滑动平均模型与自回归滑动平均模型(时间序列分析-北

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1、A?mS?1n? w?.?g8w?.?o?”w?.?.?MA(q)?.?MA(q)S?S?MA(q)X?4?O?MA(q)?.?g8w?.ARMA(p, q)?.9?)ARMA(p, q)S?g?ARMA(p, q)?.?O5ARMAS?_5fA?mS?1n? w?.?g8w?.?o?”2005cGA?mS?1n? w?.?g8w?.?o?”w?.?.?MA(q)?.?MA(q)S?S?MA(q)X?4?O?MA(q)?.?g8w?.ARMA(p, q)?.9?)ARMA(p, q)S?g?ARMA(p, q)?.?O5ARMAS?_5f?8w?.g8w?.A?mS?1n? w?.?g8w?.

2、?o?”w?.?.?MA(q)?.?MA(q)S?S?MA(q)X?4?O?MA(q)?.?g8w?.ARMA(p, q)?.9?)ARMA(p, q)S?g?ARMA(p, q)?.?O5ARMAS?_5f?!8w?.?.?MA(q)?.?MA(q)S?S?MA(q)X?4?O?MA(q)?.?g8w?.A?mS?1n? w?.?g8w?.?o?”w?.?.?MA(q)?.?MA(q)S?S?MA(q)X?4?O?MA(q)?.?g8w?.ARMA(p, q)?.9?)ARMA(p, q)S?g?ARMA(p, q)?.?O5ARMAS?_5fq?IS?Xt?g?e?vq6= 0,k= 0,

3、k qKXtq?Ik?5?kq?5?w?.A?mS?1n? w?.?g8w?.?o?”w?.?.?MA(q)?.?MA(q)S?S?MA(q)X?4?O?MA(q)?.?g8w?.ARMA(p, q)?.9?)ARMA(p, q)S?g?ARMA(p, q)?.?O5ARMAS?_5fq?IS?Xt?g?e?vq6= 0,k= 0,k qKXtq?Ik?5?kq?5?w?.A?mS?1n? w?.?g8w?.?o?”w?.?.?MA(q)?.?MA(q)S?S?MA(q)X?4?O?MA(q)?.?g8w?.ARMA(p, q)?.9?)ARMA(p, q)S?g?ARMA(p, q)?.?O

4、5ARMAS?_5fw?.?fIz?2?P?z?A?mS?xt,t = 1,.,197I?yt= xt xt1,t = 2,.,197Iyt?g?X?y?5I?Yt= t+bt1,t Z(1.1)I?.A:k 1?A?mS?1n? w?.?g8w?.?o?”w?.?.?MA(q)?.?MA(q)S?S?MA(q)X?4?O?MA(q)?.?g8w?.ARMA(p, q)?.9?)ARMA(p, q)S?g?ARMA(p, q)?.?O5ARMAS?_5fw?.?fIz?2?P?z?A?mS?xt,t = 1,.,197I?yt= xt xt1,t = 2,.,197Iyt?g?X?y?5I?Y

5、t= t+bt1,t Z(1.1)I?.A:k 1?A?mS?1n? w?.?g8w?.?o?”w?.?.?MA(q)?.?MA(q)S?S?MA(q)X?4?O?MA(q)?.?g8w?.ARMA(p, q)?.9?)ARMA(p, q)S?g?ARMA(p, q)?.?O5ARMAS?_5fw?.?fIz?2?P?z?A?mS?xt,t = 1,.,197I?yt= xt xt1,t = 2,.,197Iyt?g?X?y?5I?Yt= t+bt1,t Z(1.1)I?.A:k 1?A?mS?1n? w?.?g8w?.?o?”w?.?.?MA(q)?.?MA(q)S?S?MA(q)X?4?O

6、?MA(q)?.?g8w?.ARMA(p, q)?.9?)ARMA(p, q)S?g?ARMA(p, q)?.?O5ARMAS?_5fw?.?fIz?2?P?z?A?mS?xt,t = 1,.,197I?yt= xt xt1,t = 2,.,197Iyt?g?X?y?5I?Yt= t+bt1,t Z(1.1)I?.A:k 1?A?mS?1n? w?.?g8w?.?o?”w?.?.?MA(q)?.?MA(q)S?S?MA(q)X?4?O?MA(q)?.?g8w?.ARMA(p, q)?.9?)ARMA(p, q)S?g?ARMA(p, q)?.?O5ARMAS?_5fw?.?fIz?2?P?z?

7、A?mS?xt,t = 1,.,197I?yt= xt xt1,t = 2,.,197Iyt?g?X?y?5I?Yt= t+bt1,t Z(1.1)I?.A:k 1?A?mS?1n? w?.?g8w?.?o?”w?.?.?MA(q)?.?MA(q)S?S?MA(q)X?4?O?MA(q)?.?g8w?.ARMA(p, q)?.9?)ARMA(p, q)S?g?ARMA(p, q)?.?O5ARMAS?_5fMA(q)?.?MA(q)S?I1.1?t WN(0,2), XJb1,b2,.,bq(bq6= 0)?B(z) = 1 +qXj=1bjzj6= 0, |z| 1,KXt= t+qXj=1

8、bjtj,t Z,(1.2)q ?w?., ?MA(q)?.;Id(1.2)?S?Xtw?S?, ?MA(q)S?.IXJ?“B(z)3?vk:B(z) 6= 0 ?|z| 1, K(1.2)?_?MA(q)?., ?A?S?_?MA(q)S?.A?mS?1n? w?.?g8w?.?o?”w?.?.?MA(q)?.?MA(q)S?S?MA(q)X?4?O?MA(q)?.?g8w?.ARMA(p, q)?.9?)ARMA(p, q)S?g?ARMA(p, q)?.?O5ARMAS?_5fMA(q)?.?MA(q)S?I1.1?t WN(0,2), XJb1,b2,.,bq(bq6= 0)?B(z

9、) = 1 +qXj=1bjzj6= 0, |z| 1,KXt= t+qXj=1bjtj,t Z,(1.2)q ?w?., ?MA(q)?.;Id(1.2)?S?Xtw?S?, ?MA(q)S?.IXJ?“B(z)3?vk:B(z) 6= 0 ?|z| 1, K(1.2)?_?MA(q)?., ?A?S?_?MA(q)S?.A?mS?1n? w?.?g8w?.?o?”w?.?.?MA(q)?.?MA(q)S?S?MA(q)X?4?O?MA(q)?.?g8w?.ARMA(p, q)?.9?)ARMA(p, q)S?g?ARMA(p, q)?.?O5ARMAS?_5fMA(q)?.?MA(q)S?

