"Frontmatter". In: Analysis of Financial Time Series

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390 MULTIVARIATE VOLATILITY MODELSfrml rh = exp(rh1(t))/(1+exp(rh1(t)))frml gdet = -0.5*(log(h1(t)=gvar1(t))+log(h2(t)=gvar2(t)) $+log(1.0-(rho(t)=rh(t))**2))frml gln = gdet(t)-0.5/(1.0-rho(t)**2)*((a1t(t)**2/h1(t)) $+(a2t(t)**2/h2(t))-2*rho(t)*a1t(t)*a2t(t)/sqrt(h1(t)*h2(t)))smpl 4 888compute c1 = 1.4, c2 = 0.7, p1 = 0.1, p3 = -0.1compute a0 = 2.95, a1 = 0.08, b1 = 0.87, f1 = -.03compute a00 = 2.05, a11 = 0.05compute b11 = 0.92, f11=-.06, d11=.04, q0 = -2.0compute q1 = 3.0, q2 = 0.1nlpar(criterion=value,cvcrit=0.00001)maximize(method=bhhh,recursive,iterations=150) glnset fv1 = gvar1(t)set resi1 = a1t(t)/sqrt(fv1(t))set residsq = resi1(t)*resi1(t)* Checking standardized residuals *cor(qstats,number=16,span=4) resi1* Checking squared standardized residuals *cor(qstats,number=16,span=4) residsqset fv2 = gvar2(t)set resi2 = a2t(t)/sqrt(fv2(t))set residsq = resi2(t)*resi2(t)* Checking standardized residuals *cor(qstats,number=16,span=4) resi2* Checking squared standardized residuals *cor(qstats,number=16,span=4) residsq* Last few observations needed for computing forecasts *set rhohat = rho(t)set shock1 = a1t(t)set shock2 = a2t(t)print 885 888 shock1 shock2 fv1 fv2 rhohat(C): Estimation of the time-varying correlation model in Example 9.2 usingCholesky decomposition.all 0 888:1open data m-ibmspln.datdata(org=obs) / r1 r2set h1 = 45.0set h2 = 20.0set q = 0.8nonlin a0 a1 b1 a00 a11 b11 d11 f11 c1 c2 p1 p3 t0 t1 t2frml a1t = r1(t)-c1-p1*r1(t-1)-p3*r2(t-2)frml a2t = r2(t)-c2frml v1 = a0+a1*a1t(t-1)**2+b1*h1(t-1)frml qt = t0 + t1*q(t-1) + t2*a2t(t-1)frml bt = a2t(t) - (q(t)=qt(t))*a1t(t)frml v2 = a00+a11*bt(t-1)**2+b11*h2(t-1)+f11*h1(t-1) $+d11*a1t(t-1)**2frml gdet = -0.5*(log(h1(t) = v1(t))+ log(h2(t)=v2(t)))frml garchln = gdet-0.5*(a1t(t)**2/h1(t)+bt(t)**2/h2(t))smpl 5 888compute c1 = 1.4, c2 = 0.7, p1 = 0.1, p3 = -0.1
APPENDIX A. SOME REMARKS ON ESTIMATION 391compute a0 = 1.0, a1 = 0.08, b1 = 0.87compute a00 = 2.0, a11 = 0.05, b11 = 0.8compute d11=.04, f11=-.06, t0 =0.2, t1 = 0.1, t2 = 0.1nlpar(criterion=value,cvcrit=0.00001)maximize(method=bhhh,recursive,iterations=150) garchlnset fv1 = v1(t)set resi1 = a1t(t)/sqrt(fv1(t))set residsq = resi1(t)*resi1(t)* Checking standardized residuals *cor(qstats,number=16,span=4) resi1* Checking squared standardized residuals *cor(qstats,number=16,span=4) residsqset fv2 = v2(t)+qt(t)**2*v1(t)set resi2 = a2t(t)/sqrt(fv2(t))set residsq = resi2(t)*resi2(t)* Checking standardized residuals *cor(qstats,number=16,span=4) resi2* Checking squared standardized residuals *cor(qstats,number=16,span=4) residsq* Last few observations needed for forecasts *set rhohat = qt(t)*sqrt(v1(t)/fv2(t))set shock1 = a1t(t)set shock2 = a2t(t)set g22 = v2(t)set q21 = qt(t)set b2t = bt(t)print 885 888 shock1 shock2 fv1 fv2 rhohat g22 q21 b2t(D): Estimation of the three-dimensional time-varying correlation volatilitymodel in Example 9.3 using Cholesky decomposition. Initial estimates are obtainedby a sequential modeling procedure.all 0 2275:1open data d-cscointc.datdata(org=obs) / r1 r2 r3set h1 = 1.0set h2 = 4.0set h3 = 3.0set q21 = 0.8set q31 = 0.3set q32 = 0.3nonlin c1 c2 c3 p3 p21 p22 p31 p33 a0 a1 a2 t0 t1 t2 b0 b1 $b2 u0 u1 u2 w0 w1 w2 d0 d1 d2 d5frml a1t = r1(t)-c1-p3*r1(t-3)frml a2t = r2(t)-c2-p21*r1(t-2)-p22*r2(t-2)frml a3t = r3(t)-c3-p31*r1(t-1)-p33*r3(t-1)frml v1 = a0+a1*a1t(t-1)**2+a2*h1(t-1)frml q1t = t0 + t1*q21(t-1) + t2*a2t(t-1)frml bt = a2t(t) - (q21(t)=q1t(t))*a1t(t)frml v2 = b0+b1*bt(t-1)**2+b2*h2(t-1)frml q2t = u0 + u1*q31(t-1) + u2*a3t(t-1)frml q3t = w0 + w1*q32(t-1) + w2*a2t(t-1)frml b1t = a3t(t)-(q31(t)=q2t(t))*a1t(t)-(q32(t)=q3t(t))*bt(t)frml v3 = d0+d1*b1t(t-1)**2+d2*h3(t-1)+d5*h2(t-1)
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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
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
Page 365 and 366: Analysis of Financial Time Series.
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 and 386: VECTOR VOLATILITY MODELS 377using t
Page 387 and 388: VECTOR VOLATILITY MODELS 379large-c
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: 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)
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
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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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