"Frontmatter". In: Analysis of Financial Time Series

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346 VECTOR TIME SERIESTable 8.9. Factor Analysis of the Monthly Log Stock Returns of IBM,Hewlett-Packard, Intel, Merrill Lynch, and Morgan Stanley Dean Witter.The Returns Include Dividends and Are from January 1990 to December1999. The Analysis Is Based on the Sample Cross-Correlation Matrix andAssumes Two Common Factors.(a) The principal component methodEstimates ofRotatedfactor loadings factor loadings CommunalitiesVariable f 1 f 2 f1 ∗ f2 ∗ 1 − iIBM 0.536 0.561 0.011 0.776 0.602HWP 0.744 0.335 0.317 0.752 0.665INTC 0.607 0.433 0.151 0.730 0.556MER 0.788 −0.515 0.928 0.158 0.887MWD 0.791 −0.514 0.930 0.161 0.891Variance 2.4563 1.1448 1.8502 1.7509 3.6011Proportion 0.491 0.229 0.370 0.350 0.720(b) The maximum likelihood methodEstimates ofRotatedfactor loadings factor loadings CommunalitiesVariable f 1 f 2 f1 ∗ f2 ∗ 1 − iIBM 0.191 0.496 0.087 0.524 0.282HWP 0.394 0.689 0.247 0.755 0.630INTC 0.250 0.511 0.141 0.551 0.323MER 0.800 0.072 0.769 0.232 0.645MWD 0.994 −0.015 0.976 0.186 0.988Variance 1.8813 0.9866 1.6324 1.2355 2.8679Proportion 0.376 0.197 0.326 0.247 0.574• The two factors identified by the principal component method explain morevariability than those identified by the maximum likelihood method.• Based on the rotated factor loadings, the two estimation methods identify essentiallythe same two common factors for the data. The financial stocks (MerrillLynch and Morgan Stanley Dean Witter) load heavily on the first factor,whereas the technology stocks (IBM, Hewlett-Packard, and Intel) load highlyon the second factor. These two rotated factors jointly differentiate the industrialsectors.• In this particular instance, the varimax rotation does not change much the twofactors identified by the maximum likelihood method. Yet the first unrotatedfactor identified by the principal component method was destroyed by the rotation.This is not surprising in view of the idea behind the varimax criterion.• The specific variances of IBM and Intel stock returns are relatively large basedon the maximum likelihood method, indicating that these two stocks have theirown features that are worth further investigation.
FACTOR ANALYSIS 347Example 8.9. In this example, we consider the monthly log returns of U.S.bond indexes with maturities in 30 years, 20 years, 10 years, 5 years, and 1 year.The data are described in Example 8.2, but have been transformed into log returns.There are 696 observations. As shown in Example 8.2, there is serial dependencein the data. However, removing serial dependence by fitting a VARMA(2, 1) modelhas hardly any effects on the concurrent correlation matrix. As a matter of fact, thecorrelation matrixes before and after fitting a VARMA(2, 1) model are⎡⎤1.0.98 1.0ˆρ o =⎢.92 .91 1.0⎥⎣.85 .86 .90 1.0 ⎦ ,.63 .64 .67 .81 1.0⎡⎤1.0ˆρ = .98 1.0⎢.92 .92 1.0⎥⎣.85 .86 .90 1.0 ⎦ ,.66 .67 .71 .84 1.0Table 8.10. Factor Analysis of the Monthly Log Returns of U.S. BondIndexes With Maturities in 30 Years, 20 Years, 10 Years, 5 Years, and 1Year. The Data Are from January 1942 to December 1999. The Analysis IsBased on the Sample Cross-Correlation Matrix and Assumes Two CommonFactors.(a) The principal component methodEstimates ofRotatedfactor loadings factor loadings CommunalitiesVariable f 1 f 2 f1 ∗ f2 ∗ 1 − i30 years 0.952 0.253 0.927 0.333 0.97020 years 0.954 0.240 0.922 0.345 0.96810 years 0.956 0.140 0.866 0.429 0.9345 years 0.955 −0.142 0.704 0.660 0.9311 year 0.800 −0.585 0.325 0.936 0.982Variance 4.2812 0.5039 3.0594 1.7256 4.7851Proportion 0.856 0.101 0.612 0.345 0.957(b) The maximum likelihood methodEstimates ofRotatedfactor loadings factor loadings CommunalitiesVariable f 1 f 2 f1 ∗ f2 ∗ 1 − i30 years 0.849 −0.513 0.895 0.430 0.98520 years 0.857 −0.486 0.876 0.451 0.97010 years 0.896 −0.303 0.744 0.584 0.8955 years 1.000 0.000 0.547 0.837 1.0001 year 0.813 0.123 0.342 0.747 0.675Variance 3.91783 0.6074 2.5378 1.9874 4.5252Proportion 0.784 0.121 0.508 0.397 0.905
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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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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 and 346: 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: FACTOR ANALYSIS 345matrix P to tran
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 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 and 398: APPENDIX A. SOME REMARKS ON ESTIMAT
Page 399 and 400: APPENDIX A. SOME REMARKS ON ESTIMAT
Page 401 and 402: 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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