"Frontmatter". In: Analysis of Financial Time Series

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12 FINANCIAL TIME SERIES AND THEIR CHARACTERISTICSthere is no lower bound for r t , and the lower bound for R t is satisfied using 1 + R t =exp(r t ). However, the lognormal assumption is not consistent with all the propertiesof historical stock returns. In particular, many stock returns exhibit a positive excesskurtosis.Stable DistributionThe stable distributions are a natural generalization of normal in that they are stableunder addition, which meets the need of continuously compounded returns r t .Furthermore, stable distributions are capable of capturing excess kurtosis shown byhistorical stock returns. However, non-normal stable distributions do not have a finitevariance, which is in conflict with most finance theories. In addition, statistical modelingusing non-normal stable distributions is difficult. An example of non-normalstable distributions is the Cauchy distribution, which is symmetric with respect to itsmedian, but has infinite variance.Scale Mixture of Normal DistributionsRecent studies of stock returns tend to use scale mixture or finite mixture of normaldistributions. Under the assumption of scale mixture of normal distributions, the logreturn r t is normally distributed with mean µ and variance σ 2 [i.e., r t ∼ N(µ, σ 2 )].However, σ 2 is a random variable that follows a positive distribution (e.g., σ −2 followsa Gamma distribution). An example of finite mixture of normal distributionsisr t ∼ (1 − X)N(µ, σ 2 1 ) + XN(µ, σ 2 2 ),where 0 ≤ α ≤ 1, σ1 2 is small and σ 2 2 is relatively large. For instance, with α =0.05, the finite mixture says that 95% of the returns follow N(µ, σ1 2 ) and 5% followN(µ, σ2 2). The large value of σ 2 2 enables the mixture to put more mass at the tails ofits distribution. The low percentage of returns that are from N(µ, σ2 2 ) says that themajority of the returns follow a simple normal distribution. Advantages of mixturesof normal include that they maintain the tractability of normal, have finite higherorder moments, and can capture the excess kurtosis. Yet it is hard to estimate themixture parameters (e.g., the α in the finite-mixture case).Figure 1.1 shows the probability density functions of a finite mixture of normal,Cauchy, and standard normal random variable. The finite mixture of normalis 0.95N(0, 1) + 0.05N(0, 16) and the density function of Cauchy isf (x) =1π(1 + x 2 , −∞ < x < ∞.)It is seen that Cauchy distribution has fatter tails than the finite mixture of normal,which in turn has fatter tails than the standard normal.
DISTRIBUTIONAL PROPERTIES OF RETURNS 13f(x)0.0 0.1 0.2 0.3 0.4MixtureCauchyNormal-4 -2 0 2 4xFigure 1.1. Comparison of finite-mixture, stable, and standard normal density functions.1.2.3 Multivariate ReturnsLet r t = (r 1t ,...,r Nt ) ′ be the log returns of N assets at time t. The multivariateanalyses of Chapters 8 and 9 are concerned with the joint distribution of {r t }t=1 T .This joint distribution can be partitioned in the same way as that of Eq. (1.15). Theanalysis is then focused on the specification of the conditional distribution functionF(r t | r t−1 ,...,r 1 , θ). In particular, how the conditional expectation and conditionalcovariance matrix of r t evolve over time constitute the main subjects of Chapters 8and 9.The mean vector and covariance matrix of a random vector X = (X 1 ,...,X p ) aredefined asE(X) = µ x =[E(X 1 ),...,E(X p )] ′ ,Cov(X) = Σ x = E[(X − µ x )(X − µ x ) ′ ]provided that the expectations involved exist. When the data {x 1 ,...,x T } of X areavailable, the sample mean and covariance matrix are defined aŝµ x = 1 TT∑x t ,t=1̂Σ x = 1 TT∑(x t − ̂µ x )(x t − ̂µ x ) ′ .t=1
Page 1 and 2: Analysis of FinancialTime SeriesFin
Page 3 and 4: ContentsPrefacexi1. Financial Time
Page 5 and 6: CONTENTSix6.6 Black-Scholes Pricing
Page 7 and 8: PrefaceThis book grew out of an MBA
Page 9 and 10: Analysis of Financial Time Series.
Page 11 and 12: ASSET RETURNS 3Multiperiod Simple R
Page 13 and 14: ASSET RETURNS 5r t [k] =ln(1 + R t
Page 15 and 16: DISTRIBUTIONAL PROPERTIES OF RETURN
Page 18 and 19: 10 FINANCIAL TIME SERIES AND THEIR
Page 26: 18 FINANCIAL TIME SERIES AND THEIR
Page 29 and 30: REFERENCES 21Federal Reserve Bank o
Page 31 and 32: CORRELATION AND AUTOCORRELATION FUN
Page 33 and 34: CORRELATION AND AUTOCORRELATION FUN
Page 35 and 36: WHITE NOISE AND LINEAR TIME SERIES
Page 37 and 38: SIMPLE AUTOREGRESSIVE MODELS 29whic
Page 39 and 40: SIMPLE AUTOREGRESSIVE MODELS 31wher
Page 41 and 42: SIMPLE AUTOREGRESSIVE MODELS 33wher
Page 43 and 44: SIMPLE AUTOREGRESSIVE MODELS 35expa
Page 45 and 46: SIMPLE AUTOREGRESSIVE MODELS 37Tabl
Page 47 and 48: SIMPLE AUTOREGRESSIVE MODELS 39Mode
Page 49 and 50: SIMPLE AUTOREGRESSIVE MODELS 41The
Page 51 and 52: SIMPLE MOVING-AVERAGE MODELS 43wher
Page 53 and 54: SIMPLE MOVING-AVERAGE MODELS 45s-rt
Page 55 and 56: SIMPLE MOVING-AVERAGE MODELS 47The
Page 57 and 58: SIMPLE ARMA MODELS 49A time series
Page 59 and 60: SIMPLE ARMA MODELS 51operator, the
Page 61 and 62: SIMPLE ARMA MODELS 53Table 2.5. Sam
Page 63 and 64: SIMPLE ARMA MODELS 55This represent
Page 65 and 66: UNIT-ROOT NONSTATIONARITY 57ˆp h (
Page 67 and 68: UNIT-ROOT NONSTATIONARITY 59log-pri
Page 69 and 70: 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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Analysis of Financial Time Series.
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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
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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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