"Frontmatter". In: Analysis of Financial Time Series

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282 VALUEATRISKnext trading day. If the probability is 5%, which means that with probability 0.95 theloss will be less than or equal to the VaR for the next trading day, then the resultsobtained are1. $302,500 for the RiskMetrics,2. $287,200 for a Gaussian AR(2)-GARCH(1, 1) model,3. $283,520 for an AR(2)-GARCH(1, 1) model with a standardized Student-tdistribution with 5 degrees of freedom,4. $216,030 for using the empirical quantile, and5. $184,127 for applying the traditional extreme value theory using monthly minima(i.e., subperiod length n = 21).If the probability is 1%, then the VaR is1. $426,500 for the RiskMetrics,2. $409,738 for a Gaussian AR(2)-GARCH(1, 1) model,3. $475,943 for an AR(2)-GARCH(1, 1) model with a standardized Student-tdistribution with 5 degrees of freedom,4. $365,709 for using the empirical quantile, and5. $340,013 for applying the traditional extreme value theory using monthly minima(i.e., subperiod length n = 21).If the probability is 0.1%, then the VaR becomes1. $566,443 for the RiskMetrics,2. $546,641 for a Gaussian AR(2)-GARCH(1, 1) model,3. $836,341 for an AR(2)-GARCH(1, 1) model with a standardized Student-tdistribution with 5 degrees of freedom,4. $780,712 for using the empirical quantile, and5. $666,590 for applying the traditional extreme value theory using monthly minima(i.e., subperiod length n = 21).There are substantial differences among different approaches. This is not surprisingbecause there exists substantial uncertainty in estimating tail behavior of a statisticaldistribution. Since there is no true VaR available to compare the accuracy of differentapproaches, we recommend that one applies several methods to gain insight into therange of VaR.The choice of tail probability also plays an important role in VaR calculation. Forthe daily IBM stock returns, the sample size is 9190 so that the empirical quantilesof 5% and 1% are decent estimates of the quantiles of the return distribution. In thiscase, we can treat the results based on empirical quantiles as conservative estimatesof the true VaR (i.e., lower bounds). In this view, the approach based on the traditionalextreme value theory seems to underestimate the VaR for the daily log returns of IBM
EXTREME VALUE TO VAR 283stock. The conditional approach of extreme value theory discussed in the next sectionovercomes this weakness.When the tail probability is small (e.g., 0.1%), the empirical quantile is a lessreliable estimate of the true quantile. The VaR based on empirical quantiles can nolonger serve as a lower bound of the true VaR. Finally, the earlier results show clearlythe effects of using a heavy-tail distribution in VaR calculation when the tail probabilityis small. The VaR based on either a Student-t distribution with 5 degrees offreedom or the extreme value distribution is greater than that based on the normalassumption when the probability is 0.1%.7.6.2 Multiperiod VaRThe square root of time rule of the RiskMetrics methodology becomes a special caseunder the extreme value theory. The proper relationship between l-day and 1-dayhorizons isVaR(l) = l 1/α VaR = l −k VaR,where α is the tail index and k is the shape parameter of the extreme value distribution;see Danielsson and de Vries (1997a). This relationship is referred to as theα-root of time rule. Here α = 1 k , not the scale parameter α n.For illustration, consider the daily log returns of IBM stock in Example 7.6. If weuse p = 0.05 and the results of n = 21, then for a 30-day horizon we haveVaR(30) = (30) 0.335 VaR = 3.125 × $184,127 = $575,397.Because l 0.335 0fork n ̸= 0.
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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
Page 239 and 240: DISTRIBUTIONS OF PRICE AND RETURN 2
Page 241 and 242: BLACK-SCHOLES EQUATION 233and G t =
Page 243 and 244: BLACK-SCHOLES FORMULA 235risk prefe
Page 245 and 246: BLACK-SCHOLES FORMULA 237The stock
Page 247 and 248: BLACK-SCHOLES FORMULA 239(a) Call o
Page 249 and 250: AN EXTENSION OF ITO’S LEMMA 2410
Page 251 and 252: STOCHASTIC INTEGRAL 243using the It
Page 253 and 254: JUMP DIFFUSION MODELS 245where w t
Page 255 and 256: JUMP DIFFUSION MODELS 247where it i
Page 257 and 258: JUMP DIFFUSION MODELS 249sudden jum
Page 259 and 260: APPENDIX A 251Pricing formulas for
Page 261 and 262: EXERCISES 253which involves the CDF
Page 263 and 264: REFERENCES 255Bakshi, G., Cao, C.,
Page 265 and 266: VALUEATRISK 257log return-0.2 -0.1
Page 267 and 268: RISKMETRICS 259we use log returns r
Page 269 and 270: RISKMETRICS 261investor is$10,000,0
Page 271 and 272: ECONOMETRIC APPROACH TO VAR 263Cons
Page 273 and 274: ECONOMETRIC APPROACH TO VAR 265The
Page 275 and 276: QUANTILE ESTIMATION 267Using the fo
Page 277 and 278: QUANTILE ESTIMATION 269ˆx 0.01 = p
Page 279 and 280: EXTREME VALUE THEORY 271= 1 −= 1
Page 281 and 282: EXTREME VALUE THEORY 273shows that
Page 283 and 284: EXTREME VALUE THEORY 275{ [] }r n(i
Page 285 and 286: EXTREME VALUE THEORY 2777.5.3 Appli
Page 287 and 288: EXTREME VALUE TO VAR 2797.6 AN EXTR
Page 289: EXTREME VALUE TO VAR 281VaR =−2.5
Page 293 and 294: A NEW APPROACH TO VAR 285series (e.
Page 295 and 296: A NEW APPROACH TO VAR 287In Eq. (7.
Page 297 and 298: A NEW APPROACH TO VAR 289Example 7.
Page 299 and 300: A NEW APPROACH TO VAR 291feature is
Page 301 and 302: A NEW APPROACH TO VAR 293(a) Homoge
Page 303 and 304: A NEW APPROACH TO VAR 295Table 7.4.
Page 305 and 306: REFERENCES 297a Gaussian GARCH(1, 1
Page 307 and 308: Analysis of Financial Time Series.
Page 309 and 310: CROSS-CORRELATION 301correlation co
Page 311 and 312: CROSS-CORRELATION 303where ̂D is t
Page 313 and 314: CROSS-CORRELATION 305Table 8.1. Sum
Page 315 and 316: 30yr20yr10yr5yr1yr-10 0 510-10 0 51
Page 317 and 318: VAR MODELS 3098.2 VECTOR AUTOREGRES
Page 319 and 320: VAR MODELS 311where G = Cov(b t ).
Page 321 and 322: VAR MODELS 313where φ 0 and a t ar
Page 323 and 324: VAR MODELS 315To specify the order
Page 325 and 326: VAR MODELS 317the S&P 500 index. Th
Page 327 and 328: VECTOR MA MODELS 319[ ] [ ] [ ] [r1
Page 329 and 330: VECTOR MA MODELS 321turn used to co
Page 331 and 332: VECTOR ARMA MODELS 323For the VAR(1
Page 333 and 334: 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
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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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