10、I1.1?t WN(0,2), XJb1,b2,.,bq(bq6= 0)?B(z) = 1 +qXj=1bjzj6= 0, |z| 0)t= B1(B)Xt=Xj=0jXtj(1.4)A?mS?1n? w?.?g8w?.?o?”w?.?.?MA(q)?.?MA(q)S?S?MA(q)X?4?O?MA(q)?.?g8w?.ARMA(p, q)?.9?)ARMA(p, q)S?g?ARMA(p, q)?.?O5ARMAS?_5fMA?A?I$?MA?AR?1w(w?)?I5?Ip?MA?AR?A?I?fr?.?Xt= B(B)t,t Z(1.3)Iu?_MAB1(z)kTaylor“B1(z) =

11、Xj=0jzj,|z| 1 + ( 0)t= B1(B)Xt=Xj=0jXtj(1.4)A?mS?1n? w?.?g8w?.?o?”w?.?.?MA(q)?.?MA(q)S?S?MA(q)X?4?O?MA(q)?.?g8w?.ARMA(p, q)?.9?)ARMA(p, q)S?g?ARMA(p, q)?.?O5ARMAS?_5fMA?A?I$?MA?AR?1w(w?)?I5?Ip?MA?AR?A?I?fr?.?Xt= B(B)t,t Z(1.3)Iu?_MAB1(z)kTaylor“B1(z) =Xj=0jzj,|z| 1 + ( 0)t= B1(B)Xt=Xj=0jXtj(1.4)A?mS

12、?1n? w?.?g8w?.?o?”w?.?.?MA(q)?.?MA(q)S?S?MA(q)X?4?O?MA(q)?.?g8w?.ARMA(p, q)?.9?)ARMA(p, q)S?g?ARMA(p, q)?.?O5ARMAS?_5fMA?A?I$?MA?AR?1w(w?)?I5?Ip?MA?AR?A?I?fr?.?Xt= B(B)t,t Z(1.3)Iu?_MAB1(z)kTaylor“B1(z) =Xj=0jzj,|z| 1 + ( 0)t= B1(B)Xt=Xj=0jXtj(1.4)A?mS?1n? w?.?g8w?.?o?”w?.?.?MA(q)?.?MA(q)S?S?MA(q)X?

13、4?O?MA(q)?.?g8w?.ARMA(p, q)?.9?)ARMA(p, q)S?g?ARMA(p, q)?.?O5ARMAS?_5fMA?A?I$?MA?AR?1w(w?)?I5?Ip?MA?AR?A?I?fr?.?Xt= B(B)t,t Z(1.3)Iu?_MAB1(z)kTaylor“B1(z) =Xj=0jzj,|z| 1 + ( 0)t= B1(B)Xt=Xj=0jXtj(1.4)A?mS?1n? w?.?g8w?.?o?”w?.?.?MA(q)?.?MA(q)S?S?MA(q)X?4?O?MA(q)?.?g8w?.ARMA(p, q)?.9?)ARMA(p, q)S?g?AR

14、MA(p, q)?.?O5ARMAS?_5fMAS?g?IPb0= 1KMA(q)S?kEXt= 0k=E(XtXt+k)=(2Pqkj=0bjbj+k,0 k q0,k q(1.5)A?mS?1n? w?.?g8w?.?o?”w?.?.?MA(q)?.?MA(q)S?S?MA(q)X?4?O?MA(q)?.?g8w?.ARMA(p, q)?.9?)ARMA(p, q)S?g?ARMA(p, q)?.?O5ARMAS?_5fMAS?Innn1.1MA(q)S?Xt?g?q?q= 2bq6= 0,k= 0, |k| q.(1.6)?k?f () =22|B(ei)|2=12qXk=qkeik,

15、,.(1.7)I(1.7)?1?5?n?y1?g?“?A?mS?1n? w?.?g8w?.?o?”w?.?.?MA(q)?.?MA(q)S?S?MA(q)X?4?O?MA(q)?.?g8w?.ARMA(p, q)?.9?)ARMA(p, q)S?g?ARMA(p, q)?.?O5ARMAS?_5fMAS?Innn1.1MA(q)S?Xt?g?q?q= 2bq6= 0,k= 0, |k| q.(1.6)?k?f () =22|B(ei)|2=12qXk=qkeik, ,.(1.7)I(1.7)?1?5?n?y1?g?“?A?mS?1n? w?.?g8w?.?o?”w?.?.?MA(q)?.?MA

16、(q)S?S?MA(q)X?4?O?MA(q)?.?g8w?.ARMA(p, q)?.9?)ARMA(p, q)S?g?ARMA(p, q)?.?O5ARMAS?_5fMA(q)S?IMA(q)S?g?q?e?Xtg?q?K?7?MA(q)S?Innn1.3 ?S?Xtkg?k,KXtMA(q)S?7?q6= 0, k= 0, |k| q.I?I?y5IyI?EC?nA?mS?1n? w?.?g8w?.?o?”w?.?.?MA(q)?.?MA(q)S?S?MA(q)X?4?O?MA(q)?.?g8w?.ARMA(p, q)?.9?)ARMA(p, q)S?g?ARMA(p, q)?.?O5AR

17、MAS?_5fMA(q)S?IMA(q)S?g?q?e?Xtg?q?K?7?MA(q)S?Innn1.3 ?S?Xtkg?k,KXtMA(q)S?7?q6= 0, k= 0, |k| q.I?I?y5IyI?EC?nA?mS?1n? w?.?g8w?.?o?”w?.?.?MA(q)?.?MA(q)S?S?MA(q)X?4?O?MA(q)?.?g8w?.ARMA(p, q)?.9?)ARMA(p, q)S?g?ARMA(p, q)?.?O5ARMAS?_5fMA(q)S?IMA(q)S?g?q?e?Xtg?q?K?7?MA(q)S?Innn1.3 ?S?Xtkg?k,KXtMA(q)S?7?q6

18、= 0, k= 0, |k| q.I?I?y5IyI?EC?nA?mS?1n? w?.?g8w?.?o?”w?.?.?MA(q)?.?MA(q)S?S?MA(q)X?4?O?MA(q)?.?g8w?.ARMA(p, q)?.9?)ARMA(p, q)S?g?ARMA(p, q)?.?O5ARMAS?_5fMA(q)S?IMA(q)S?g?q?e?Xtg?q?K?7?MA(q)S?Innn1.3 ?S?Xtkg?k,KXtMA(q)S?7?q6= 0, k= 0, |k| q.I?I?y5IyI?EC?nA?mS?1n? w?.?g8w?.?o?”w?.?.?MA(q)?.?MA(q)S?S?M

19、A(q)X?4?O?MA(q)?.?g8w?.ARMA(p, q)?.9?)ARMA(p, q)S?g?ARMA(p, q)?.?O5ARMAS?_5f?n1.2I?nnn1.2?cj?cq6= 0 ?g() =12qXj=qcjeij 0, ,Kk?X?“?B(z) = 1 +qXj=1bjzj6= 0, |z| 1, bq6= 0.(1.8)?g() =22|B(ei)|2.p2?,?.Iy?-G(z) =Pqj=qcjzj+qKG(z)?2q ?z1?7kz11?z16= 1?z116= z1r?3?S=?|B(.)A?mS?1n? w?.?g8w?.?o?”w?.?.?MA(q)?.?

20、MA(q)S?S?MA(q)X?4?O?MA(q)?.?g8w?.ARMA(p, q)?.9?)ARMA(p, q)S?g?ARMA(p, q)?.?O5ARMAS?_5f?n1.2I?nnn1.2?cj?cq6= 0 ?g() =12qXj=qcjeij 0, ,Kk?X?“?B(z) = 1 +qXj=1bjzj6= 0, |z| 1, bq6= 0.(1.8)?g() =22|B(ei)|2.p2?,?.Iy?-G(z) =Pqj=qcjzj+qKG(z)?2q ?z1?7kz11?z16= 1?z116= z1r?3?S=?|B(.)A?mS?1n? w?.?g8w?.?o?”w?.?

