARC
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MARKOV SWITCHING MODELS 433β i ∼
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MARKOV SWITCHING MODELS 435(a) mont
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MARKOV SWITCHING MODELS 437α ij ar
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FORECASTING 439garchrtn-sq0 200 400
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EXERCISES 441eter uncertainty in pr
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REFERENCES 443Justel, A., Peña, D.
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446 INDEXData (cont.)equal-weighted
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448 INDEXPoisson process, 244inhomo
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