"Frontmatter". In: Analysis of Financial Time Series

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338 VECTOR TIME SERIESThe result of Eq. (8.36) says thatVar(y i )∑ ki=1Var(r i ) = λ i.λ 1 +···+λ kConsequently, the proportion of total variance in r explained by the ith principal componentis simply the ratio between the ith eigenvalue and the sum of all eigenvaluesof Σ r . One can also compute the cumulative proportion of total variance explainedby the first i principal components [i.e., ( ∑ ij=1 λ j )/( ∑ kj=1 λ j )]. In practice, oneselects a small i such that the prior cumulative proportion is large.Since tr(ρ r ) = k, the proportion of variance explained by the ith principal componentbecomes λ i /k when the correlation matrix is used to perform the PCA.A byproduct of the PCA is that a zero eigenvalue of Σ r ,orρ r , indicates theexistence of an exact linear relationship between the components of r. For instance,if the smallest eigenvalue λ k = 0, then by the prior Result Var(y k ) = 0. Therefore,y k = ∑ kj=1 e kj r j is a constant and there are only k − 1 random quantities in r. Inthis case, the dimension of r can be reduced. For this reason, PCA has been used inthe literature as a tool for dimension reduction.8.7.2 Empirical PCAIn application, the covariance matrix Σ r and the correlation matrix ρ r of the returnvector r are unknown, but they can be estimated consistently by the sample covarianceand correlation matrixes under some regularity conditions. Assuming that thereturns are weakly stationary and the data consist of {r t | t = 1,...,T }, we have thefollowing estimateŝΣ r ≡[ˆσ ij,r ]= 1T − 1T∑(r t −¯r)(r t −¯r) ′ , ¯r = 1 Tt=1T∑r t (8.37)t=1ˆρ r = ̂D −1̂Σ r̂D −1 (8.38)where ̂D = diag{ √ ˆσ 11,r ,..., √ ˆσ kk,r } is the diagonal matrix of sample standarderrors of r t . Methods to compute eigenvalues and eigenvectors of a symmetric matrixcan then be used to perform the PCA. Most statistical packages now have the capabilityto perform principal component analysis.Example 8.7. Consider the monthly log returns of International BusinessMachines, Hewlett-Packard, Intel Corporation, Merrill Lynch, and Morgan StanleyDean Witter from January 1990 to December 1999. The returns are in percentagesand include dividends. The data set has 120 observations. Figure 8.10 shows thetime plots of these five monthly return series. As expected, returns of companies inthe same industrial sector tend to exhibit similar patterns.Denote the returns by r ′ = (IBM, HWP, INTC, MER, MWD). The sample meanvector of the returns is (1.47, 1.97, 3.05, 2.30, 2.36) ′ and the sample covariance and
PRINCIPAL COMPONENT ANALYSIS 339(a) IBMrtnrtnrtn-30-1010-20 010-30-10101990 1992 1994year1996 1998 2000(b) HWP1990 1992 1994year1996 1998 2000(c) INTC1990 1992 1994year1996 1998 2000(d) MERrtnrtn-40-200 20-40 0 20401990 1992 1994 1996 1998 2000year(e) MWD1990 1992 1994 1996 1998 2000yearFigure 8.10. Time plots of monthly log returns in percentages and including dividends forInternational Business Machines, Hewlett-Packard, Intel, Merrill Lynch, and Morgan StanleyDean Witter from January 1990 to December 1999.correlation matrixes are⎡⎤73.1036.48 103.60̂Σ r =⎢27.08 48.86 113.96⎥⎣16.06 37.59 27.96 105.56 ⎦ ,16.33 40.72 26.86 85.47 109.91⎡⎤1.000.42 1.00ˆρ r =⎢0.30 0.45 1.00⎥⎣0.18 0.36 0.26 1.00 ⎦ .0.18 0.38 0.24 0.79 1.00Table 8.8 gives the results of PCA using both the covariance and correlationmatrixes. Also given are eigenvalues, eigenvectors, and proportions of variabilities
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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.
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
Page 341 and 342: THRESHOLD CO-INTEGRATION 333ously b
Page 343 and 344: PRINCIPAL COMPONENT ANALYSIS 3358.6
Page 345: PRINCIPAL COMPONENT ANALYSIS 337(c
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
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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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