金融时间序列分析

Lecture Notes of Bus41202(Spring2010)Analysis of Financial Time SeriesRuey S.TsaySimple AR models:(Regression with lagged variables.) Motivating example:The growth rate of U.S.quarterly real GNP from1947to1991.Recall that the model discussed before isr t=0.005+0.35r t−1+0.18r t−2−0.14r t−3+a t,ˆσa=0.01.This is called an AR(3)model because the growth rate r t depends on the growth rates of the past three quarters.How do we specify this model from the data?Is it adequate for the data?What are the implications of the model?These are the questions we shall address in this lecture.Another example:U.S.monthly unemployment rate.AR(1)model:1.Form:r t=φ0+φ1r t−1+a t,whereφ0andφ1are real numbers,which are referred to as“parameters”(to be estimated from the data in an application).For example,r t=0.005+0.2r t−1+a t2.Stationarity:necessary and sufficient condition|φ1|<1.Why?3.Mean:E(r t)=φ01−φ1Timeg n p19501960197019801990−0.02−0.010.000.010.020.030.04U.S. quarterly real GNP growth rate: 1947.II to 1991.IFigure 1:U.S.quarterly growth rate of real GNP:1947-19914.Alternative representation:Let E (r t )=µbe the mean of r t so that µ=φ0/(1−φ1).Equivalently,φ0=µ(1−φ1).Plugging in the model,we have(r t −µ)=φ1(r t −1−µ)+a t .(1)This model also has two parameters (µand φ1).It explicitly uses the mean of the series.It is less commonly used in the literature,but is the model representation used in R.5.Variance:Var(r t )=σ2a 1−φ21.6.Autocorrelations:ρ1=φ1,ρ2=φ21,etc.In general,ρk =φk1and ACF ρk decays exponentially as k increases,Timey19501960197019801990200020104681U.S. monthly unemployment rate 1948.1 to 2010.2Figure 2:U.S.monthly unemployment rate (total civilian,16and older)from January 1948to February 20107.Forecast(minimum squared error):Suppose the forecast origin is n.For simplicity,we shall use the model representation in(1) and write x t=r t−µ.The model then becomes x t=φ1x t−1+a t. Note that forecast of r t is simply the forecast of x t plusµ.(a)1-step ahead forecast at time n:ˆx n(1)=φ1x n(b)1-step ahead forecast error:e n(1)=x n+1−ˆx n(1)=a n+1Thus,a n+1is the un-predictable part of x n+1.It is the shock at time n+1!(c)Variance of1-step ahead forecast error:Var[e n(1)]=Var(a n+1)=σ2a.(d)2-step ahead forecast:ˆx n(2)=φ1ˆx n(1)=φ21x n.(e)2-step ahead forecast error:e n(2)=x n+2−ˆx n(2)=a n+2+φ1a n+1(f)Variance of2-step ahead forecast error:Var[e n(2)]=(1+φ21)σ2awhich is greater than or equal to Var[e n(1)],implying that uncertainty in forecasts increases as the number of steps in-creases.(g)Behavior of multi-step ahead forecasts.In general,for the-step ahead forecast at n,we haveˆx n( )=φ 1x n,the forecast errore n( )=a n+ +φ1a n+ −1+···+φ −11a n+1,and the variance of forecast errorVar[e n( )]=(1+φ21+···+φ2( −1)1)σ2a.In particular,as →∞,ˆx n( )→0,i.e.,ˆr n( )→µ.This is called the mean-reversion of the AR(1)process.The variance of forecast error approachesVar[e n( )]=11−φ21σ2a=Var(r t).In practice,it means that for the long-term forecasts serialdependence is not important.The forecast is just the samplemean and the uncertainty is simply the uncertainty about