21、.?MA(q)?.?MA(q)S?S?MA(q)X?4?O?MA(q)?.?g8w?.ARMA(p, q)?.9?)ARMA(p, q)S?g?ARMA(p, q)?.?O5ARMAS?_5fn1.3yIdg?“?f () =12qXk=qkeikId?nf () =22|B(ei|2,B(z)?Svk?.IXJB(z)3?S?vk?K?t= B1(B)Xt?5?“?t?xD(?I?Uk?Hilbert?m?y(?Brockwell & Davis proposition 3.2.1,P.89, ?P.70)A?mS?1n? w?.?g8w?.?o?”w?.?.?MA(q)?.?MA(q)S?

22、S?MA(q)X?4?O?MA(q)?.?g8w?.ARMA(p, q)?.9?)ARMA(p, q)S?g?ARMA(p, q)?.?O5ARMAS?_5fn1.3yIdg?“?f () =12qXk=qkeikId?nf () =22|B(ei|2,B(z)?Svk?.IXJB(z)3?S?vk?K?t= B1(B)Xt?5?“?t?xD(?I?Uk?Hilbert?m?y(?Brockwell & Davis proposition 3.2.1,P.89, ?P.70)A?mS?1n? w?.?g8w?.?o?”w?.?.?MA(q)?.?MA(q)S?S?MA(q)X?4?O?MA(q

23、)?.?g8w?.ARMA(p, q)?.9?)ARMA(p, q)S?g?ARMA(p, q)?.?O5ARMAS?_5fn1.3yIdg?“?f () =12qXk=qkeikId?nf () =22|B(ei|2,B(z)?Svk?.IXJB(z)3?S?vk?K?t= B1(B)Xt?5?“?t?xD(?I?Uk?Hilbert?m?y(?Brockwell & Davis proposition 3.2.1,P.89, ?P.70)A?mS?1n? w?.?g8w?.?o?”w?.?.?MA(q)?.?MA(q)S?S?MA(q)X?4?O?MA(q)?.?g8w?.ARMA(p,

24、q)?.9?)ARMA(p, q)S?g?ARMA(p, q)?.?O5ARMAS?_5fn1.3yIdg?“?f () =12qXk=qkeikId?nf () =22|B(ei|2,B(z)?Svk?.IXJB(z)3?S?vk?K?t= B1(B)Xt?5?“?t?xD(?I?Uk?Hilbert?m?y(?Brockwell & Davis proposition 3.2.1,P.89, ?P.70)A?mS?1n? w?.?g8w?.?o?”w?.?.?MA(q)?.?MA(q)S?S?MA(q)X?4?O?MA(q)?.?g8w?.ARMA(p, q)?.9?)ARMA(p, q)

25、S?g?ARMA(p, q)?.?O5ARMAS?_5f?S?I1.2?Xt: t ZS?. HxLXt?)?Hilbert?m, Hx(s)LdXt: t 6= s?)?Hilbert?m. XJHx6= Hx(s),?s Z, KXt?S?.IS?y,?sHx(s) 6= HxKk?sHx(x) 6= HxI?S?z?Xt?k?Xsvk?&EI?5?S?(,Gn?)?S?Innn1.4(7)?S?Xtk?f (),KXt?S?7?Zdf () .(1.10)A?mS?1n? w?.?g8w?.?o?”w?.?.?MA(q)?.?MA(q)S?S?MA(q)X?4?O?MA(q)?.?g8w

26、?.ARMA(p, q)?.9?)ARMA(p, q)S?g?ARMA(p, q)?.?O5ARMAS?_5f?S?I1.2?Xt: t ZS?. HxLXt?)?Hilbert?m, Hx(s)LdXt: t 6= s?)?Hilbert?m. XJHx6= Hx(s),?s Z, KXt?S?.IS?y,?sHx(s) 6= HxKk?sHx(x) 6= HxI?S?z?Xt?k?Xsvk?&EI?5?S?(,Gn?)?S?Innn1.4(7)?S?Xtk?f (),KXt?S?7?Zdf () .(1.10)A?mS?1n? w?.?g8w?.?o?”w?.?.?MA(q)?.?MA(q

27、)S?S?MA(q)X?4?O?MA(q)?.?g8w?.ARMA(p, q)?.9?)ARMA(p, q)S?g?ARMA(p, q)?.?O5ARMAS?_5f?S?I1.2?Xt: t ZS?. HxLXt?)?Hilbert?m, Hx(s)LdXt: t 6= s?)?Hilbert?m. XJHx6= Hx(s),?s Z, KXt?S?.IS?y,?sHx(s) 6= HxKk?sHx(x) 6= HxI?S?z?Xt?k?Xsvk?&EI?5?S?(,Gn?)?S?Innn1.4(7)?S?Xtk?f (),KXt?S?7?Zdf () .(1.10)A?mS?1n? w?.?

28、g8w?.?o?”w?.?.?MA(q)?.?MA(q)S?S?MA(q)X?4?O?MA(q)?.?g8w?.ARMA(p, q)?.9?)ARMA(p, q)S?g?ARMA(p, q)?.?O5ARMAS?_5f?S?I1.2?Xt: t ZS?. HxLXt?)?Hilbert?m, Hx(s)LdXt: t 6= s?)?Hilbert?m. XJHx6= Hx(s),?s Z, KXt?S?.IS?y,?sHx(s) 6= HxKk?sHx(x) 6= HxI?S?z?Xt?k?Xsvk?&EI?5?S?(,Gn?)?S?Innn1.4(7)?S?Xtk?f (),KXt?S?7?

29、Zdf () .(1.10)A?mS?1n? w?.?g8w?.?o?”w?.?.?MA(q)?.?MA(q)S?S?MA(q)X?4?O?MA(q)?.?g8w?.ARMA(p, q)?.9?)ARMA(p, q)S?g?ARMA(p, q)?.?O5ARMAS?_5f?S?I1.2?Xt: t ZS?. HxLXt?)?Hilbert?m, Hx(s)LdXt: t 6= s?)?Hilbert?m. XJHx6= Hx(s),?s Z, KXt?S?.IS?y,?sHx(s) 6= HxKk?sHx(x) 6= HxI?S?z?Xt?k?Xsvk?&EI?5?S?(,Gn?)?S?Inn

30、n1.4(7)?S?Xtk?f (),KXt?S?7?Zdf () .(1.10)A?mS?1n? w?.?g8w?.?o?”w?.?.?MA(q)?.?MA(q)S?S?MA(q)X?4?O?MA(q)?.?g8w?.ARMA(p, q)?.9?)ARMA(p, q)S?g?ARMA(p, q)?.?O5ARMAS?_5f?S?III?Y?S?IAR(p)S?S?I?_?MA(q)S?Yk?e.?S?I?k?MA(q)S?f ()k|1 ei(j)|2= | j|2?d?_?MA(q)S?S?I?S?Un?5?g?A?mS?1n? w?.?g8w?.?o?”w?.?.?MA(q)?.?MA(

31、q)S?S?MA(q)X?4?O?MA(q)?.?g8w?.ARMA(p, q)?.9?)ARMA(p, q)S?g?ARMA(p, q)?.?O5ARMAS?_5f?S?III?Y?S?IAR(p)S?S?I?_?MA(q)S?Yk?e.?S?I?k?MA(q)S?f ()k|1 ei(j)|2= | j|2?d?_?MA(q)S?S?I?S?Un?5?g?A?mS?1n? w?.?g8w?.?o?”w?.?.?MA(q)?.?MA(q)S?S?MA(q)X?4?O?MA(q)?.?g8w?.ARMA(p, q)?.9?)ARMA(p, q)S?g?ARMA(p, q)?.?O5ARMAS?

