"Frontmatter". In: Analysis of Financial Time Series

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150 NONLINEAR TIME SERIESf 2 (r t−1 ) = exp(39.25 − 22.17r t−1 − 17.34r t−2 − 5.98r t−3 )1 + exp(39.25 − 22.17r t−1 − 17.34r t−2 − 5.98r t−3 ) .The standard error of the residuals for the prior model is 6.56. For comparison, wealso built an AR model for the data and obtainedr t = 1.101 + 0.077r t−1 + a t , σ a = 6.61. (4.34)The residual standard error is slightly greater than that of the feed-forward model inEq. (4.33).Forecast ComparisonThe monthly returns of IBM stock in 1998 and 1999 form the second subsample andare used to evaluate the out-of-sample forecasting performance of neural networks.As a benchmark for comparison, we use the sample mean of r t in the first subsampleas the 1-step ahead forecast for all the monthly returns in the second subsample. Thiscorresponds to assuming that the log monthly price of IBM stock follows a randomwalk with a drift. The mean squared forecast error (MSE) of this benchmark modelis 91.85. For the AR(1) model in Eq. (4.34), the MSE of 1-step ahead forecasts is91.70. Thus, the AR(1) model outperforms slightly the benchmark. For the 3-2-1feed-forward network in Eq. (4.33), the MSE is 91.74, which is essentially the sameas that of the AR(1) model.Remark: The estimation of feed-forward networks is done by using theS-Plus program with default starting weights; see Venables and Ripley (1999) formore information. Our limited experience shows that the estimation results vary.For the IBM stock returns used in Example 4.5, the out-of-sample MSE for a 3-2-1network can be as low as 89.46 and as high as 93.65. If we change the number ofnodes in the hidden layer, the range for the MSE becomes even wider. The S-Pluscommands used in Example 4.5 are given in Appendix B.Example 4.6. Nice features of the feed-forward networks include its flexibilityand wide applicability. For illustration, we use the network with a Heavisideactivation function for the output layer to forecast the direction of price movementfor IBM stock considered in Example 4.5. Define a direction variable asd t ={ 1 ifrt ≥ 00 ifr t < 0.We use eight input nodes consisting of the first four lagged values of both r t andd t and four nodes in the hidden layer to build an 8-4-1 feed-forward network ford t in the first subsample. The resulting network is then used to compute the 1-stepahead probability of an “upward movement” (i.e., a positive return) for the followingmonth in the second subsample. Figure 4.6 shows a typical output of probability
NONLINEAR MODELS 151prob0.0 0.2 0.4 0.6 0.8 1.0ooooo o o o o o o o o oo oo oo oo o o o1998.5 1999.0 1999.5 2000.0monthFigure 4.6. One-step ahead probability forecasts for a positive monthly return for IBM stockusing an 8-4-1 feed-forward neural network. The forecasting period is from January 1998 toDecember 1999.forecasts and the actual directions in the second subsample with the latter denotedby “o.” A horizontal line of 0.5 is added to the plot. If we take a rigid approach byletting ˆd t = 1 if the probability forecast is greater than or equal to 0.5 and ˆd t = 0otherwise, then the neural network has a successful rate of 0.58. The success rate ofthe network varies substantially from one estimation to another, and the network uses49 parameters. To gain more insight, we did a simulation study of running the 8-4-1feed-forward network 500 times and computed the number of errors in predictingthe upward and downward movement using the same method as before. The meanand median of errors over the 500 runs are 11.28 and 11, respectively, whereas themaximum and minimum numbers of errors are 18 and 4. For comparison, we alsodid a simulation with 500 runs using a random walk with a drift—that is,{ 1 if ˆrt = 1.19 + ɛˆd t =t ≥ 00 otherwise,where 1.19 is the average monthly log return for IBM stock from January 1926 toDecember 1997 and {ɛ t } is a sequence of iid N(0, 1) random variables. The mean andmedian of the number of forecast errors become 10.53 and 11, whereas the maximumand minimum numbers of errors are 17 and 5, respectively. Figure 4.7 shows thehistograms of the number of forecast errors for the two simulations. The results show
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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)
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
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: NONLINEAR MODELS 1494.1.9.2 Trainin
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 and 174: APPLICATION 165unem. rate4 6 8 10
Page 175 and 176: APPLICATION 167Table 4.4. Out-of-Sa
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
Page 183 and 184: Analysis of Financial Time Series.
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
Page 191 and 192: n(trades)0 20406080 1200 1000 2000
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
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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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