"Frontmatter". In: Analysis of Financial Time Series

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216 HIGH-FREQUENCY DATAbyh(x) = f (x)S(x)(5.57)where f (.) and S(.) are the pdf and survival function of X, respectively.Example 5.6. For the Weibull distribution with parameters α and β, the survivalfunction and hazard function are:[ ( ) x α ]S(x | α, β) = exp − ,βh(x | α, β) = α β α xα−1 , x > 0.In particular, when α = 1, we have h(x | β) = 1/β. Therefore, for an exponentialdistribution, the hazard function is constant. For a Weibull distribution, the hazard isa monotone function. If α>1, then the hazard function is monotonously increasing.If α 0.αAPPENDIX C.SOME RATS PROGRAMS FOR DURATION MODELSThe data used are adjusted time durations of intraday transactions of IBM stock fromNovember 1 to November 9, 1990. The file name is “ibm1to5.dat” and it has 3534observations.A. Program for Estimating a WACD(1, 1) Modelall 0 3534:1open data ibm1to5.datdata(org=obs) / x r1set psi = 1.0nonlin a0 a1 b1 alfrml gvar = a0+a1*x(t-1)+b1*psi(t-1)frml gma = %LNGAMMA(1.0+1.0/al)frml gln =al*gma(t)+log(al)-log(x(t)) $+al*log(x(t)/(psi(t)=gvar(t)))-(exp(gma(t))*x(t)/psi(t))**al
THE PCD MODEL 217smpl 2 3534compute a0 = 0.2, a1 = 0.1, b1 = 0.1, al = 0.8maximize(method=bhhh,recursive,iterations=150) glnset fv = gvar(t)set resid = x(t)/fv(t)set residsq = resid(t)*resid(t)cor(qstats,number=20,span=10) residcor(qstats,number=20,span=10) residsqB. Program for Estimating a GACD(1, 1) Modelsall 0 3534:1open data ibm1to5.datdata(org=obs) / x r1set psi = 1.0nonlin a0 a1 b1 al kafrml cv = a0+a1*x(t-1)+b1*psi(t-1)frml gma = %LNGAMMA(ka)frml lam = exp(gma(t))/exp(%LNGAMMA(ka+(1.0/al)))frml xlam = x(t)/(lam(t)*(psi(t)=cv(t)))frml gln =-gma(t)+log(al/x(t))+ka*al*log(xlam(t))-(xlam(t))**alsmpl 2 3534compute a0 = 0.238, a1 = 0.075, b1 = 0.857, al = 0.5, ka = 4.0nlpar(criterion=value,cvcrit=0.00001)maximize(method=bhhh,recursive,iterations=150) glnset fv = cv(t)set resid = x(t)/fv(t)set residsq = resid(t)*resid(t)cor(qstats,number=20,span=10) residcor(qstats,number=20,span=10) residsqC. A program for estimating a Tar-WACD(1, 1) model. The threshold 3.79 isprespecified.all 0 3534:1open data ibm1to5.datdata(org=obs) / x rtset psi = 1.0nonlin a1 a2 al b0 b2 blfrml u = ((x(t-1)-3.79)/abs(x(t-1)-3.79)+1.0)/2.0frml cp1 = a1*x(t-1)+a2*psi(t-1)frml gma1 = %LNGAMMA(1.0+1.0/al)frml cp2 = b0+b2*psi(t-1)frml gma2 = %LNGAMMA(1.0+1.0/bl)frml cp = cp1(t)*(1-u(t))+cp2(t)*u(t)frml gln1 =al*gma1(t)+log(al)-log(x(t)) $+al*log(x(t)/(psi(t)=cp(t)))-(exp(gma1(t))*x(t)/psi(t))**alfrml gln2 =bl*gma2(t)+log(bl)-log(x(t)) $+bl*log(x(t)/(psi(t)=cp(t)))-(exp(gma2(t))*x(t)/psi(t))**blfrml gln = gln1(t)*(1-u(t))+gln2(t)*u(t)smpl 2 3534compute a1 = 0.2, a2 = 0.85, al = 0.9
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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
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 and 182: REFERENCES 173Fan, J. (1993), “Lo
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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: THE PCD MODEL 215Generalized Gamma
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
Page 233 and 234: STOCHASTIC PROCESSES 2253. stationa
Page 235 and 236: ITO’S LEMMA 2276.3.2 Stochastic D
Page 237 and 238: 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
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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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