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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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12 FINANCIAL TIME SERIES AND THEIR
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14 FINANCIAL TIME SERIES AND THEIR
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16 FINANCIAL TIME SERIES AND THEIR
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18 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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Analysis of Financial Time Series.
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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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- 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
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- 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
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- Page 185 and 186: NONSYNCHRONOUS TRADING 177t − k t
- Page 187 and 188: BID-ASK SPREAD 179The lag-1 autocov
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- 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
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- Page 207 and 208: DURATION MODELS 199actions might no
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- Page 211 and 212: DURATION MODELS 203reduces to that
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- Page 215 and 216: THE PCD MODEL 207where w(α) denote
- Page 217 and 218: THE PCD MODEL 209(a) All transactio
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- Page 221 and 222: THE PCD MODEL 213⎧⎪⎨1f (x |
- Page 223 and 224: THE PCD MODEL 215Generalized Gamma
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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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Analysis of Financial Time Series.
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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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Analysis of Financial Time Series.
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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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GARCH MODELS FOR BIVARIATE RETURNS
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GARCH MODELS FOR BIVARIATE RETURNS
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GARCH MODELS FOR BIVARIATE RETURNS
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GARCH MODELS FOR BIVARIATE RETURNS
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GARCH MODELS FOR BIVARIATE RETURNS
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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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Analysis of Financial Time Series.
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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