theseries.8.A compact form:(1−φ1B)r t=φ0+a t.Half-life:A common way to quantify the speed of mean reversion is the half-life,which is defined as the number of periods needed sothat the magnitude of the forecast becomes half of that of the forecast origin.For an AR(1)model,this meanx n (k )=12x n .Thus,φk 1x n =12x n .Consequently,the half-life of the AR(1)modelis k =ln(0.5)ln(|φ1|).For example,if φ1=0.5,the k =1.If φ1=0.9,thenk ≈6.58.AR(2)model:1.Form:r t =φ0+φ1r t −1+φ2r t −2+a t ,or(1−φ1B −φ2B 2)r t =φ0+a t .2.Stationarity condition:(factor of polynomial)3.Characteristic equation:(1−φ1x −φ2x 2)=04.Mean:E (r t )=φ01−φ1−φ25.ACF:ρ0=1,ρ1=φ11−φ2,ρ =φ1ρ −1+φ2ρ −1,≥2.6.Stochastic business cycle:if φ21+4φ2<0,then r t shows char-acteristics of business cycles with average lengthk =2πcos −1[φ1/(2√−φ2)],where the cosine inverse is stated in radian.If we denote thesolutions of the polynomial as a ±bi ,where i =√−1,then wehaveφ1=2a andφ2=−(a2+b2)so thatk=2πcos−1(a/√a2+b2).In R or S-Plus,one can obtain √a2+b2using the commandMod.7.Forecasts:Similar to AR(1)modelsSimulation in R:Use the command arima.sim1.y1=arima.sim(model=list(ar=c(1.3,-.4)),1000)2.y2=arima.sim(model=list(ar=c(.8,-.7)),1000) Check the ACF and PACF of the above two simulated series. Building an AR model•Order specification1.Partial ACF:(naive,but effective)–Use consecutivefittings–See Text(p.40)for details–Key feature:PACF cuts offat lag p for an AR(p)model.–Illustration:See the PACF of the U.S.quarterly growthrate of GNP.2.Akaike information criterionAIC( )=ln(˜σ2 )+2 T ,for an AR( )model,where˜σ2 is the MLE of residual vari-ance.Find the AR order with minimum AIC for ∈[0,···,P].3.BIC criterion:BIC( )=ln(˜σ2 )+ ln(T) T.•Needs a constant term?Check the sample mean.•Estimation:least squares method or maximum likelihood method •Model checking:1.Residual:obs minus thefit,i.e.1-step ahead forecast errorsat each time point.2.Residual should be close to white noise if the model is ade-e Ljung-Box statistics of residuals,but degrees offreedom is m−g,where g is the number of AR coefficientsused in the model.Example:Analysis of U.S.GNP growth rate series.R demonstration:>setwd("C:/Users/rst/teaching/bs41202/sp2010")>library(fBasics)>da=read.table("dgnp82.dat")>x=da[,1]>par(mfcol=c(2,2))%put4plots on a page>plot(x,type=’l’)%first plot>plot(x[1:175],x[2:176])%2nd plot>plot(x[1:174],x[3:176])%3rd plot>acf(x,lag=12)%4th plot>pacf(x,lag.max=12)%Compute PACF(not shown in this handout)>Box.test(x,lag=10,type=’Ljung’)%Compute Q(10)statistics Box-Ljung testdata:xX-squared=43.2345,df=10,p-value=4.515e-06>m1=ar(x,method=’mle’)%Automatic AR fitting using AIC criterion. >m1Call:ar(x=x,method="mle")Coefficients:123%An AR(3)is specified.0.34800.1793-0.1423Order selected3sigma^2estimated as9.427e-05>names(m1)[1]"order""ar""var.pred""x.mean""aic" [6]"ed""order.max""partialacf""resid""method"[11]"series""frequency""call""asy.var.coef">plot(m1$resid,type=’l’)%Plot residuals of the fitted model(not shown) >Box.test(m1$resid,lag=10,type=’Ljung’)%Model checkingBox-Ljung testdata:m1$residX-squared=7.0808,df=10,p-value=0.7178>m2=arima(x,order=c(3,0,0))%Another approach with order given.