09-长记忆时间序列

1、第九章 长记忆时间序列模型,Long memory Long range dependence,短记忆与长记忆,长记忆时间序列的应用,monthly unemployment rate of US males. US money supply and monetary aggregates. monthly IBM revenue data. Monthly UK inflation rates. exchange rates. spot prices. consumer goods.,长记忆序列的定义,ARFIMA (FARIMA)模型 Granger and Joyeux（1980），Ho

2、sking（1981）,其中 |d|0.5时, 该过程是可逆的，并有一个线性表示。 当d0时, 该过程是长记忆的，ACF以双曲速率递减,ARFIMA(p,d,q)的性质,FI(d) process,Library(fracdiff) frac.simulation=fracdiff.sim(1000, d = .3)$series ts.plot(frac.simulation),ACF,spectrum,ARFIMA model=FI(d)+ARMA, frac.simulation=fracdiff.sim(1000,ar=0.2,ma=0.3, d = .3)$series acf(fr

3、ac.simulation) spectrum(frac.simulation),ACF,spectrum,Parameter Estimation,Use package “fracdiff” Write the code,Estimation method,用GPH方法、周期图方法、叠加方差法、差分叠加方差法、R/S法、小波方法等估计方法对长记忆参数d进行估计 这些估计结果基本上比较接近 Finite Sample Behavior Comparison,Parameteric estimators: MLE in the time domain MLE in the frequency

4、domain Semiparametric estimators: Log-periodogram regression Local Whittle approach Wavelet estimtors: Wavelet OLS estimator Wavelet MLE,R functions,fdGPH() fdSperio() aggvarFit() diffvarFit() rsFit() perFit() whittleFit() waveletFit() pengFit() fracdiff(),例子,ts.test - fracdiff.sim( 5000, ar = .2, m

5、a = -.4, d = .3) fracdiff( ts.test$series, nar = length(ts.test$ar), nma = length(ts.test$ma),Simulation,fracdiff.sim() fracdiff.sim(100, ar = .2, ma = .4, d = .3) r - fracdiff.sim(n=1500, ar=-0.9, d= 0.3) plot(as.ts(r$series),Hassler model,S=4(quarterly data), 12(monthly data). R语言中，目前没有现成的simulati

6、on,estimation code,S=4(quarterly data), 12(monthly data).,R语言中，目前没有现成的simulation,estimation code,Gegenbauer process,is Gegenbauer frequency,R语言中，目前没有现成的simulation,estimation code,Giraitis, L. and Leipus, R. (1995),If , what happens?,About Gegenbauer coefficients,定理： 一个平稳的 Gegenbauer 过程是长记忆的，如果满足 0d0

7、.5 and or if and,K-factor Gegenbauer process,is Gegenbauer frequency,R语言中，目前没有现成的simulation,estimation code,Woodward,Cheng ,Gray (1998),is Gegenbauer frequency,R语言中，目前没有现成的simulation,estimation code,Reference,Granger, C.W.J. and Joyeux, R. (1980). An introduction to long memory time series models an

8、d fractional differencing. J. Time Series Analysis,1,15-29. Hosking, J.R.M. (1981). Fractional differencing. Biometrika, 68(1), 165-176. Giraitis, L. and Leipus, R. (1995). A generalized fractionally differencing approach in long memory modeling. Lithuanian Mathematical Journal, 35, 65-81. Woodward, W.A., Cheng, Q.C. and Gray, H.L.(1998). A k-factor GARMA long-memory model. J. Time Series Analysis, 19(5), 48