"Frontmatter". In: Analysis of Financial Time Series

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166 NONLINEAR TIME SERIESa decrease or a minor increase. Here the economy should be stable, and essentiallythe change in the rate follows a simple AR(1) model because the lag-2 coefficient isinsignificant. In the second regime, there is a substantial jump in the unemploymentrate (0.1 or larger). This typically corresponds to the contraction phase in the businesscycle. It is also the period during which government interventions and industrialrestructuring are likely to occur. Here x t follows an AR(2) model with a positiveconstant, indicating an upward trend in x t . The AR(2) polynomial contains twocomplex characteristic roots, which indicate possible cyclical behavior in x t . Consequently,the chance of having a turning point in x t increases, suggesting that theperiod of large increases in x t should be short. This implies that the contractionphases in the U.S. economy tend to be shorter than the expansion phases.Applying a Markov Chain Monte Carlo method, Montgomery et al. (1998) obtainthe following Markov switching model for x t :{ −0.07 + 0.38xt−1 − 0.05x t−2 + ɛ 1t if s t = 1x t =0.16 + 0.86x t−1 − 0.38x t−2 + ɛ 2t if s t = 2.(4.52)The conditional means of x t are −0.10 for s t = 1 and 0.31 for s t = 2. Thus,the first state represents the expansionary periods in the economy, and the secondstate represents the contractions. The sample variances of ɛ 1t and ɛ 2t are 0.031 and0.192, respectively. The standard errors of the three parameters in state s t = 1are0.03, 0.14, and 0.11, and those of state s t = 2 are 0.04, 0.13, and 0.14, respectively.The state transition probabilities are P(s t = 2 | s t−1 = 1) = 0.084(0.060) andP(s t = 1 | s t−1 = 2) = 0.126(0.053), where the number in parentheses is thecorresponding standard error. This model implies that in the second state the unemploymentrate x t has an upward trend with an AR(2) polynomial possessing complexcharacteristic roots. This feature of the model is similar to the second regime of theTAR model in Eq. (4.51). In the first state, the unemployment rate x t has a slightlydecreasing trend with a much weaker autoregressive structure.Forecasting PerformanceA rolling procedure was used by Montgomery et al. (1998) to forecast the unemploymentrate x t . The procedure works as follows:1. Begin with forecast origin T = 83, corresponding to 1968.II which was usedin the literature to monitor performance of various econometric models in forecastingunemployment rate. Estimate the linear, TAR, and MSA models usingthe data from 1948.I to the forecast origin (inclusive).2. Perform 1-quarter to 5-quarter ahead forecasts and compute the forecast errorsof each model. Forecasts of nonlinear models used are computed by using theparametric bootstrap method of Section 4.4.3. Advance the forecast origin by 1 and repeat the estimation and forecastingprocesses until all data are employed.4. Use MSE and mean forecast error to compare performance of the models.