>m2Call:arima(x=x,order=c(3,0,0))Coefficients:ar1ar2ar3intercept%Fitted model is0.34800.1793-0.14230.0077%y(t)=0.348y(t-1)+0.179y(t-2) s.e.0.07450.07780.07450.0012%-0.142y(t-3)+a(t),%where y(t)=x(t)-0.0077sigma^2estimated as9.427e-05:log likelihood=565.84,aic=-1121.68 >names(m2)[1]"coef""sigma2""var.coef""mask""loglik""aic" [7]"arma""residuals""call""series""code""n.cond"[13]"model">Box.test(m2$residuals,lag=10,type=’Ljung’)Box-Ljung testdata:m2$residualsX-squared=7.0169,df=10,p-value=0.7239>plot(m2$residuals,type=’l’)%Residual plot>tsdiag(m2)%obtain3plots of model checking(not shown in handout).>>p1=c(1,-m2$coef[1:3])%Further analysis of the fitted model.>roots=polyroot(p1)>roots[1] 1.590253+1.063882e+00i-1.920152-3.530887e-17i 1.590253-1.063882e+00i>Mod(roots)[1]1.9133081.9201521.913308>k=2*pi/acos(1.590253/1.913308)>k[1]10.65638>predict(m2,8)%Prediction1-step to8-step ahead.$predTime Series:Start=177End=184Frequency=1[1]0.0012362540.0045555190.0074549060.007958518[5]0.0081814420.0079368450.0078200460.007703826$seTime Series:Start=177End=184Frequency=1[1]0.0097093220.010*******.010*******.010688994[5]0.010*******.010*******.010*******.010696190Another example:Monthly U.S.unemployment rate from Jan-uary1948to February2010.Demonstration:in class,including the R scripts fore,foreplot, and backtest.>source(‘‘fore.R’’)>source(‘‘foreplot.R’’)>source(‘‘backtest.R’’)>x=read.table("m-unrate.txt",header=T)>dim(x)[1]7464>head(x)Mon Day Year VALUE1111948 3.42211948 3.83311948 4.04411948 3.95511948 3.56611948 3.6>y=x$VALUE>plot(y)>y1=ts(y,frequency=12,start=c(1948,1))>plot(y1)>acf(diff(y))>m1=ar(diff(y),method=’mle’)>m1Call:ar(x=diff(y),method="mle")Coefficients:123456780.01000.22540.15910.10000.1274-0.0034-0.03220.008191011120.0015-0.10540.0223-0.1208Order selected12sigma^2estimated as0.03849>length(y)[1]746>t.test(diff(y))One Sample t-testdata:diff(y)t=1.0673,df=744,p-value=0.2862alternative hypothesis:true mean is not equal to095percent confidence interval:-0.0070982570.024011008sample estimates:mean of x0.008456376>m1=arima(y,order=c(12,1,0))>m1Call:arima(x=y,order=c(12,1,0))Coefficients:ar1ar2ar3ar4ar5ar6ar7ar8ar90.01050.22590.15940.10040.1276-0.0032-0.03210.00840.0018 s.e.0.03650.03660.03730.03790.03810.03840.03840.03830.0381ar10ar11ar12-0.10510.0226-0.1206s.e.0.03760.03690.0369sigma^2estimated as0.03851:log likelihood=155.65,aic=-285.29>tsdiag(m1,gof=24)>m2=arima(y,order=c(2,1,1),seasonal=list(order=c(1,0,1),period=12))>m2Call:arima(x=y,order=c(2,1,1),seasonal=list(order=c(1,0,1),period=12)) Coefficients:ar1ar2ma1sar1sma10.57930.2467-0.58070.5624-0.8199s.e.0.06220.03950.05750.07330.0536sigma^2estimated as0.03715:log likelihood=167.76,aic=-323.53>tsdiag(m2,gof=24)>p2=fore(m2,y,740,6)Time Series:Start=741End=746Frequency=1[1]9.7831779.8410139.8648439.7974719.8199089.802541Time Series:Start=741End=746Frequency=1[1]0.19288600.27282860.36345510.45141650.53947100.6263706>foreplot(p2,y,740,720)>p1=fore(m1,y,740,6)Time Series:Start=741End=746Frequency=1[1]9.7354539.7550069.7512179.6593909.6393419.545857Time