"Frontmatter". In: Analysis of Financial Time Series

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378 MULTIVARIATE VOLATILITY MODELS(a) S&P 500 indexln-rtn-6 -4 -2 0 2 41992 1994 1996 1998 2000year(b) Cisco systemsln-rtn-20 -10 0 10ln-rtn-15-10-5 0 5 101992 1994 1996year1998 2000(c) Intel corp1992 1994 1996year1998 2000Figure 9.8. Time plots of daily log returns in percentages of the S&P 500 index and stocks ofCisco Systems and Intel Corporation from January 2, 1991 to December 31, 1999.of the table shows that (a) the daily log return of S&P 500 index does not depend onthe past returns of Cisco or Intel, (b) the log return of Cisco stock has some serialcorrelations and depends on the past return of S&P 500 index (see lags 2 and 5), and(c) the log return of Intel stock depends on the past returns of S&P 500 index (seelags 1 and 5). These observations are similar to that between the returns of IBM stockand S&P 500 index analyzed in Chapter 8. They suggest that returns of individualTable 9.1. Sample Cross-Correlation Matrixes of Daily Log Returns ofthe S&P 500 Index and the Stocks of Cisco Systems and Intel Corporationfrom January 2, 1991 to December 31, 1999.Lag1 2 3 4 5 6. . .. . .− . .. . .. − .. . .− . .. . .. . .. . .. . .. . .− . .− . .− . .. . .. − .. . .
VECTOR VOLATILITY MODELS 379large-cap companies tend to be affected by the past behavior of the market. However,the market return is not significantly affected by the past returns of individualcompanies.Turning to volatility modeling and following the suggested procedure, we startwith the log return of S&P 500 index and obtain the modelr 1t = 0.078 + 0.042r 1,t−1 − 0.062r 1,t−3 − 0.048r 1,t−4 − 0.052r 1,t−5 + a 1tσ 11,t = 0.013 + 0.092a 2 1,t−1 + 0.894σ 11,t−1, (9.29)where standard errors of the parameters in the mean equation are 0.016, 0.023,0.020, 0.022, and 0.020, respectively, and those of the parameters in the volatilityequation are 0.002, 0.006, and 0.007, respectively. Univariate Ljung–Box statisticsof the standardized residuals and their squared series fail to detect any remainingserial correlation or conditional heteroscedasticity in the data. Indeed, we haveQ(10) = 7.38(0.69) for the standardized residuals and Q(10) = 3.14(0.98) for thesquared series.Augmenting the daily log returns of Cisco stock to the system, we build a bivariatemodel with mean equations given byr 1t = 0.065 − 0.046r 1,t−3 + a 1tr 2t = 0.325 + 0.195r 1,t−2 − 0.091r 2,t−2 + a 2t , (9.30)where all of the estimates are statistically significant at the 1% level. Using the notationof Cholesky decomposition, we obtain the volatility equations asg 11,t = 0.006 + 0.051b 2 1,t−1 + 0.943g 11,t−1q 21,t = 0.331 + 0.790q 21,t−1 − 0.041a 2,t−1 (9.31)g 22,t = 0.177 + 0.082b2,t−1 2 + 0.890g 22,t−1,where b 1t = a 1t , b 2t = a 2t − q 21,t b 1t , standard errors of the parameters in the equationof g 11,t are 0.001, 0.005, and 0.006, those of the parameters in the equation ofq 21,t are 0.156, 0.099, and 0.011, and those of the parameters in the equation of g 22,tare 0.029, 0.008, and 0.011, respectively. The bivariate Ljung–Box statistics of thestandardized residuals fail to detect any remaining serial dependence or conditionalheteroscedasticity. The bivariate model is adequate. Comparing with Eq. (9.29), wesee that the difference between the marginal and univariate models of r 1t is small.The next and final step is to augment the daily log returns of Intel stock to thesystem. The mean equations becomer 1t = 0.065 − 0.043r 1,t−3 + a 1tr 2t = 0.326 + 0.201r 1,t−2 − 0.089r 2,t−2 + a 2t (9.32)r 3t = 0.192 − 0.264r 1,t−1 + 0.059r 3,t−1 + a 3t ,
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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
Page 335 and 336: VECTOR ARMA MODELS 327interest-rate
Page 337 and 338: CO-INTEGRATION 329Series : x1ACF0.0
Page 339 and 340: CO-INTEGRATION 331Stock Exchange; o
Page 341 and 342: THRESHOLD CO-INTEGRATION 333ously b
Page 343 and 344: PRINCIPAL COMPONENT ANALYSIS 3358.6
Page 345 and 346: PRINCIPAL COMPONENT ANALYSIS 337(c
Page 347 and 348: PRINCIPAL COMPONENT ANALYSIS 339(a)
Page 349 and 350: FACTOR ANALYSIS 341(a) 5 stock retu
Page 351 and 352: FACTOR ANALYSIS 343andCov(r, F) = E
Page 353 and 354: FACTOR ANALYSIS 345matrix P to tran
Page 355 and 356: FACTOR ANALYSIS 347Example 8.9. In
Page 357 and 358: APPENDIX A. REVIEW OF VECTORS AND M
Page 359 and 360: APPENDIX A. REVIEW OF VECTORS AND M
Page 361 and 362: APPENDIX B. MULTIVARIATE NORMAL DIS
Page 363 and 364: REFERENCES 355(b) Build a bivariate
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Page 367 and 368: REPARAMETERIZATION 359To illustrate
Page 369 and 370: REPARAMETERIZATION 361where ⊥ den
Page 371 and 372: GARCH MODELS FOR BIVARIATE RETURNS
Page 385: VECTOR VOLATILITY MODELS 377using t
Page 389 and 390: VECTOR VOLATILITY MODELS 381(a) S&P
Page 391 and 392: FACTOR-VOLATILITY MODELS 383conside
Page 393 and 394: APPLICATION 385[ ]σ11,t=σ 22,t⎡
Page 395 and 396: MULTIVARIATE t DISTRIBUTION 387σ 1
Page 397 and 398: APPENDIX A. SOME REMARKS ON ESTIMAT
Page 399 and 400: APPENDIX A. SOME REMARKS ON ESTIMAT
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)
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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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