32、_5f?S?III?Y?S?IAR(p)S?S?I?_?MA(q)S?Yk?e.?S?I?k?MA(q)S?f ()k|1 ei(j)|2= | j|2?d?_?MA(q)S?S?I?S?Un?5?g?A?mS?1n? w?.?g8w?.?o?”w?.?.?MA(q)?.?MA(q)S?S?MA(q)X?4?O?MA(q)?.?g8w?.ARMA(p, q)?.9?)ARMA(p, q)S?g?ARMA(p, q)?.?O5ARMAS?_5f?S?III?Y?S?IAR(p)S?S?I?_?MA(q)S?Yk?e.?S?I?k?MA(q)S?f ()k|1 ei(j)|2= | j|2?d?_

33、?MA(q)S?S?I?S?Un?5?g?A?mS?1n? w?.?g8w?.?o?”w?.?.?MA(q)?.?MA(q)S?S?MA(q)X?4?O?MA(q)?.?g8w?.ARMA(p, q)?.9?)ARMA(p, q)S?g?ARMA(p, q)?.?O5ARMAS?_5f?S?III?Y?S?IAR(p)S?S?I?_?MA(q)S?Yk?e.?S?I?k?MA(q)S?f ()k|1 ei(j)|2= | j|2?d?_?MA(q)S?S?I?S?Un?5?g?A?mS?1n? w?.?g8w?.?o?”w?.?.?MA(q)?.?MA(q)S?S?MA(q)X?4?O?MA(

34、q)?.?g8w?.ARMA(p, q)?.9?)ARMA(p, q)S?g?ARMA(p, q)?.?O5ARMAS?_5fMA(q)X?O?IMA(q)S?X(b1,b2,.,bq)92?0,1,.,q?(I?z5?O?.A?mS?1n? w?.?g8w?.?o?”w?.?.?MA(q)?.?MA(q)S?S?MA(q)X?4?O?MA(q)?.?g8w?.ARMA(p, q)?.9?)ARMA(p, q)S?g?ARMA(p, q)?.?O5ARMAS?_5fMA(q)X?O?IMA(q)S?X(b1,b2,.,bq)92?0,1,.,q?(I?z5?O?.A?mS?1n? w?.?g8

35、w?.?o?”w?.?.?MA(q)?.?MA(q)S?S?MA(q)X?4?O?MA(q)?.?g8w?.ARMA(p, q)?.9?)ARMA(p, q)S?g?ARMA(p, q)?.?O5ARMAS?_5fMA(q)X?O?IIPA =01000001000000100000qq,C =10.0q1k=12k23k+1qq+1q+k1,q=12.q(1.11)Kk?bq=12(q AC),2= 0 CC,(1.12)? = limkk1kk.(1.13)“(1.12),(1.13)?*?OMA(q)?.?e?:.A?mS?1n? w?.?g8w?.?o?”w?.?.?MA(q)?.?M

36、A(q)S?S?MA(q)X?4?O?MA(q)?.?g8w?.ARMA(p, q)?.9?)ARMA(p, q)S?g?ARMA(p, q)?.?O5ARMAS?_5fMA(1)I?_MA(1)Xt= t+ bt1,t WN(0,2),|b| 1Ig?g?0=2(1 + b2)1=2bk=0,k 2(1=b1+b2k=0,k 2A?mS?1n? w?.?g8w?.?o?”w?.?.?MA(q)?.?MA(q)S?S?MA(q)X?4?O?MA(q)?.?g8w?.ARMA(p, q)?.9?)ARMA(p, q)S?g?ARMA(p, q)?.?O5ARMAS?_5fMA(1)I?_MA(1

37、)Xt= t+ bt1,t WN(0,2),|b| 1,|b2| 1,|b2| 1,|b2| 2Ig?X1=b1+ b1b21 + b21+ b22,2=b21 + b21+ b22k=0,k 2I?f () =22|1 + b1ei+ b2ei 2|2A?mS?1n? w?.?g8w?.?o?”w?.?.?MA(q)?.?MA(q)S?S?MA(q)X?4?O?MA(q)?.?g8w?.ARMA(p, q)?.9?)ARMA(p, q)S?g?ARMA(p, q)?.?O5ARMAS?_5fMA(2)IIIg?0=2(1 + b21+ b22)1=2(b1+ b1b2)2=2b2k=0,k

38、2Ig?X1=b1+ b1b21 + b21+ b22,2=b21 + b21+ b22k=0,k 2I?f () =22|1 + b1ei+ b2ei 2|2A?mS?1n? w?.?g8w?.?o?”w?.?.?MA(q)?.?MA(q)S?S?MA(q)X?4?O?MA(q)?.?g8w?.ARMA(p, q)?.9?)ARMA(p, q)S?g?ARMA(p, q)?.?O5ARMAS?_5fMA(2)IIIg?0=2(1 + b21+ b22)1=2(b1+ b1b2)2=2b2k=0,k 2Ig?X1=b1+ b1b21 + b21+ b22,2=b21 + b21+ b22k=0

39、,k 2I?f () =22|1 + b1ei+ b2ei 2|2A?mS?1n? w?.?g8w?.?o?”w?.?.?MA(q)?.?MA(q)S?S?MA(q)X?4?O?MA(q)?.?g8w?.ARMA(p, q)?.9?)ARMA(p, q)S?g?ARMA(p, q)?.?O5ARMAS?_5fMA(2)SfIMA(2)?Sf?Xt= t 0.36t1+ 0.85t2IA?1.084652ei1.374297I0= 2(1 + b21+ b22) = 7.40841= 2(b1+ b1b2)= 2.6642= 2b2= 3.4k= 0,k 2I(1,2) = (0.3596,0,

40、4589).I?9X?)?A?mS?1n? w?.?g8w?.?o?”w?.?.?MA(q)?.?MA(q)S?S?MA(q)X?4?O?MA(q)?.?g8w?.ARMA(p, q)?.9?)ARMA(p, q)S?g?ARMA(p, q)?.?O5ARMAS?_5fMA(2)SfIMA(2)?Sf?Xt= t 0.36t1+ 0.85t2IA?1.084652ei1.374297I0= 2(1 + b21+ b22) = 7.40841= 2(b1+ b1b2)= 2.6642= 2b2= 3.4k= 0,k 2I(1,2) = (0.3596,0,4589).I?9X?)?A?mS?1n

41、? w?.?g8w?.?o?”w?.?.?MA(q)?.?MA(q)S?S?MA(q)X?4?O?MA(q)?.?g8w?.ARMA(p, q)?.9?)ARMA(p, q)S?g?ARMA(p, q)?.?O5ARMAS?_5fMA(2)SfIMA(2)?Sf?Xt= t 0.36t1+ 0.85t2IA?1.084652ei1.374297I0= 2(1 + b21+ b22) = 7.40841= 2(b1+ b1b2)= 2.6642= 2b2= 3.4k= 0,k 2I(1,2) = (0.3596,0,4589).I?9X?)?A?mS?1n? w?.?g8w?.?o?”w?.?.