Series:Start=741End=746Frequency=1[1]0.19637340.27921720.36979860.46142790.55377860.6530188>foreplot(p1,y,740,720)>>backtest(m1,y,740,1)[1]"RMSE of out-of-sample forecasts"[1]0.181902>backtest(m2,y,740,1)[1]"RMSE of out-of-sample forecasts"[1]0.1752202>backtest(m1,y,720,1)[1]"RMSE of out-of-sample forecasts"[1]0.2149663>backtest(m2,y,720,1)[1]"RMSE of out-of-sample forecasts"[1]0.2048295Moving-average(MA)modelModel withfinite memory!Some daily stock returns have minor serial correlations and can be modeled as MA or AR models.MA(1)model•Form:r t=µ+a t−θa t−1•Stationarity:always stationary.•Mean(or expectation):E(r t)=µ•Variance:Var(r t)=(1+θ2)σ2a.•Autocovariance:g1:Cov(r t,r t−1)=−θσ2ag :Cov(r t,r t− )=0for >1.Thus,r t is not related to r t−2,r t−3,···.•ACF:ρ1=−θ,ρ =0for >1.1+θ2Finite memory!MA(1)models do not remember what happen two time periods ago.•Forecast(at origin t=n):1.1-step ahead:ˆr n(1)=µ−θa n.Why?Because at time n,a n is known,but a n+1is not.2.1-step ahead forecast error:e n(1)=a n+1with varianceσ2a.3.Multi-step ahead:ˆr n( )=µfor >1.Thus,for an MA(1)model,the multi-step ahead forecasts are just the mean of the series.Why?Because the model has memory of1time period.4.Multi-step ahead forecast error:e n( )=a n+ −θa n+ −15.Variance of multi-step ahead forecast error:(1+θ2)σ2a=variance of r t.•Invertibility:–Concept:r t is a proper linear combination of a t and the past observations{r t−1,r t−2,···}.–Why is it important?It provides a simple way to obtain the shock a t.For an invertible model,the dependence of r t on r t− con-verges to zero as increases.–Condition:|θ|<1.–Invertibility of MA models is the dual property of stationarityfor AR models.MA(2)model•Form:r t=µ+a t−θ1a t−1−θ2a t−2.orr t=µ+(1−θ1B−θ2B2)a t.•Stationary with E(r t)=µ.•Variance:Var(r t)=(1+θ21+θ22)σ2a.•ACF:ρ2=0,butρ =0for >2.•Forecasts go the the mean after2periods.Building an MA model•Specification:Use sample ACFSample ACFs are all small after lag q for an MA(q)series.(See test of ACF.)•Constant term?Check the sample mean.•Estimation:use maximum likelihood method–Conditional:Assume a t=0for t≤0–Exact:Treat a t with t≤0as parameters,estimate them toobtain the likelihood function.Exact method is preferred,but it is more computing intensive.•Model checking:examine residuals(to be white noise)•Forecast:use the residuals as{a t}(which can be obtained from the data andfitted parameters)to perform forecasts.Model form in R:R parameterizes the MA(q)model asr t=µ+a t+θ1a t−1+···+θq a t−q,instead of the usual minus sign inθ.Consequently,care needs to be exercised in writing down afitted MA parameter in R.For instance, an estimateˆθ1=−0.5of an MA(1)in R indicates the model is r t=a t−0.5a t−1.Example:Daily log return of the value-weighted indexR demonstration>setwd("C:/Users/rst/teaching/bs41202/sp2010")>library(fBasics)>da=read.table("d-ibmvwew6202.txt")>dim(da)[1]101944>vw=log(1+da[,3])*100%Compute percentage log returns of the vw index.>acf(vw,lag.max=10)%ACF plot is not shon in this handout.