"Frontmatter". In: Analysis of Financial Time Series

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316 VECTOR TIME SERIESTable 8.3. Order-Specification Statistics for the Monthly LogReturns of IBM Stock and the S&P 500 Index from January1926 to December 1999. The 5% and 1% Critical Values ofa chi-Squared Distribution with 4 Degrees of Freedom are 9.5and 13.3.Order 1 2 3 4 5 6M(i) 9.81 8.93 12.57 6.08 9.56 2.80AIC 6.757 6.756 6.750 6.753 6.751 6.756statistics are nonsignificant at the 1% level, confirming the previous observation thatthe dynamic linear dependence between the two return series is weak.Estimation and Model CheckingFor a specified VAR model, one can estimate the parameters using either the ordinaryleast squares method or the maximum likelihood method. The two methodsare asymptotically equivalent. Under some regularity conditions, the estimates areasymptotically normal; see Reinsel (1993). A fitted model should then be checkedcarefully for any possible inadequacy. The Q k (m) statistic can be applied to theresidual series to check the assumption that there are no serial or cross-correlationsin the residuals. For a fitted VAR(p) model, the Q k (m) statistic of the residuals isasymptotically a chi-squared distribution with k 2 m − g degrees of freedom, where gis the number of estimated parameters in the AR coefficient matrixes.Example 8.4. (continued) Table 8.4(a) shows the estimation results of aVAR(3) model for the bivariate series of monthly log returns of IBM stock andTable 8.4. Estimation Results of a VAR(3) Model for the Monthly Log Returns, in Percentages,of IBM Stock and the S&P 500 Index from January 1926 to December 1999.Param. φ 0 Φ 1 Φ 3 Σ(a) Full modelEstimate1.200.580.011 0.108−0.013 0.0840.039 −0.112−0.007 −0.10544.44 23.5123.51 31.29St. Error0.230.190.043 0.0510.036 0.0430.044 0.0520.037 0.044(b) Simplified modelEstimate1.240.570 0.1170 0.0730 −0.0830 −0.10944.48 23.5123.51 31.29St. Error0.230.19− 0.040− 0.033− 0.040− 0.033
VAR MODELS 317the S&P 500 index. The specified model is in the formr t = φ 0 + Φ 1 r t−1 + Φ 3 r t−3 + a t , (8.20)where the first component of r t denotes IBM stock returns. For this particularinstance, we only use AR coefficient matrixes at lags 1 and 3 because of the weakserial dependence of the data. In general, when the M(i) statistics and the AICcriterion specify a VAR(3) model, all three AR lags should be used. Table 8.4(b)shows the estimation results after some statistically insignificant parameters are setto zero. The Q k (m) statistics of the residual series for the fitted model in Table 8.4(b)give Q 2 (4) = 18.17 and Q 2 (8) = 41.26. Since the fitted VAR(3) model has fourparameters in the AR coefficient matrixes, these two Q k (m) statistics are distributedasymptotically as a chi-squared distribution with degrees of freedom 12 and 28,respectively. The p values of the test statistics are 0.111 and 0.051, and hence thefitted model is adequate at the 5% significance level. As shown by the univariateanalysis, the return series are likely to have conditional heteroscedasticity. Wediscuss multivariate volatility in Chapter 9.From the fitted model in Table 8.4(b), we make the following observations.(a) The concurrent correlation coefficient between the two innovational series is23.51/ √ 44.48 × 31.29 = 0.63, which, as expected, is close to the sample correlationcoefficient between r 1t and r 2t . (b) The two log return series have positive andsignificant means, implying that the log prices of the two series had an upward trendover the data span. (c) The model shows thatIBM t = 1.24 + 0.117SP5 t−1 − 0.083SP5 t−3 + a 1tSP5 t = 0.57 + 0.073SP5 t−1 − 0.109SP5 t−3 + a 2t .Consequently, at the 5% significant level, there is a unidirectional dynamic relationshipfrom the monthly S&P 500 index return to the IBM return. If the S&P 500 indexrepresents the U.S. stock market, then IBM return is affected by the past movementsof the market. However, past movements of IBM stock returns do not significantlyaffect the U.S. market, even though the two returns have substantial concurrent correlation.Finally, the fitted model can be written as[ ] IBMt=SP5 t[ ] 1.24+0.57[ ] 0.117SP50.073 t−1 −indicating that SP5 t is the driving factor of the bivariate series.[ ] [ ]0.083a1tSP50.109 t−3 + ,a 2tForecastingTreating a properly built model as the true model, one can apply the same techniquesas those in the univariate analysis to produce forecasts and standard deviations ofthe associated forecast errors. For a VAR(p) model, the 1-step ahead forecast at thetime origin h is r h (1) = φ 0 + ∑ pi=1 Φ ir h+1−i , and the associated forecast error
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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
Page 273 and 274: ECONOMETRIC APPROACH TO VAR 265The
Page 275 and 276: QUANTILE ESTIMATION 267Using the fo
Page 277 and 278: QUANTILE ESTIMATION 269ˆx 0.01 = p
Page 279 and 280: EXTREME VALUE THEORY 271= 1 −= 1
Page 281 and 282: EXTREME VALUE THEORY 273shows that
Page 283 and 284: EXTREME VALUE THEORY 275{ [] }r n(i
Page 285 and 286: EXTREME VALUE THEORY 2777.5.3 Appli
Page 287 and 288: EXTREME VALUE TO VAR 2797.6 AN EXTR
Page 289 and 290: EXTREME VALUE TO VAR 281VaR =−2.5
Page 291 and 292: EXTREME VALUE TO VAR 283stock. The
Page 293 and 294: 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: VAR MODELS 315To specify the order
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 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 373 and 374: GARCH MODELS FOR BIVARIATE RETURNS
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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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