42、?MA(q)?.?MA(q)S?S?MA(q)X?4?O?MA(q)?.?g8w?.ARMA(p, q)?.9?)ARMA(p, q)S?g?ARMA(p, q)?.?O5ARMAS?_5fMA(2)SfIMA(2)?Sf?Xt= t 0.36t1+ 0.85t2IA?1.084652ei1.374297I0= 2(1 + b21+ b22) = 7.40841= 2(b1+ b1b2)= 2.6642= 2b2= 3.4k= 0,k 2I(1,2) = (0.3596,0,4589).I?9X?)?A?mS?1n? w?.?g8w?.?o?”w?.?.?MA(q)?.?MA(q)S?S?MA

43、(q)X?4?O?MA(q)?.?g8w?.ARMA(p, q)?.9?)ARMA(p, q)S?g?ARMA(p, q)?.?O5ARMAS?_5fMA(2)SfIMA(2)?Sf?Xt= t 0.36t1+ 0.85t2IA?1.084652ei1.374297I0= 2(1 + b21+ b22) = 7.40841= 2(b1+ b1b2)= 2.6642= 2b2= 3.4k= 0,k 2I(1,2) = (0.3596,0,4589).I?9X?)?A?mS?1n? w?.?g8w?.?o?”w?.?.?MA(q)?.?MA(q)S?S?MA(q)X?4?O?MA(q)?.?g8w

44、?.ARMA(p, q)?.9?)ARMA(p, q)S?g?ARMA(p, q)?.?O5ARMAS?_5f?!8w?.g8w?.ARMA(p,q)?.9?)ARMA(p,q)S?g?ARMA(p,q)?.?O5ARMAS?_5fA?mS?1n? w?.?g8w?.?o?”w?.?.?MA(q)?.?MA(q)S?S?MA(q)X?4?O?MA(q)?.?g8w?.ARMA(p, q)?.9?)ARMA(p, q)S?g?ARMA(p, q)?.?O5ARMAS?_5fARMA?.I2.1?t WN(0,2), X?“A(z)?B(z)vk?, ?vb0= 1, apbq6= 0 ?A(z)

45、 =1 pXj=1ajzj6= 0,|z| 1,B(z) =qXj=0bjzj6= 0, |z| 1,(2.1)?Xt=pXj=1ajXtj+qXj=0bjtj,t Z,(2.2)?g8w?., ?ARMA(p,q)?.?v(2.2)?S?Xt?)ARMA(p,q)S?.A?mS?1n? w?.?g8w?.?o?”w?.?.?MA(q)?.?MA(q)S?S?MA(q)X?4?O?MA(q)?.?g8w?.ARMA(p, q)?.9?)ARMA(p, q)S?g?ARMA(p, q)?.?O5ARMAS?_5fARMA?.)I?.?A(B)Xt= B(B)t,t Z(2.3)IA1(z)B(z

46、)3|z| )?(1 A(B)?k2.2?n12?A(B)(B)t= A(B)A1(B)B(B)t= B(B)t=(B)tARMA(p,q)?.(2.2)?)A?mS?1n? w?.?g8w?.?o?”w?.?.?MA(q)?.?MA(q)S?S?MA(q)X?4?O?MA(q)?.?g8w?.ARMA(p, q)?.9?)ARMA(p, q)S?g?ARMA(p, q)?.?O5ARMAS?_5fARMA?.)I?.?A(B)Xt= B(B)t,t Z(2.3)IA1(z)B(z)3|z| )?(1 A(B)?k2.2?n12?A(B)(B)t= A(B)A1(B)B(B)t= B(B)t=

47、(B)tARMA(p,q)?.(2.2)?)A?mS?1n? w?.?g8w?.?o?”w?.?.?MA(q)?.?MA(q)S?S?MA(q)X?4?O?MA(q)?.?g8w?.ARMA(p, q)?.9?)ARMA(p, q)S?g?ARMA(p, q)?.?O5ARMAS?_5fARMA?.)I?.?A(B)Xt= B(B)t,t Z(2.3)IA1(z)B(z)3|z| )?(1 A(B)?k2.2?n12?A(B)(B)t= A(B)A1(B)B(B)t= B(B)t=(B)tARMA(p,q)?.(2.2)?)A?mS?1n? w?.?g8w?.?o?”w?.?.?MA(q)?.

48、?MA(q)S?S?MA(q)X?4?O?MA(q)?.?g8w?.ARMA(p, q)?.9?)ARMA(p, q)S?g?ARMA(p, q)?.?O5ARMAS?_5fARMA?.)III?eYt(2.2)?)3(2.2)?A1(B)=?A1(B)A(B)Yt= Yt= A1(B)B(B)t= (B)tI=Xt= A1(B)B(B)t= (B)t(2.6)ARMA(p,q)?.(2.2)?)I(2.6)?j?Xt?WoldXInnn2.1d(2.6)?S?XtARMA(p,q)?.(2.2)?).A?mS?1n? w?.?g8w?.?o?”w?.?.?MA(q)?.?MA(q)S?S?M

49、A(q)X?4?O?MA(q)?.?g8w?.ARMA(p, q)?.9?)ARMA(p, q)S?g?ARMA(p, q)?.?O5ARMAS?_5fARMA?.)III?eYt(2.2)?)3(2.2)?A1(B)=?A1(B)A(B)Yt= Yt= A1(B)B(B)t= (B)tI=Xt= A1(B)B(B)t= (B)t(2.6)ARMA(p,q)?.(2.2)?)I(2.6)?j?Xt?WoldXInnn2.1d(2.6)?S?XtARMA(p,q)?.(2.2)?).A?mS?1n? w?.?g8w?.?o?”w?.?.?MA(q)?.?MA(q)S?S?MA(q)X?4?O?M

50、A(q)?.?g8w?.ARMA(p, q)?.9?)ARMA(p, q)S?g?ARMA(p, q)?.?O5ARMAS?_5fARMA?.)III?eYt(2.2)?)3(2.2)?A1(B)=?A1(B)A(B)Yt= Yt= A1(B)B(B)t= (B)tI=Xt= A1(B)B(B)t= (B)t(2.6)ARMA(p,q)?.(2.2)?)I(2.6)?j?Xt?WoldXInnn2.1d(2.6)?S?XtARMA(p,q)?.(2.2)?).A?mS?1n? w?.?g8w?.?o?”w?.?.?MA(q)?.?MA(q)S?S?MA(q)X?4?O?MA(q)?.?g8w?