>m1=arima(vw,order=c(0,0,1))%fits an MA(1)model>m1Call:arima(x=vw,order=c(0,0,1))Coefficients:ma1intercept0.14650.0396%The model is vw(t)=0.0396+a(t)+0.1465a(t-1).s.e.0.00990.0100sigma^2estimated as0.7785:log likelihood=-13188.48,aic=26382.96>tsdiag(m1)>predict(m1,5)$predTime Series:Start=10195End=10199Frequency=1[1]0.050362980.039608870.039608870.039608870.03960887$seTime Series:Start=10195End=10199Frequency=1[1]0.88232900.89175230.89175230.89175230.8917523Mixed ARMA model:A compact form forflexible models. Focus on the ARMA(1,1)model for1.simplicityeful for understanding GARCH models in Ch.3for volatilitymodeling.ARMA(1,1)model•Form:(1−φ1B)r t=φ0+(1−θB)a t orr t=φ1r t−1+φ0+a t−θ1a t−1.A combination of an AR(1)on the LHS and an MA(1)on theRHS.•Stationarity:same as AR(1)•Invertibility:same as MA(1)•Mean:as AR(1),i.e.E(r t)=φ01−φ1•Variance:given in the text•ACF:Satisfiesρk=φ1ρk−1for k>1,butρ1=φ1−[θ1σ2a/Var(r t)]=φ1.This is the difference between AR(1)and ARMA(1,1)models.•PACF:does not cut offatfinite lags.Building an ARMA(1,1)model•Specification:use EACF or AIC•What is EACF?How to use it?[See text].•Estimation:cond.or exact likelihood method•Model checking:as before•Forecast:MA(1)affects the1-step ahead forecast.Others are similar to those of AR(1)models.Three model representations:•ARMA form:compact,useful in estimation and forecasting•AR representation:(by long division)r t=φ0+a t+π1r t−1+π2r t−2+···It tells how r t depends on its past values.•MA representation:(by long division)r t=µ+a t+ψ1a t−1+ψ2a t−2+···It tells how r t depends on the past shocks.For a stationary series,ψi converges to zero as i→∞.Thus,the effect of any shock is transitory.The MA representation is particularly useful in computing variances of forecast errors.For a -step ahead forecast,the forecast error ise n( )=a n+ +ψ1a n+ −1+···+ψ −1a n+1.The variance of forecast error isVar[e n( )]=(1+ψ21+···+ψ2 −1)σ2a.Unit-root NonstationarityRandom walk•Form p t=p t−1+a t•Unit root?It is an AR(1)model with coefficientφ1=1.•Nonstationary:Why?Because the variance of r t diverges to infinity as t increases.•Strong memory:sample ACF approaches1for anyfinite lag.•Repeated substitution showsp t=∞i=0a t−i=∞i=0ψi a t−iwhereψi=1for all i.Thus,ψi does not converge to zero.The effect of any shock is permanent.Random walk with drift•Form:p t=µ+p t−1+a t,µ=0.•Has a unit root•Nonstationary•Strong memory•Has a time trend with slopeµ.Why?differencing•1st difference:r t=p t−p t−1If p t is the log price,then the1st difference is simply the log return.Typically,1st difference means the“change”or“incre-ment”of the original series.•Seasonal difference:y t=p t−p t−s,where s is the periodicity,e.g.s=4for quarterly series and s=12for monthly series.If p t denotes quarterly earnings,then y t is the change in earning from the same quarter one year before.Meaning of the constant term in a model•MA model:mean•AR model:related to mean•1st differenced:time slope,etc.Practical implication infinancial time seriesExample:Monthly log returns of General Electrics(GE)from1926 to1999(74years)Sample mean:1.04%,std(ˆµ)=0.26Very significant!is about12.45%a year$1investment in the beginning of1926is worth•annual compounded payment:$5907•quarterly