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66 LINEAR TIME SERIES ANALYSIS AND ITS APPLICATIONSpoint forecasts are shown by dots, and the dashed lines show 95% interval forecasts.The forecasts show a strong seasonal pattern and are close to the observed data.When the seasonal pattern of a time series is stable over time (e.g., close to adeterministic function), dummy variables may be used to handle the seasonality. Thisapproach is taken by some analysts. However, deterministic seasonality is a specialcase of the multiplicative seasonal model discussed before. Specifically, if = 1,then model (2.37) contains a deterministic seasonal component. Consequently, thesame forecasts are obtained by using either dummy variables or a multiplicative seasonalmodel when the seasonal pattern is deterministic. Yet use of dummy variablescan lead to inferior forecasts if the seasonal pattern is not deterministic. In practice,we recommend that the exact likelihood method should be used to estimate a multiplicativeseasonal model, especially when the sample size is small or when there isthe possibility of having a deterministic seasonal component.2.9 REGRESSION MODELS WITH TIME SERIES ERRORSIn many applications, the relationship between two time series is of major interest.The Market Model in finance is an example that relates the return of an individualstock to the return of a market index. The term structure of interest rates is anotherexample in which the time evolution of the relationship between interest rates withdifferent maturities is investigated. These examples lead to the consideration of alinear regression in the formr 1t = α + βr 2t + e t , (2.39)where r 1t and r 2t are two time series and e t denotes the error term. The least squares(LS) method is often used to estimate model (2.39). If {e t } is a white noise series,then the LS method produces consistent estimates. In practice, however, it is commonto see that the error term e t is serially correlated. In this case, we have a regressionmodel with time series errors, and the LS estimates of α and β may not be consistent.Regression model with time series errors is widely applicable in economics andfinance, but it is one of the most commonly misused econometric models becausethe serial dependence in e t is often overlooked. It pays to study the model carefully.We introduce the model by considering the relationship between two U.S. weeklyinterest rate series:1. r 1t : The 1-year Treasury constant maturity rate.2. r 3t : The 3-year Treasury constant maturity rate.Both series have 1967 observations from January 5, 1962 to September 10, 1999 andare measured in percentages. The series are obtained from the Federal Reserve Bankof St Louis. Figure 2.12 shows the time plots of the two interest rates with solidline denoting the 1-year rate and dashed line the 3-year rate. Figure 2.13(a) plots r 1tversus r 3t , indicating that, as expected, the two interest rates are highly correlated.
• ••• ••••••••••••••percent5 10 151970 1980 1990 2000yearFigure 2.12. Time plots of U.S. weekly interest rates (in percentages) from January 5, 1962to September 10, 1999. The solid line is the Treasury 1-year constant maturity rate and thedashed line the Treasury 3-year constant maturity rate.3-year4 6 8 10 12 14 16• •••••• •••• ••• • • •••••• • • • • •• •• •••• • •••••• ••••••• ••••• ••• •• • •• •• •• •• • • •• • • • ••• •••• • •••••••••••••••• •• •••••••••• ••••• •••••• ••••• ••••••••••• ••••••••••• •••• ••• •••••••• •••• •••• •• ••••• • •• • ••• • •• ••••• ••••••• ••••••• •••••••••• ••••• ••••••••••••••••• ••••••••••••••••••••••• ••••• •••••• ••• ••• ••••• • •• •••••••• • •••••••••• •••• ••••••• ••••• • • • •••• •• •••••• • •••••• • •• • ••••••• ••• • ••••••••••• •••••• • ••••• • chg 3-yr-1.0 -0.5 0.0 0.5 1.0 1.5••• •••••••••••••••• ••••••••••• ••••••• ••••• ••••••• ••••••••• ••••• •••••••••••••••• •••••••• •••• •••••• • •• •••••••••• •• ••4 6 8 10 12 14 161-year-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5chg 1yrFigure 2.13. Scatterplots of U.S. weekly interest rates from January 5, 1962 to September 10,1999: (a) 3-year rate versus 1-year rate, and (b) changes in 3-year rate versus changes in 1-yearrate.67
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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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Page 24 and 25: 16 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
Page 71 and 72: SEASONAL MODELS 63Series : xSeries
Page 73: SEASONAL MODELS 65models use the sa
Page 77 and 78: REGRESSION MODELS WITH TIME SERIES
Page 79 and 80: REGRESSION MODELS WITH TIME SERIES
Page 81 and 82: LONG-MEMORY MODELS 73=(k − d −
Page 83 and 84: APPENDIX A: SOME SCA COMMANDS 75est
Page 85 and 86: EXERCISES 77relation. Draw your con
Page 87 and 88: Analysis of Financial Time Series.
Page 89 and 90: STRUCTURE OF A MODEL 813.2 STRUCTUR
Page 91 and 92: THE ARCH MODEL 83corrected asset re
Page 93 and 94: THE ARCH MODEL 85Series : xACF0.0 0
Page 95 and 96: THE ARCH MODEL 87sample mean from t
Page 97 and 98: THE ARCH MODEL 89where Ɣ(x) is the
Page 99 and 100: THE ARCH MODEL 91Series : resiSerie
Page 101 and 102: THE GARCH MODEL 93other softwares a
Page 103 and 104: THE GARCH MODEL 95ahead forecast ca
Page 105 and 106: THE GARCH MODEL 97The fitted AR(3)
Page 107 and 108: THE GARCH MODEL 99Series : stdatACF
Page 109 and 110: THE GARCH-M MODEL 101two models. Th
Page 111 and 112: THE EXPONENTIAL GARCH MODEL 103To b
Page 113 and 114: THE EXPONENTIAL GARCH MODEL 105eros
Page 115 and 116: THE CHARMA MODEL 107ˆσ 2 864 (3)
Page 117 and 118: RANDOM COEFFICIENT AUTOREGRESSIVE M
Page 119 and 120: THE LONG-MEMORY STOCHASTIC VOLATILI
Page 121 and 122: AN ALTERNATIVE APPROACH 113where Va
Page 123 and 124: 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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