金融时间序列分析 第3部分 Matlab时间序列分析.ppt

金融时间序列分析,陆贵斌2012年10月,基本思路,作为引入，先搞清楚时间序列；分析时间序列的目的：挖掘背后的规律，以利于预测未来；,内容,第3部分Matlab时间序列分析,Matlab时间序列分析工具箱,统计工具箱Statisticstoolbox计量工具箱Econometrictoolbox金融工具箱Financetoolbox,3.1统计工具箱Statisticstoolbox,单变量随机分布单变量模拟抽样多变量随机分布Copulas模拟（暂时不讲）回归分析,单变量随机分布,disttool演示各种随机分布图,t分布与Z分布,也可直接用程序，呈现随机分布x=-5:0.1:5;y=tpdf(x,5);%t分布z=normpdf(x,0,1);%标准正态分布plot(x,y,-,x,z,-.),重复抽样,randtool%可保存随机数%后面可用,z1=random(Normal,0,1,2,4)%2行4列x1=1.16500.0751-0.69650.05910.62680.35161.69611.7971,多变量随机分布,mu=23;SIGMA=11.5;1.53;%协方差矩阵r=mvnrnd(mu,SIGMA,100);%随机数对plot(r(:,1),r(:,2),+),数据的随机拟合：单变量,dfittool,多变量随机数模拟,MU=12;-3-5;SIGMA=cat(3,20;0.5,10;01);p=ones(1,2)/2;obj=gmdistribution(MU,SIGMA,p);Y=random(obj,1000);scatter(Y(:,1),Y(:,2),10,.),Copulas模拟,Copulas函数：描述多变量间相关性；描述边际分布。对于非标准多元随机分布，一大难题：相互依赖性如，MonteCarlo模拟相关性可能都需要从实际数据中获取。,rho=.7;SigmaDep=sigma.2.*1rho;rho1ZDep=mvnrnd(00,SigmaDep,n);XDep=exp(ZDep);lot(XDep(:,1),XDep(:,2),.)axis(0505)axisequalxlabel(X1)ylabel(X2),传统模型,用多元正态、Gamma、极值分布来描述缺点：边际分布都是同一族类的,n=1000;z=normrnd(0,1,n,1);hist(z,-3.75:.5:3.75),Copulas的目标,生产多元分布，存在相互依赖；通过一定的转变方法，将各自的分布、及依赖结构分离出来。Z：标准正态分布；：Z的分布函数U：Unif(0,1)分布,u=normcdf(z);hist(u,.05:.1:.95),Copulas核心,将F分布的逆分布函数应用于(0，1)均匀随机变量上，仍然是F随机分布。,可以将它转变为gamma(2,1)分布,x=gaminv(u,2,1);hist(x,.25:.5:9.75),n=1000;rho=.7;Z=mvnrnd(00,1rho;rho1,n);U=normcdf(Z);X=gaminv(U(:,1),2,1)tinv(U(:,2),5);scatterhist(X(:,1),X(:,2),Direction,out),Copulas重要理解,Sklar(1959)定理：对于多元分布H，F、G为边际分布，则C函数存在，且唯一。C(u,v),u=F(x),v=G(y)u,v:(0,1)2,C:(0,1)FarlieGumbelMorgensternfamilycopulas,若X,Y相互独立，则C=uv若Y是X的确定函数，则FrchetHoeffdingbounds递减型：W(u,v)=max(0,u+v1)递增型：M(u,v)=min(u,v)W(u,v)C(u,v)M(u,v),转换transform,twoincreasingtransformations,为了描述这种一致性，引入rankpairs(R*,S*)经验copulas,Deheuvels(1979)可以证明：Cn是C的无偏估计量，服从正态分布。,Copulas研究现状,copulas模型的推断统计仍有待深入；面向终端用户的应用太少，不够直观;金融方面：FreesandValdez1998；Cherubinietal.2004,StatisticsToolboxfunctionscompute,Probabilitydensityfunctions(copulapdf)，thecumulativedistributionfunctions(copulacdf)forGaussiancopulasRankcorrelationsfromlinearcorrelations(copulastat)andviceversa(copulaparam)Randomvectors(copularnd)Parametersforcopulasfittodata(copulafit),ProbabilityDistributionsUsedforMultivariateModeling,Copulas:GenerateCorrelatedSamplesDeterminingDependenceBetweenSimulationInputsConstructingDependentBivariateDistributionsUsingRankCorrelationCoefficientsUsingBivariateCopulasHigherDimensionCopulasArchimedeanCopulasSimulatingDependentMultivariateDataUsingCopulasExample:FittingCopulastoData,回归分析,robustdemo,3.2计量工具箱Econometrictoolbox,amodel：面向对象（类似于结构变量）,modelselection,amodel：adequatelydescribesyourdata.