compounded payment:$8720•monthly compounded payment:$9570•Continuously compounded?Unit-root testLet p t be the log price of an asset.To test that p t is not predictable (i.e.has a unit root),two models are commonly employed:p t=φ1p t−1+e tp t=φ0+φ1p t−1+e t.The hypothesis of interest is H o:φ1=1vs H a:φ1<1.Dickey-Fuller test is the usual t-ratio of the OLS estimate ofφ1being 1.This is the DF unit-root test.The t-ratio,however,has a non-standard limiting distribution.Let∆p t=p t−p t−1.Then,the augmented DF unit-root test for an AR(p)model is based on∆p t=c t+βp t−1+p−1i=1φi∆p t−i+e t.The t-ratio of the OLS estimate ofβis the ADF unit-root test statis-tic.Again,the statistic has a non-standard limiting distribution. Example:Consider the log series of U.S.quaterly real GDP se-ries from1947.I to2009.IV.(data from Federal Reserve Bank of St. Louis).See q-gdpc96.txt on the course web.R demonstration>library(fUnitRoots)>help(UnitrootTests)%See the tests available>da=read.table(‘‘q-gdpc96.txt’’,header=T)>gdp=log(da[,4])>adfTest(gdp,lag=4,type=c("c"))#Assume an AR(4)model for the series.Title:Augmented Dickey-Fuller TestTest Results:PARAMETER:Lag Order:4STATISTIC:Dickey-Fuller:-1.7433P VALUE:0.4076#cannot reject the null hypothesis of a unit root.***A more careful analysis>x=diff(gdp)>ord=ar(x)#identify an AR model for the differenced series.>ordCall:ar(x=x)Coefficients:1230.34290.1238-0.1226Order selected3sigma^2estimated as8.522e-05>#An AR(3)for the differenced data is confirmed.#Our previous analysis is justified.Discussion:The command arima on R.1.dealing with the constant term.If there is any differencing,noconstant is used.2.fixing some e subcommand fixed in arima.S-Plus demonstration>da=read.table("q-gdp05.txt")>dim(da)[1]2364>plot(da[,4],type=’l’)>module(finmetrics)>gdp=log(da[,4])>plot(gdp,type=’l’)>x=diff(gdp)%take the first difference>ord=ar(x)>ord$order:[1]4>adf=unitroot(gdp,trend=’c’,lags=5,method=’adf’)>adfTest for Unit Root:Augmented DF TestNull Hypothesis:there is a unit rootType of Test:t-testTest Statistic:-1.12P-value:0.7083Coefficients:lag1lag2lag3lag4lag5constant-0.00120.29540.1358-0.0864-0.11080.0168Degrees of freedom:231total;225residualResidual standard error:0.009283S-Plus demonstration>module(finmetrics)>gnp=scan(file=’dgnp82.dat’)>plot(gnp,type=’l’)>acf(gnp,lag.max=12)Call:acf(x=gnp,lag.max=12)%Plot not shown in the handout. Autocorrelation matrix:lag gnp10 1.0000210.3769320.2539430.012554-0.085965-0.107176-0.057587-0.018298-0.0772109-0.070211100.01041211-0.02301312-0.0967>acf(gnp,lag.max=12,type=’partial’)%Compute PACFCall:acf(x=gnp,lag.max=12,type="partial")Partial Correlation matrix:lag gnp110.3769220.130433-0.142144-0.098855-0.0199660.0325770.012088-0.110699-0.041510100.09811111-0.03701212-0.1533>ord=ar(gnp,order.max=10)%Perform order selection via AIC>ord$aic[1]27.5691310 2.6081086 1.58955500.00000000.2734771 2.2034466[7] 4.0171066 5.9916210 5.82648337.52300257.8223499>ord$order[1]3>m1=arima.mle(gnp,model=list(order=c(3,0,0)))%This fit misses the mean.