51、.ARMA(p, q)?.9?)ARMA(p, q)S?g?ARMA(p, q)?.?O5ARMAS?_5fARMA?.)III?eYt(2.2)?)3(2.2)?A1(B)=?A1(B)A(B)Yt= Yt= A1(B)B(B)t= (B)tI=Xt= A1(B)B(B)t= (B)t(2.6)ARMA(p,q)?.(2.2)?)I(2.6)?j?Xt?WoldXInnn2.1d(2.6)?S?XtARMA(p,q)?.(2.2)?).A?mS?1n? w?.?g8w?.?o?”w?.?.?MA(q)?.?MA(q)S?S?MA(q)X?4?O?MA(q)?.?g8w?.ARMA(p, q)

52、?.9?)ARMA(p, q)S?g?ARMA(p, q)?.?O5ARMAS?_5fARMA?.?)I?.(2.2)?)?Yt= Xt+kXj=1r(j)1Xl=0Vl,jtltjcos(jt l,j),t Z,(2.7)?Xt?)(2.6)z1,z2,.,zk?A(z)?Np?:, zj= jeijkr(j). ?C?Vj,l,l,jdY0 X0,Y1 X1,.,Yp1 Xp1?.A?mS?1n? w?.?g8w?.?o?”w?.?.?MA(q)?.?MA(q)S?S?MA(q)X?4?O?MA(q)?.?g8w?.ARMA(p, q)?.9?)ARMA(p, q)S?g?ARMA(p,

53、q)?.?O5ARMAS?_5fARMA?.?)I|Yt Xt| kXj=1r(j)1Xl=0|Vl,j|tltj,t (2.8)I?d?ARMA?.: ?Y(p1)= = Y1= Y0= 04?Yt=pXj=1ajYtj+qXj=0bjtj,t = 1,2,.,m + n?m?Yt,t = m + 1,m + 2,.,m + n?ARMA(p,q)?.?I?A(z)k?C?m?A?mS?1n? w?.?g8w?.?o?”w?.?.?MA(q)?.?MA(q)S?S?MA(q)X?4?O?MA(q)?.?g8w?.ARMA(p, q)?.9?)ARMA(p, q)S?g?ARMA(p, q)?

54、.?O5ARMAS?_5fARMA?.?)I|Yt Xt| kXj=1r(j)1Xl=0|Vl,j|tltj,t (2.8)I?d?ARMA?.: ?Y(p1)= = Y1= Y0= 04?Yt=pXj=1ajYtj+qXj=0bjtj,t = 1,2,.,m + n?m?Yt,t = m + 1,m + 2,.,m + n?ARMA(p,q)?.?I?A(z)k?C?m?A?mS?1n? w?.?g8w?.?o?”w?.?.?MA(q)?.?MA(q)S?S?MA(q)X?4?O?MA(q)?.?g8w?.ARMA(p, q)?.9?)ARMA(p, q)S?g?ARMA(p, q)?.?O

55、5ARMAS?_5fARMA?.?)I|Yt Xt| kXj=1r(j)1Xl=0|Vl,j|tltj,t (2.8)I?d?ARMA?.: ?Y(p1)= = Y1= Y0= 04?Yt=pXj=1ajYtj+qXj=0bjtj,t = 1,2,.,m + n?m?Yt,t = m + 1,m + 2,.,m + n?ARMA(p,q)?.?I?A(z)k?C?m?A?mS?1n? w?.?g8w?.?o?”w?.?.?MA(q)?.?MA(q)S?S?MA(q)X?4?O?MA(q)?.?g8w?.ARMA(p, q)?.9?)ARMA(p, q)S?g?ARMA(p, q)?.?O5AR

56、MAS?_5fARMAS?g?Ik?dWoldXL:k=Xj=0jj+k,k = 0,1,2,.(2.10)Iduj= o(j),j d(2.10)?k= o(j),j A?mS?1n? w?.?g8w?.?o?”w?.?.?MA(q)?.?MA(q)S?S?MA(q)X?4?O?MA(q)?.?g8w?.ARMA(p, q)?.9?)ARMA(p, q)S?g?ARMA(p, q)?.?O5ARMAS?_5fARMAS?g?Ik?dWoldXL:k=Xj=0jj+k,k = 0,1,2,.(2.10)Iduj= o(j),j d(2.10)?k= o(j),j A?mS?1n? w?.?g8

57、w?.?o?”w?.?.?MA(q)?.?MA(q)S?S?MA(q)X?4?O?MA(q)?.?g8w?.ARMA(p, q)?.9?)ARMA(p, q)S?g?ARMA(p, q)?.?O5ARMAS?_5fARMAS?WoldX4?“IPbj= 0,j q, b0= 1j= 0,j 0Idap= (a1,.,ap)T, bp= (b1,.,bq)TO?j?4?j=(1,j = 0,bj+Ppj=1akjk,j = 1,2,.(2.11)A?mS?1n? w?.?g8w?.?o?”w?.?.?MA(q)?.?MA(q)S?S?MA(q)X?4?O?MA(q)?.?g8w?.ARMA(p,

58、 q)?.9?)ARMA(p, q)S?g?ARMA(p, q)?.?O5ARMAS?_5fARMAS?WoldX4?“IPbj= 0,j q, b0= 1j= 0,j 0A?mS?1n? w?.?g8w?.?o?”w?.?.?MA(q)?.?MA(q)S?S?MA(q)X?4?O?MA(q)?.?g8w?.ARMA(p, q)?.9?)ARMA(p, q)S?g?ARMA(p, q)?.?O5ARMAS?_5fARMAS?Y-W?IIIaqAR?.?ARMA?.?Y-W?: 3?.?Xtk?E(XtXtk) =pXj=1ajE(XtjXtk) +qXj=0bjE(tjXtk)=k=pXj=1

59、ajkj+qXj=0bjE(tjXl=0ltkl)=pXj=1ajkj+qXj=0bjjk,k ZI?q ?Xtk?E(XtXtk) =pXj=1ajE(XtjXtk) +qXj=0bjE(tjXtk)=k=pXj=1ajkj+qXj=0bjE(tjXl=0ltkl)=pXj=1ajkj+qXj=0bjjk,k ZI?q k?jk= 0,j = 0,1,.,q?“?k=pXj=1ajkj,k q + 1A?mS?1n? w?.?g8w?.?o?”w?.?.?MA(q)?.?MA(q)S?S?MA(q)X?4?O?MA(q)?.?g8w?.ARMA(p, q)?.9?)ARMA(p, q)S?g

60、?ARMA(p, q)?.?O5ARMAS?_5fARMAS?Y-W?IIIIo? pXj=1ajkj=2Pqj=max(0,k)bjjk,k q(2.14)Ik q?Y-W?/“?q+1q+2.q+p=qq1qp+1q+1qqp+2.q+p1q+p2qa1a2.ap(2.15)A?mS?1n? w?.?g8w?.?o?”w?.?.?MA(q)?.?MA(q)S?S?MA(q)X?4?O?MA(q)?.?g8w?.ARMA(p, q)?.9?)ARMA(p, q)S?g?ARMA(p, q)?.?O5ARMAS?_5fARMAS?Y-W?IIIIo? pXj=1ajkj=2Pqj=max(0,

61、k)bjjk,k q(2.14)Ik q?Y-W?/“?q+1q+2.q+p=qq1qp+1q+1qqp+2.q+p1q+p2qa1a2.ap(2.15)A?mS?1n? w?.?g8w?.?o?”w?.?.?MA(q)?.?MA(q)S?S?MA(q)X?4?O?MA(q)?.?g8w?.ARMA(p, q)?.9?)ARMA(p, q)S?g?ARMA(p, q)?.?O5ARMAS?_5fARMAS?Y-W?IIIIrX?P?p,q:p,q=(|q+ij|)i,j=1,2,.,p=qq1qp+1q+1qqp+2.q+p1q+p2qI?p,q?_K?)?a1,.,apA?mS?1n? w?

