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442 MCMC METHODS4. Consider the monthly log returns of General Motors stock from 1950 to 1999 with600 observations: (a) build a GARCH model for the series, (b) build a stochasticvolatility model for the series, and (c) compare and discuss the two volatilitymodels.5. Build a stochastic volatility model for the daily log return of Cisco Systems stockfrom January 1991 to December 1999. You may download the data from CRSPdatabase or the file “d-csco9199.dat.” Use the model to obtain a predictive distributionfor 1-step ahead volatility forecast at the forecast origin December 1999.Finally, use the predictive distribution to compute the Value at Risk of a longposition worth $1 million with probability 0.01 for the next trading day.6. Build a bivariate stochastic volatility model for the monthly log returns of GeneralMotors stock and the S&P 500 index for the sample period from January 1950 toDecember 1999. Discuss the relationship between the two volatility processes andcompute the time-varying beta for GM stock.REFERENCESBox, G. E. P., and Tiao, G. C. (1973), Bayesian Inference in Statistical Analysis. Addison-Wesley: Reading, MA.Chang, I., Tiao, G. C., and Chen, C. (1988), “Estimation of time series parameters in thepresence of outliers,” Technometrics, 30, 193–204.Carlin, B. P., and Louis, T. A. (2000), Bayes and Empirical Bayes Methods for Data Analysis,2nd ed., Chapman and Hall: London.DeGroot, M. H. (1970), Optimal Statistical Decisions, McGraw-Hill: New York.Dempster, A. P., Laird, N. M., and Rubin, D. B. (1977), “Maximum likelihood from incompletedata via the EM algorithm” (with discussion), Journal of the Royal Statistical Society,Series B, 39, 1–38.Elerian, O., Chib, S., and Shephard, N. (2001), “Likelihood inference for discretely observednonlinear diffusions,” Econometrica, 69, 959–993.Eraker, B. (2001), “Markov Chain Monte Carlo analysis of diffusion models with applicationto finance,” Journal of Business & Economic Statistics 19, 177–191.Gelfand, A. E., and Smith, A. F. M. (1990), “Sampling-based approaches to calculatingmarginal densities,” Journal of the American Statistical Association, 85, 398–409.Gelman, A., Carlin, J. B., Stern, H. S., and Rubin, D. B. (1995), Bayesian Data Analysis, CRCPress: London.Geman, S., and Geman, D. (1984), “Stochastic relaxation, Gibbs distributions, and theBayesian restoration of images,” IEEE transactions on Pattern Analysis and MachineIntelligence, 6, 721–741.Hasting, W. K. (1970), “Monte Carlo sampling methods using Markov chains and their applications,”Biometrika, 57,97–109.Jacquier, E., Polson, N. G., and Rossi, P. E. (1994), “Bayesian analysis of stochastic volatilitymodels” (with discussion), Journal of Business & Economic Statistics, 12, 371–417.Jones, R. H.(1980), “Maximum likelihood fitting of ARMA models to time series with missingobservations,” Technometrics, 22, 389–395.
REFERENCES 443Justel, A., Peña, D., and Tsay, R. S. (2001), “Detection of outlier patches in autoregressivetime series,” Statistica Sinica, 11 (to appear).Liu, J., Wong, W. H., and Kong, A. (1994), “Correlation structure and convergence rate ofthe Gibbs samplers I: Applications to the comparison of estimators and augmentationschemes,” Biometrika, 81,27–40.McCulloch, R. E., and Tsay, R. S. (1994), “Bayesian analysis of autoregressive time series viathe Gibbs sampler,” Journal of Time Series Analysis, 15, 235–250.McCulloch, R. E., and Tsay, R. S. (1994), “Statistical analysis of economic time series viaMarkov switching models,” Journal of Time Series Analysis, 15, 523–539.Metropolis, N., and Ulam, S. (1949), “The Monte Carlo method,” Journal of the AmericanStatistical Association, 44, 335–341.Metropolis, N., Rosenbluth, A. W., Rosenbluth, M. N., Teller, A. H., and Teller, E. (1953),“Equation of state calculations by fast computing machines,” Journal of Chemical Physics,21, 1087–1092.Tanner, M. A. (1996), Tools for Statistical Inference: Methods for the exploration of posteriordistributions and likelihood functions, 3rd ed., Springer-Verlag: New York.Tanner, M. A., and Wong, W. H. (1987), “The calculation of posterior distributions by dataaugmentation” (with discussion), Journal of the American Statistical Association, 82, 528–550.Tierney, L. (1994), “Markov chains for exploring posterior distributions” (with discussion),Annals of Statistics, 22, 1701–1762.Tsay, R. S. (1988), “Outliers, level shifts, and variance changes in time series,” Journal ofForecasting, 7,1–20.Tsay, R. S., Peña, D., and Pankratz, A. (2000), “Outliers in multivariate time series,”Biometrika, 87, 789–804.Zhang, M. Y., Russell, J. R. and Tsay, R. S. (2000), “Determinants of bid and ask quotes andimplications for the cost of trading,” Working paper, Statistics Research Center, GraduateSchool of Business, University of Chicago.