>summary(m1)Call:arima.mle(x=gnp,model=list(order=c(3,0,0)))Method:Maximum Likelihood with likelihood conditional on3observationsARIMA order:300Value Std.Error t-value%No intercept because the program assumes it is zero. ar(1)0.454200.07597 5.9780ar(2)0.266800.08095 3.2960ar(3)-0.038170.07597-0.5024Variance-Covariance Matrix:ar(1)ar(2)ar(3)ar(1)0.005771926-0.002566306-0.001441892ar(2)-0.0025663060.006552753-0.002566306ar(3)-0.001441892-0.0025663060.005771926Estimated innovations variance:0.0001Optimizer has convergedConvergence Type:relative function convergenceAIC:-1085.0397>x=gnp-mean(gnp)%Remove sample mean.>m1=arima.mle(x,model=list(order=c(3,0,0)))>summary(m1)Call:arima.mle(x=x,model=list(order=c(3,0,0)))Method:Maximum Likelihood with likelihood conditional on3observationsARIMA order:300Value Std.Error t-valuear(1)0.35090.07523 4.664%Fitted model isar(2)0.18090.07863 2.301%x(t)=0.351x(t-1)+0.181x(t-2)-0.144x(t-3)+a(t). ar(3)-0.14430.07523-1.919Variance-Covariance Matrix:ar(1)ar(2)ar(3)ar(1)0.0056599161-0.001877448-0.0007529176ar(2)-0.00187744800.006182526-0.0018774480ar(3)-0.0007529176-0.0018774480.0056599161Estimated innovations variance:0.0001Optimizer has convergedConvergence Type:relative function convergenceAIC:-1104.1574>names(m1)[1]"model""var.coef""method""series""aic"[6]"loglik""sigma2""ed""n.cond""converged"[11]"conv.type""call">names(m1$model)[1]"order""ar""ndiff">m1$model$ar[1]0.35091070.1809056-0.1443412>>arima.diag(m1)%Model checking,plots not shown.>p1=c(1,-m1$model$ar)%Further analysis of the fitted model. >roots=polyroot(p1)>roots[1] 1.582837+1.057071e+000i-1.912355-6.609277e-017i[3] 1.582837-1.057071e+000i>Mod(roots)[1]1.9033591.9123551.903359>k=2*pi/acos(1.582837/1.903359)>k[1]10.67098>>arima.forecast(x,m1$model,8)%prediction$mean:[1]-0.00651901645-0.00317061250-0.000236329850.00028445018 [5]0.000514713150.000266189120.000145465240.00002490612 $std.err:[1]0.0097793140.010*******.010*******.010*******.010785783 [6]0.010*******.010*******.010792592S-Plus demonstration>vw=d6202[,3]%Identify the vw-index returns.>lnvw=log(1+vw)%compute log returns.>acf(lnvw,lag.max=10)%ACF plot i snot shown in this handout. Call:acf(x=lnvw,lag.max=10)Autocorrelation matrix:lag lnvw10 1.0000210.140232-0.012043-0.0027540.0029650.007576-0.014987-0.006698-0.0034109-0.00851110-0.0074>length(lnvw)[1]10194>x1=rep(1,10194)%Create a constant to handle non-zero mean>m1=arima.mle(lnvw,xreg=x1,model=list(order=c(0,0,1)))>summary(m1)Call:arima.mle(x=lnvw,model=list(order=c(0,0,1)),xreg=x1) Method:Maximum Likelihood with likelihood conditional on0observationsARIMA order:001Value Std.Error t-valuema(1)-0.14650000.009797-14.96x10.0003962NA NA%Model is vw=.000396+a(t)+0.1465a(t-1) Variance-Covariance Matrix:ma(1)ma(1)0.00009599039Estimated innovations variance:0.0001Optimizer has convergedConvergence Type:relative function convergenceAIC:-67509.2476>arima.diag(m1)%Plots not shown in this handout.>arima.forecast(lnvw,model=m1$model,6)$mean:[1]0.00015816540.00000000000.00000000000.00000000000.0000000000[6]0.0000000000%Need to add the constant0.000396to the forecast. $std.err:[1]0.0088300560.0089243610.0089243610.0089243610.008924361[6]0.008924361。