"Frontmatter". In: Analysis of Financial Time Series

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76 LINEAR TIME SERIES ANALYSIS AND ITS APPLICATIONS-- compute residual acf.acf r4. maxl 10.-- exit SCAstopEXERCISES1. Suppose that the simple return of a monthly bond index follows the MA(1)modelR t = a t + 0.2a t−1 , σ a = 0.025.Assume that a 100 = 0.01. Compute the 1-step and 2-step ahead forecasts of thereturn at the forecast origin t = 100. What are the standard deviations of theassociated forecast errors? Also compute the lag-1 and lag-2 autocorrelations ofthe return series.2. Suppose that the daily log return of a security follows the modelr t = 0.01 + 0.2r t−2 + a t ,where {a t } is a Gaussian white noise series with mean zero and variance 0.02.What are the mean and variance of the return series r t ? Compute the lag-1 andlag-2 autocorrelations of r t . Assume that r 100 =−0.01, and r 99 = 0.02. Computethe 1- and 2-step ahead forecasts of the return series at the forecast origint = 100. What are the associated standard deviations of the forecast errors?3. The file “bnd.dat” contains simple returns of monthly indexes of U.S. governmentbonds with maturities in 30 years, 20 years, 10 years, 5 years, and 1 year(in column order). The data are obtained from CRSP, and the sample period isfrom January 1942 to December 1999. Build an AR or MA model for the simplereturn of bond index with maturity 5 years. Is the fitted model adequate?4. Consider the sampling period from January 1990 to December 1999. Are thedaily log returns of ALCOA stock predictable? You may test the hypothesisusing (a) the first 5 lags of the autocorrelation function, and (b) the first10lagsofthe autocorrelation function. Draw your conclusion by using the 5% significancelevel. The data are available from CRSP.5. Consider the daily log returns of Hewlett-Packard stock, value-weighted index,equal-weighted index, and S&P 500 index from January 1980 to December 1999for 5056 observations. The returns include all distributions and are in percentages.The data can be obtained from CRSP or from the file “d-hwp3dx8099.dat,”which has four columns with the same ordering as stated before. For each returnseries, test the hypothesis H o : ρ 1 = ··· = ρ 10 = 0 versus the alternativehypothesis H a : ρ i ̸= 0forsomei ∈{1,...,10}, whereρ i is the lag-i autocor-
EXERCISES 77relation. Draw your conclusion based on the 5% significance level. Compare theresults between returns of individual stocks and market indexes.6. Consider the monthly log returns of CRSP equal-weighted index from January1962 to December 1999 for 456 observations. You may obtain the data fromCRSP directly or from the file “m-ew6299.dat” on the Web.• Build an AR model for the series and check the fitted model.• Build an MA model for the series and check the fitted model.• Compute 1- and 2-step ahead forecasts of the AR and MA models built in theprevious two questions.• Compare the fitted AR and MA models.7. Column 3 of the file “d-hwp3dx8099.dat” contains the daily log returns of theCRSP equal-weighted index from January 1980 to December 1999.• Build an AR model for the series and check the fitted model.• Build an ARMA model for the series and check the fitted model.• Use the fitted AR model to compute 1-step to 7-step ahead forecasts at theforecast origin December 27, 1999 (i.e., h = 5052). Note that for this particularinstance the lag-5 coefficient is statistically significant. This might be dueto the weekend effects.8. Again, consider the daily log return of CRSP equal-weighted index from January1980 to December 1999. Create indicator variables for Mondays, Tuesdays,Wednesdays, and Thursdays and use a regression model, possibly withtime series errors, to study the effects of trading days on the index return. Whatis the fitted model? Are there serial correlations in the residuals?9. This problem is concerned with the dynamic relationship between the spotand futures prices of the S&P500 index. The data file “sp5may.dat” has threecolumns: log(futures price), log(spot price), and cost-of-carry (×100). The datawere obtained from the Chicago Mercantile Exchange for the S&P 500 stockindex in May 1993 and its June futures contract. The time interval is 1 minute(intraday). Several authors used the data to study index futures arbitrage. Herewe focus on the first two columns. Let f t and s t be the log prices of futuresand spot, respectively. Consider y t = f t − f t−1 and x t = s t − s t−1 . Build aregression model with time series errors between {y t } and {x t }, with y t beingthe dependent variable.10. The data file “qunemrate.dat” contains the U.S. quarterly unemployment rate,seasonally adjusted, from 1948 to the second quarter of 1991. Consider thechange series y t = x t − x t−1 ,wherex t is the quarterly unemployment rate.
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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
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 and 74: SEASONAL MODELS 65models use the sa
Page 75 and 76: • ••• •••••••
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: APPENDIX A: SOME SCA COMMANDS 75est
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
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
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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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