APPLICATION 167Table 4.4. Out-of-Sample Forecast Comparison Among Linear, TAR, and MSA Modelsfor the U.S. Quarterly Unemployment Rate. The Starting Forecast Origin is 1968.II,Where the Row Marked by “MSE” Shows the MSE of the Benchmark Linear Model.(A)Relative MSE of forecastModel 1-step 2-step 3-step 4-step 5-step(a) Overall comparisonLinear 1.00 1.00 1.00 1.00 1.00TAR 1.00 1.04 0.99 0.98 1.03MSA 1.19 1.39 1.40 1.45 1.61MSE 0.08 0.31 0.67 1.13 1.54(b) Forecast origins in economic contractionsLinear 1.00 1.00 1.00 1.00 1.00TAR 0.85 0.91 0.83 0.72 0.72MSA 0.97 1.03 0.96 0.86 1.02MSE 0.22 0.97 2.14 3.38 3.46(c) Forecast origins in economic expansionsLinear 1.00 1.00 1.00 1.00 1.00TAR 1.06 1.13 1.10 1.15 1.17MSA 1.31 1.64 1.73 1.84 1.87MSE 0.06 0.21 0.45 0.78 1.24(B)Mean of forecast errorsModel 1-step 2-step 3-step 4-step 5-step(a) Overall comparisonLinear .03 .09 .17 .25 .33TAR −.10 −.02 −.03 −.03 −.01MSA .00 −.02 −.04 −.07 −.12(b) Forecast origins in economic contractionsLinear 0.31 0.68 1.08 1.41 1.38TAR 0.24 0.56 0.87 1.01 0.86MSA 0.20 0.41 0.57 0.52 0.14(c) Forecast origins in economic expansionsLinear −.01 .00 .03 .08 .17TAR −.05 −.11 −.17 −.19 −.14MSA −.03 −.08 −.13 −.17 −.16
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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
Page 123 and 124: APPLICATION 115(a) IBMlog-rtn-30 -1
Page 125 and 126: APPLICATION 117equationThe fitted m
Page 127 and 128: KURTOSIS OF GARCH MODELS 119where K
Page 129 and 130: SOME RATS PROGRAMS FOR ESTIMATING V
Page 131 and 132: EXERCISES 123• Is there evidence
Page 133 and 134: REFERENCES 125Melino, A., and Turnb
Page 135 and 136: NONLINEAR TIME SERIES 127To put non
Page 137 and 138: NONLINEAR MODELS 129Example 4.1. Co
Page 139 and 140: NONLINEAR MODELS 131jumps when it b
Page 141 and 142: NONLINEAR MODELS 133the series and
Page 143 and 144: NONLINEAR MODELS 135els to the mont
Page 145 and 146: NONLINEAR MODELS 137growth-2 -1 0 1
Page 147 and 148: NONLINEAR MODELS 139average of y t
Page 149 and 150: NONLINEAR MODELS 141indicating that
Page 151 and 152: NONLINEAR MODELS 143s T,2t=1T∑K h
Page 153 and 154: NONLINEAR MODELS 145f i (.) of Eq.
Page 155 and 156: NONLINEAR MODELS 1474.1.9.1 Feed-Fo
Page 157 and 158: NONLINEAR MODELS 1494.1.9.2 Trainin
Page 159 and 160: NONLINEAR MODELS 151prob0.0 0.2 0.4
Page 161 and 162: NONLINEARITY TESTS 153idea behind v
Page 163 and 164: NONLINEARITY TESTS 155C l (δ, T )
Page 165 and 166: NONLINEARITY TESTS 157and obtain th
Page 167 and 168: NONLINEARITY TESTS 159• Step 2: C
Page 169 and 170: FORECASTING 161returns. In summary,
Page 171 and 172: FORECASTING 163in m 11 and m 22 ind
Page 173: APPLICATION 165unem. rate4 6 8 10
Page 177 and 178: APPENDIX A 169compute a0 = 0.1, a1
Page 179 and 180: REFERENCES 171P(s t = 2 | s t−1 =
Page 181 and 182: REFERENCES 173Fan, J. (1993), “Lo
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Page 185 and 186: NONSYNCHRONOUS TRADING 177t − k t
Page 187 and 188: BID-ASK SPREAD 179The lag-1 autocov
Page 189 and 190: EMPIRICAL CHARACTERISTICS 181The ma
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Page 193 and 194: EMPIRICAL CHARACTERISTICS 185after-
Page 195 and 196: MODELS FOR PRICE CHANGES 187deep un
Page 197 and 198: MODELS FOR PRICE CHANGES 189σ 2i (
Page 199 and 200: MODELS FOR PRICE CHANGES 191A i = 1
Page 201 and 202: MODELS FOR PRICE CHANGES 193( )piln
Page 203 and 204: DURATION MODELS 195For the IBM data
Page 205 and 206: DURATION MODELS 197(a) is the avera
Page 207 and 208: DURATION MODELS 199actions might no
Page 209 and 210: DURATION MODELS 201Series : xACF0.0
Page 211 and 212: DURATION MODELS 203reduces to that
Page 213 and 214: DURATION MODELS 205Series : xACF0.0
Page 215 and 216: THE PCD MODEL 207where w(α) denote
Page 217 and 218: THE PCD MODEL 209(a) All transactio
Page 219 and 220: THE PCD MODEL 2110 10 20 30 40 500
Page 221 and 222: THE PCD MODEL 213⎧⎪⎨1f (x |
Page 223 and 224: 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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