基础for：regressioninference,forecasting,andMonteCarlosimulation.Specificationtests:identifydatageneratingprocess.Modelcomparisonsthefitofcompetingmodels,withpenaltiesforcomplexity.Goodness-of-fitchecksassessthein-sampleadequacy,assumptionshold,out-of-sampleforecastperformance.,ModelObjects,Properties,andMethods,ModelObjectsarimagarchegarchgjrModelPropertiesParametricformofthemodelNumberofmodelparameters(e.g.,thedegreeofthemodel)Innovationdistribution(GaussianorStudentst)Amountofpresampledataneededtoinitializethemodel,Astaticconditionalmeanmodel:theordinarylinearregressionmodel.Adynamicconditionalmeanmodel,p207,arimaclass,modelobjectsforstationary,orunit-rootnonstationarylineartimeseriesmodels.includesmovingaverage(MA),autoregressive(AR),mixedautoregressiveandmovingaverage(ARMA),integrated(ARIMA),multiplicativeseasonalmodels.,SpecifyConditionalMeanModelsUsingarima,arima(p,D,q),(seasonal)ARIMAmodelp257stationaryARMAmodelp283MA模型modelMA=arima(Constant,0,MA,0.8,0.5,-0.1);impulse(modelMA,30)：响应值，2012a版出错,ARMAModel,modelARMA=arima(AR,0.6,-0.3,MA,0.4);,Methods,estimate：EstimateARIMAmodelparametersinfer：InferGJRmodelconditionalvariancesforecastsimulate,1）Simulate500datapointsfromtheARMA(2,1)modelsimModel=arima(AR,0.5,-0.3,MA,0.2,Constant,0,Variance,0.1);rng(5);Y=simulate(simModel,500);2）SpecifyanARMA(2,1)modelwithnoconstantandunknowncoefficientsandvariance.model=arima(2,0,1);model.Constant=03）FittheARMA(2,1)modeltoY.fit=estimate(model,Y),DataTransform,WhyTransform?Isolatetemporalcomponentsofinterest.Removetheeffectofnuisancecomponents(likeseasonality).Makeaseriesstationary.Reducespuriousregressioneffects.Stabilizevariabilitythatgrowswiththeleveloftheseries.Maketwoormoretimeseriesmoredirectlycomparable.,p122,afive-stepprocessforidentifying,selecting,andassessingconditionalmeanmodels(fordiscrete,univariatetimeseriesdata).ARMAEstablishthestationarityofyourtimeseries.ACF,PACFIdentifya(stationary)conditionalmeanmodelforyourdata.Specifythemodel,andestimatethemodelparameters.Conductgoodness-of-fitcheckstoensurethemodeldescribesyourdataadequately.usethemodeltoforecastorgenerateMonteCarlosimulations,步骤1识别：决定模型的阶数，数据的动态特征数据时间图，acf，pacf步骤2估计：参数估计，OLS,极大似然估计；步骤3检验：模型检验过度拟合法：拟合一个更大模型，应该不显著；残差诊断法：残差序列无线性关系，否则还有模型没有反映出来的动态特征；（acf，pacf，Ljung-Box法）,ARMA模型选择的信息法则,一开始实际数据比较杂乱，acf和pacf的规律不会非常简单、直观；难以选择模型；信息准则包含：残差平方和的函数；由于增加额外参数所丧失的自由度的惩罚项；增加一个新变量：残差平方和减少惩罚项增加目标：使信息准则的值最小。,最常用的信息准则,赤池AIC更具有有效性阶数较大Schwarz贝叶斯准则SBIC较强一致性渐进递归到正确的模型阶数平均波动性较大Hannan-Quinn准则HQIC没有一种准则明显有优势，各有特点。