62、.?g8w?.?o?”w?.?.?MA(q)?.?MA(q)S?S?MA(q)X?4?O?MA(q)?.?g8w?.ARMA(p, q)?.9?)ARMA(p, q)S?g?ARMA(p, q)?.?O5ARMAS?_5fARMAS?Y-W?IIIIrX?P?p,q:p,q=(|q+ij|)i,j=1,2,.,p=qq1qp+1q+1qqp+2.q+p1q+p2qI?p,q?_K?)?a1,.,apA?mS?1n? w?.?g8w?.?o?”w?.?.?MA(q)?.?MA(q)S?S?MA(q)X?4?O?MA(q)?.?g8w?.ARMA(p, q)?.9?)ARMA(p, q)S?g?A

63、RMA(p, q)?.?O5ARMAS?_5fARMAS?Y-W?IVI)?a1,.,ap?-Yt= A(B)Xt= B(B)t,t ZIKYt?MA(q)S?g?q?y(k) =E(YtYtk)=pXj=0pXl=0jlE(XtjXtl)=pXj=0pXl=0jlk+lj,0 k qI?k3.1?)?b1,.,bq,2Iu?p,q?_, KARMA(p,q)S?g?ARMA(p,q)?.?(aTp,bTq,2) ?p?A?mS?1n? w?.?g8w?.?o?”w?.?.?MA(q)?.?MA(q)S?S?MA(q)X?4?O?MA(q)?.?g8w?.ARMA(p, q)?.9?)ARMA

64、(p, q)S?g?ARMA(p, q)?.?O5ARMAS?_5fARMAS?Y-W?IVI)?a1,.,ap?-Yt= A(B)Xt= B(B)t,t ZIKYt?MA(q)S?g?q?y(k) =E(YtYtk)=pXj=0pXl=0jlE(XtjXtl)=pXj=0pXl=0jlk+lj,0 k qI?k3.1?)?b1,.,bq,2Iu?p,q?_, KARMA(p,q)S?g?ARMA(p,q)?.?(aTp,bTq,2) ?p?A?mS?1n? w?.?g8w?.?o?”w?.?.?MA(q)?.?MA(q)S?S?MA(q)X?4?O?MA(q)?.?g8w?.ARMA(p,

65、q)?.9?)ARMA(p, q)S?g?ARMA(p, q)?.?O5ARMAS?_5fARMAS?Y-W?IVI)?a1,.,ap?-Yt= A(B)Xt= B(B)t,t ZIKYt?MA(q)S?g?q?y(k) =E(YtYtk)=pXj=0pXl=0jlE(XtjXtl)=pXj=0pXl=0jlk+lj,0 k qI?k3.1?)?b1,.,bq,2Iu?p,q?_, KARMA(p,q)S?g?ARMA(p,q)?.?(aTp,bTq,2) ?p?A?mS?1n? w?.?g8w?.?o?”w?.?.?MA(q)?.?MA(q)S?S?MA(q)X?4?O?MA(q)?.?g8

66、w?.ARMA(p, q)?.9?)ARMA(p, q)S?g?ARMA(p, q)?.?O5ARMAS?_5fARMAS?Y-W?IVI)?a1,.,ap?-Yt= A(B)Xt= B(B)t,t ZIKYt?MA(q)S?g?q?y(k) =E(YtYtk)=pXj=0pXl=0jlE(XtjXtl)=pXj=0pXl=0jlk+lj,0 k qI?k3.1?)?b1,.,bq,2Iu?p,q?_, KARMA(p,q)S?g?ARMA(p,q)?.?(aTp,bTq,2) ?p?A?mS?1n? w?.?g8w?.?o?”w?.?.?MA(q)?.?MA(q)S?S?MA(q)X?4?O

67、?MA(q)?.?g8w?.ARMA(p, q)?.9?)ARMA(p, q)S?g?ARMA(p, q)?.?O5ARMAS?_5fARMA?.AR?)IXJp,q?_Kd(2.15)?)?a1,.,apInnn2.3?k?ARMA(p,q)S?Xt ?g?,Km p?m,q?_yyy:?y,?d?n2.2?gA?mS?1n? w?.?g8w?.?o?”w?.?.?MA(q)?.?MA(q)S?S?MA(q)X?4?O?MA(q)?.?g8w?.ARMA(p, q)?.9?)ARMA(p, q)S?g?ARMA(p, q)?.?O5ARMAS?_5fARMA?.AR?)IXJp,q?_Kd(

68、2.15)?)?a1,.,apInnn2.3?k?ARMA(p,q)S?Xt ?g?,Km p?m,q?_yyy:?y,?d?n2.2?gA?mS?1n? w?.?g8w?.?o?”w?.?.?MA(q)?.?MA(q)S?S?MA(q)X?4?O?MA(q)?.?g8w?.ARMA(p, q)?.9?)ARMA(p, q)S?g?ARMA(p, q)?.?O5ARMAS?_5fARMA?.AR?)IXJp,q?_Kd(2.15)?)?a1,.,apInnn2.3?k?ARMA(p,q)S?Xt ?g?,Km p?m,q?_yyy:?y,?d?n2.2?gA?mS?1n? w?.?g8w?.?

69、o?”w?.?.?MA(q)?.?MA(q)S?S?MA(q)X?4?O?MA(q)?.?g8w?.ARMA(p, q)?.9?)ARMA(p, q)S?g?ARMA(p, q)?.?O5ARMAS?_5fARMA?.AR?)IXJp,q?_Kd(2.15)?)?a1,.,apInnn2.3?k?ARMA(p,q)S?Xt ?g?,Km p?m,q?_yyy:?y,?d?n2.2?gA?mS?1n? w?.?g8w?.?o?”w?.?.?MA(q)?.?MA(q)S?S?MA(q)X?4?O?MA(q)?.?g8w?.ARMA(p, q)?.9?)ARMA(p, q)S?g?ARMA(p, q

70、)?.?O5ARMAS?_5fARMA?.AR?)y(Y)?m,q(m m?)?K?3 = (0,1,.,m1)T6= 0?m,q = 0=m1Xl=0lq+kl= 0,k = 0,1,.,m 1(2.18)5?k m?q + k l q?q+kl=Ppj=1ajq+klj?k = mkm1Xl=0lq+kl=m1Xl=0lpXj=1ajq+klj=pXj=1ajm1Xl=0lq+(kj)l=0(d(2.18)90 k j m 1)A?mS?1n? w?.?g8w?.?o?”w?.?.?MA(q)?.?MA(q)S?S?MA(q)X?4?O?MA(q)?.?g8w?.ARMA(p, q)?.9

71、?)ARMA(p, q)S?g?ARMA(p, q)?.?O5ARMAS?_5fARMA?.AR?)y(YII)4?“?k m?dm1Xl=0lq+kl= 0,k -Yt=Pm1l=0lXtlKYt?|E(YtXtqk) =m1Xl=0lq+kl= 0,k 0?Yt?g?q 1?YtMA(q 1)S?3t WN(0,s2)?m1Xl=0lXtl=q1Xj=0tj?n2.2gA?mS?1n? w?.?g8w?.?o?”w?.?.?MA(q)?.?MA(q)S?S?MA(q)X?4?O?MA(q)?.?g8w?.ARMA(p, q)?.9?)ARMA(p, q)S?g?ARMA(p, q)?.?O

72、5ARMAS?_5fARMA?.?Innn2.4?S?Xtkg?k. q?a1,a2, ,ap(ap6= 0) ?A(z) = 1 Ppj=1ajzj?v?, ,?kpXj=1ajkj=(c 6= 0,k = q,0,k q,(2.19)KXt?ARMA(p0,q0)S?, ?p0 p, q0 q.A?mS?1n? w?.?g8w?.?o?”w?.?.?MA(q)?.?MA(q)S?S?MA(q)X?4?O?MA(q)?.?g8w?.ARMA(p, q)?.9?)ARMA(p, q)S?g?ARMA(p, q)?.?O5ARMAS?_5fn2.4yyyy:?Yt= A(B)Xt= XtPpj=