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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
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EXERCISES 123• Is there evidence
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REFERENCES 125Melino, A., and Turnb
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NONLINEAR TIME SERIES 127To put non
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NONLINEAR MODELS 129Example 4.1. Co
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NONLINEAR MODELS 131jumps when it b
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NONLINEAR MODELS 133the series and
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NONLINEAR MODELS 135els to the mont
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NONLINEAR MODELS 137growth-2 -1 0 1
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NONLINEAR MODELS 139average of y t
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NONLINEAR MODELS 141indicating that
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NONLINEAR MODELS 143s T,2t=1T∑K h
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NONLINEAR MODELS 145f i (.) of Eq.
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NONLINEAR MODELS 1474.1.9.1 Feed-Fo
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NONLINEAR MODELS 1494.1.9.2 Trainin
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NONLINEAR MODELS 151prob0.0 0.2 0.4
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NONLINEARITY TESTS 153idea behind v
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NONLINEARITY TESTS 155C l (δ, T )
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NONLINEARITY TESTS 157and obtain th
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NONLINEARITY TESTS 159• Step 2: C
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FORECASTING 161returns. In summary,
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FORECASTING 163in m 11 and m 22 ind
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APPLICATION 165unem. rate4 6 8 10
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APPLICATION 167Table 4.4. Out-of-Sa
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APPENDIX A 169compute a0 = 0.1, a1
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REFERENCES 171P(s t = 2 | s t−1 =
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REFERENCES 173Fan, J. (1993), “Lo
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NONSYNCHRONOUS TRADING 177t − k t
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BID-ASK SPREAD 179The lag-1 autocov
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EMPIRICAL CHARACTERISTICS 181The ma
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n(trades)0 20406080 1200 1000 2000
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EMPIRICAL CHARACTERISTICS 185after-
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MODELS FOR PRICE CHANGES 187deep un
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MODELS FOR PRICE CHANGES 189σ 2i (
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MODELS FOR PRICE CHANGES 191A i = 1
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MODELS FOR PRICE CHANGES 193( )piln
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DURATION MODELS 195For the IBM data
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DURATION MODELS 197(a) is the avera
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DURATION MODELS 199actions might no
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DURATION MODELS 201Series : xACF0.0
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DURATION MODELS 203reduces to that
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DURATION MODELS 205Series : xACF0.0
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THE PCD MODEL 207where w(α) denote
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THE PCD MODEL 209(a) All transactio
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THE PCD MODEL 2110 10 20 30 40 500
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THE PCD MODEL 213⎧⎪⎨1f (x |
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THE PCD MODEL 215Generalized Gamma
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THE PCD MODEL 217smpl 2 3534compute
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REFERENCES 219(a) Use the data to c
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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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Page 401 and 402: REFERENCES 393(a) Compute the sampl
Page 403 and 404: Analysis of Financial Time Series.
Page 405 and 406: GIBBS SAMPLING 397mean adding auxil
Page 407 and 408: BAYESIAN INFERENCE 399distributions
Page 409 and 410: BAYESIAN INFERENCE 401normal with m
Page 411 and 412: ALTERNATIVE ALGORITHMS 403distribut
Page 413 and 414: ALTERNATIVE ALGORITHMS 40510.4.2 Me
Page 415 and 416: REGRESSION WITH SERIAL CORRELATIONS
Page 417 and 418: REGRESSION WITH SERIAL CORRELATIONS
Page 419 and 420: MISSING VALUES AND OUTLIERS 411y h
Page 421 and 422: MISSING VALUES AND OUTLIERS 413Cons
Page 423 and 424: MISSING VALUES AND OUTLIERS 415we h
Page 425 and 426: MISSING VALUES AND OUTLIERS 417c3t-
Page 427 and 428: STOCHASTIC VOLATILITY MODELS 419tha
Page 429 and 430: STOCHASTIC VOLATILITY MODELS 421whe
Page 431 and 432: STOCHASTIC VOLATILITY MODELS 423den
Page 433 and 434: STOCHASTIC VOLATILITY MODELS 425The
Page 435 and 436: STOCHASTIC VOLATILITY MODELS 427(a)
Page 437 and 438: •••MARKOV SWITCHING MODELS 42
Page 439 and 440: MARKOV SWITCHING MODELS 431(a) GARC
Page 441 and 442: MARKOV SWITCHING MODELS 433β i ∼
Page 443 and 444: MARKOV SWITCHING MODELS 435(a) mont
Page 445 and 446: MARKOV SWITCHING MODELS 437α ij ar
Page 447 and 448: FORECASTING 439garchrtn-sq0 200 400
Page 449: EXERCISES 441eter uncertainty in pr
Page 453 and 454: 446 INDEXData (cont.)equal-weighted
Page 455: 448 INDEXPoisson process, 244inhomo
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