,画出ACF，PACF,loadData_JAustralianY=Dataset.PAU;N=length(Y);figure(1)plot(Y)figure(2)subplot(2,1,1)autocorr(Y)subplot(2,1,2)parcorr(Y),Differencethedata.,ACF线性递减，显著：非平稳dY=diff(Y);差分后序列ACF快速衰减，PACF2阶截断；表现与AR(2)类似；SpecifyandfitanARIMA(2,1,0)model.model=arima(2,1,0);fit=estimate(model,Y);coefficientsaresignificantatthe0.05significancelevel,Infertheresidualsfromthefittedmodel.Checkthattheresidualsarenormallydistributedanduncorrelated.res=infer(fit,Y);plot(res./sqrt(fit.Variance)qqplot(res)：正态分布对比autocorr(res)parcorr(res),Generateforecastsandapproximate95%forecastintervalsYf,YMSE=forecast(fit,16,Y0,Y);UB=Yf+1.96*sqrt(YMSE);LB=Yf-1.96*sqrt(YMSE);h1=plot(Y,Color,.75,.75,.75);holdonh2=plot(78:93,Yf,r,LineWidth,2);h3=plot(78:93,UB,k-,LineWidth,1.5);plot(78:93,LB,k-,LineWidth,1.5);,Ljung-BoxQ-Test,P136totestforautocorrelationatmultiplelagsjointlyloadData_OvershortY=Data;N=length(Y);h,p,Qstat,crit=lbqtest(Y,Lags,5,10,15)h=111significantautocorrelation,用于检验arch效应,loadData_EquityIdx;Y=Dataset.NASDAQ;r=price2ret(Y);N=length(r);e=r-mean(r);figure(2)subplot(2,1,1)autocorr(e.2)subplot(2,1,2)parcorr(e.2)h,p=lbqtest(e.2,Lags,5,10),MaximumLikelihoodEstimationforConditionalMeanModelsp292EstimateConditionalMeanandVarianceModelp307,SpecifyingStaticTimeSeriesModels,Demo_StaticModels.mSpecifyingStaticTimeSeriesModels.pdf,Box-JenkinsDifferencingvs.ARIMAEstimation,p288,generateMsamplepaths,eachoflengthNMarkovChainMonteCarlo(MCMC)generatedependentrandomdraws;ThesimulatemethodinEconometricsToolboxgeneratesindependentrealizations.DemonstratingtheoreticalresultsForecastingfutureeventsEstimatingtheprobabilityoffutureevents,Step1.Specifyamodel.model=arima(Constant,0.5,AR,0.7,0.25,Variance,.1);Step2.Generateonesamplepathrng(default)：控制随机数生成模式Y=simulate(model,50);Step3.Generatemanysamplepaths.Y=simulate(model,50,numPaths,1000);Step4.Oversampletheprocess.Toreducetransienteffects,（短暂的）simulatepathsoflength150,anddiscardthefirst100observations.,TransientEffects,stationaryprocessestheimpulseresponsefunctiondecaystozeroovertime.thestartingpointofthesimulationiseventuallyforgotten.Oversample:generatesamplepathslongerthanneeded,anddiscardthebeginningsamplesthatshowtransienteffects.Recycle:useafirstsimulationtogeneratepresampledataforasecondsimulation.nonstationaryprocessesthestartingpointofthesimulationisneverforgotten.default:beginwith0;specifyyourownpresampledata.,没有自相关性，但是，有相关性，称为“seriallydependent”,金融条件方差模型特征,波动聚集Volatilityclustering原因：波动率的自相关性杠杆效应Leverageeffects非对称性效应：波动率对大幅下跌的反应强于大幅上升（EGARCH,GJR）,多元时间序列分析,3.3金融工具箱Financetoolbox,HotTip,HowdoIincorporatemylogotoaslidethatwillapplytoalltheotherslides?OntheViewmenu,pointtoMaster,andthenclickSlideMasterorNotesMaster.Changeimagestotheoneyoulike,thenitwillapplytoalltheotherslides.,Youcanbrieflyaddoutlineofthisslidepageinthistextbox.,Diagram,Youcanbrieflyaddoutlineofthisslidepageinthistextbox.,Diagram,Youcanbrieflyaddoutlineofthisslidepageinthistextbox.,CycleDiagram,Youcanbrieflyaddoutlineofthisslidepageinthistextbox.,Diagram,Youcanbrieflyaddoutlineofthisslidepag