73、1ajXtj. KYt?S?, ?vE(YtXtk) = kpXj=1ajkj=(c 6= 0,k = q,0,k q.ky(k) =E(YtYtk) = E?Yt(XtkpXj=1ajXtkj)?=(c 6= 0,k = q,0,k q.Yt?g?q?.A?mS?1n? w?.?g8w?.?o?”w?.?.?MA(q)?.?MA(q)S?S?MA(q)X?4?O?MA(q)?.?g8w?.ARMA(p, q)?.9?)ARMA(p, q)S?g?ARMA(p, q)?.?O5ARMAS?_5fn2.4yIIdn1.3?, Yt?MA(q)S?, =?3?Svk?q?X?“B(z)?B(0)

74、= b0= 1 ?A(B)Xt= Yt= B(B)t,t Z,(2.20)?t WN(0,2).XJA(z)?B(z)vkf, ?.?I?ARMA(p,q)?. K?fC(z), KkA(z) = C(z)A0(z), B(z) = C(z)B0(z). ?(2.20)CC(B)A0(B)Xt= C(B)B0(B)t.?C1(B)(w,C(z)?v?) ?IARMA(p0,q0)?.: A0(B)Xt= B0(B)t.A?mS?1n? w?.?g8w?.?o?”w?.?.?MA(q)?.?MA(q)S?S?MA(q)X?4?O?MA(q)?.?g8w?.ARMA(p, q)?.9?)ARMA(

75、p, q)S?g?ARMA(p, q)?.?O5ARMAS?_5fkn?IduARMAS?k?9)?5S?L?“?ARMA(p,q)S?(2.6)k?f () =12Xk=keik=22?Xj=0jeij?2=22?B(ei)A(ei)?2(2.21)/X(2.21)?kn?.A?mS?1n? w?.?g8w?.?o?”w?.?.?MA(q)?.?MA(q)S?S?MA(q)X?4?O?MA(q)?.?g8w?.ARMA(p, q)?.9?)ARMA(p, q)S?g?ARMA(p, q)?.?O5ARMAS?_5f?_?ARMA?.I2.23ARMA(p,q)?.?2.1 ,XJ?B(z)3

76、?:B(z) = 1 +qXj=1bjzj6= 0,|z| 1(2.22)KARMA(p,q)?.(2.2)?_?ARMA?., ?A?)?_?ARMA(p,q)S?.Iln1.4(?S?_?ARMA(p,q)S?S?.A?mS?1n? w?.?g8w?.?o?”w?.?.?MA(q)?.?MA(q)S?S?MA(q)X?4?O?MA(q)?.?g8w?.ARMA(p, q)?.9?)ARMA(p, q)S?g?ARMA(p, q)?.?O5ARMAS?_5f?_?ARMA?.I2.23ARMA(p,q)?.?2.1 ,XJ?B(z)3?:B(z) = 1 +qXj=1bjzj6= 0,|z|

77、 1(2.22)KARMA(p,q)?.(2.2)?_?ARMA?., ?A?)?_?ARMA(p,q)S?.Iln1.4(?S?_?ARMA(p,q)S?S?.A?mS?1n? w?.?g8w?.?o?”w?.?.?MA(q)?.?MA(q)S?S?MA(q)X?4?O?MA(q)?.?g8w?.ARMA(p, q)?.9?)ARMA(p, q)S?g?ARMA(p, q)?.?O5ARMAS?_5f?_?ARMA?.IIIu?_?ARMA(p,q)?.(2.3) duB1(z)A(z)3z : |z| ( 1) S)?, kTaylor“?B1(z)A(z) =Xj=0jzj,|z| ,(

78、2.23)?|j| = o(j),?j ,l?B1(B)A(B) =Pj=0jBj. 3(2.3)?B1(B), ?t= B1(B)A(B)Xt=Xj=0jXtj,t Z.(2.24)(2.24)(2.6) ?_=/“, L?_ARMA(p,q)S?D(S?p?5L.A?mS?1n? w?.?g8w?.?o?”w?.?.?MA(q)?.?MA(q)S?S?MA(q)X?4?O?MA(q)?.?g8w?.ARMA(p, q)?.9?)ARMA(p, q)S?g?ARMA(p, q)?.?O5ARMAS?_5fARMA2.1IARMA(4,2):a1= 0.9,a2= 1.4,a3= 0.7,a4

79、= 0.6;b1= 0.5,b2= 0.4.(2.25)IA(z)?1.1380e2.2062i, 1.1344e1.4896i, B(z)?2.3252,1.0752Id?mS?k?“(?)A?mS?1n? w?.?g8w?.?o?”w?.?.?MA(q)?.?MA(q)S?S?MA(q)X?4?O?MA(q)?.?g8w?.ARMA(p, q)?.9?)ARMA(p, q)S?g?ARMA(p, q)?.?O5ARMAS?_5fARMA2.1IARMA(4,2):a1= 0.9,a2= 1.4,a3= 0.7,a4= 0.6;b1= 0.5,b2= 0.4.(2.25)IA(z)?1.13

80、80e2.2062i, 1.1344e1.4896i, B(z)?2.3252,1.0752Id?mS?k?“(?)A?mS?1n? w?.?g8w?.?o?”w?.?.?MA(q)?.?MA(q)S?S?MA(q)X?4?O?MA(q)?.?g8w?.ARMA(p, q)?.9?)ARMA(p, q)S?g?ARMA(p, q)?.?O5ARMAS?_5fARMA2.1IARMA(4,2):a1= 0.9,a2= 1.4,a3= 0.7,a4= 0.6;b1= 0.5,b2= 0.4.(2.25)IA(z)?1.1380e2.2062i, 1.1344e1.4896i, B(z)?2.325

81、2,1.0752Id?mS?k?“(?)A?mS?1n? w?.?g8w?.?o?”w?.?.?MA(q)?.?MA(q)S?S?MA(q)X?4?O?MA(q)?.?g8w?.ARMA(p, q)?.9?)ARMA(p, q)S?g?ARMA(p, q)?.?O5ARMAS?_5fARMA2.2I?Xt ARMA(2,2), fi?(0,1, ,4) = (4.61, 1.06, 0.29, 0.69, 0.12)I?)ARMAI?A?mS?1n? w?.?g8w?.?o?”w?.?.?MA(q)?.?MA(q)S?S?MA(q)X?4?O?MA(q)?.?g8w?.ARMA(p, q)?.

82、9?)ARMA(p, q)S?g?ARMA(p, q)?.?O5ARMAS?_5fARMA2.2I?Xt ARMA(2,2), fi?(0,1, ,4) = (4.61, 1.06, 0.29, 0.69, 0.12)I?)ARMAI?A?mS?1n? w?.?g8w?.?o?”w?.?.?MA(q)?.?MA(q)S?S?MA(q)X?4?O?MA(q)?.?g8w?.ARMA(p, q)?.9?)ARMA(p, q)S?g?ARMA(p, q)?.?O5ARMAS?_5fARMA2.2I?Xt ARMA(2,2), fi?(0,1, ,4) = (4.61, 1.06, 0.29, 0.69, 0.12)I?)ARMAI?