Multivariate Time Series Analysis

Transcription

1 Mulvre me Sere Anl

2 Le { : } be Mulvre me ere. Denon: () = men vlue uncon o { : } = E[ ] or. (,) = Lgged covrnce mr o { : } = E{[ - ()][ - ()]'} or,

3 Denon: e me ere { : } onr e jon drbuon o,,, e me e jon drbuon o,,, or ll ne ube,,..., o nd ll coce o.

4 In ce en μ( ) nd ( ) E μ (,) = E{[ - ][ - ]'} = E{[ + - ][ + - ]'} = E{[ - - ][ 0 - ]'} = ( - ) or,. or.

5 Denon: e me ere { : } wel onr : or. nd or,. μ() μ (,) = ( - )

6 In ce () = E{[ + - ][ - ]'} = Cov( +, ) clled e Lgged covrnce mr o e proce { : }

7 e Cro Correlon Funcon nd e Cro Specrum

8 Noe: j () = (,j) elemen o (), cov, j nd clled e cro covrnce uncon o j nd. j j 0 jj clled e cro correlon uncon o j nd. 0

9 ) Denon: j e clled e cro pecrum o j j nd. Noe: nce j () j (-) en j () comple. ) I j () = c j () - q j () en c j () clled e Copecrum (Concden pecrl den) nd q j () clled e qudrure pecrum

10 ) I j () = A j () ep{ j ()} en A j () clled e Cro Amplude Specrum nd j () clled e Pe Specrum.

11 Denon: clled e Specrl Mr pp p p p p j p p F

12 e Mulvre Wener-Kncn Relon (p-vre) nd p p p p e Σ F d e p p p p F Σ

13 Lemm: Aume en F() : j ) Pove emdene: *F() 0 * 0, were n comple vecor. ) Hermn:F() = F*() = e Adjon o F() = e comple conjuge rnpoe o F()..e. j () =.

14 Corrollr: e c F() pove emdene lo men ll qure ubmrce long e dgonl ve pove deermnn Hence j j jj 0 nd or j jj j * j j j jj

15 Denon: K j j jj = Squred Coerenc uncon Noe: K j

16 Denon: j j rner uncon oced w j nd.

17 Applcon nd Emple o Mulvre Specrl Anl

18 Emple I - Lner Fler

19 denoe bvre me ere w zero men. Le =..., -, -, 0,,,... : Suppoe e me ere { : } conruced ollow:

20 e me ere { : } d o be conruced rom { : } b men o Lner Fler. E ' ' ' E E ' ' ' E ' ' '

21 connung ' ' ' d e ' ' ' d e ' ' ' d e e e ' ' ' d e e e ' ' '

22 connung e e d e A d u e pecrl den o e me ere { : } : e A

23 Commen A: A e clled e rner uncon o e lner ler. A clled e Gn o e ler wle rg A clled e Pe S o e ler.

24 Alo E E E

25 connung d e d e d A e

26 u cro pecrum o e bvre me ere : : A e

27 Commen B: K = Squred Coerenc uncon. A A

28 Emple II - Lner Fler w ddve noe e oupu

29 denoe bvre me ere w zero men. Le =..., -, -, 0,,,... : Suppoe e me ere { : } conruced ollow: v e noe {v : } ndependen o e ere { : } (m be we)

30 E v v E v E E ' ' ' v v E v E ' ' ' vv ' ' ' d e d e e vv

31 connung e A were A e vv d u e pecrl den o e me ere { : } : A vv

32 Alo E v E v E E

33 connung d e d e d A e

34 u cro pecrum o e bvre me ere : : A e

35 u K = Squred Coerenc uncon. A A A vv vv Noe o Sgnl Ro

36 Mulvre me Sere Anl

37 e Cro Correlon Funcon nd e Cro Specrum

38 Noe: j () = (,j) elemen o (), cov, j nd clled e cro covrnce uncon o j nd. j j 0 jj clled e cro correlon uncon o j nd. 0

39 ) Denon: j e clled e cro pecrum o j j nd. Noe: nce j () j (-) en j () comple. ) I j () = c j () - q j () en c j () clled e Copecrum (Concden pecrl den) nd q j () clled e qudrure pecrum

40 ) I j () = A j () ep{ j ()} en A j () clled e Cro Amplude Specrum nd j () clled e Pe Specrum. Noe: nd j j e j j e

41 now j j cov, cov, j cov, j j nd j j j e e j e * j

42 Denon: K j j jj = Squred Coerenc uncon Noe: K j

43 Denon: j j rner uncon oced w j nd.

44 Emple I - Lner Fler

45 denoe bvre me ere w zero men. Le =..., -, -, 0,,,... : Suppoe e me ere { : } conruced ollow:

46 u e pecrl den o e me ere { : } : e A Commen : A e clled e rner uncon o e lner ler. A clled e Gn o e ler wle rg A clled e Pe S o e ler.

47 e cro pecrum o e bvre me ere : : A e

48 Commen B: K = Squred Coerenc uncon. A A

49 Emple II - Lner Fler w ddve noe e oupu

50 denoe bvre me ere w zero men. Le =..., -, -, 0,,,... : Suppoe e me ere { : } conruced ollow: v e noe {v : } ndependen o e ere { : } (m be we)

51 e e pecrl den o e me ere { : } : A e cro pecrum o e bvre me ere : e A vv

52 Alo nd or = 0,,,..., m. ) co( ) n( I X X X b ) co( ) n( I Y Y Y b

53 Fnll b b Y X I b b b b b b X Y I b b b b I comple conjuge o

54 Noe: nd ) ep( I ) ep( C ) ep( I ) ep( C

55 Alo nd ) ep( I ) ep( C ) ep( I ) ep( C

56 Alo nd ) ep( I ) ep( C ) ep( I ) ep( C

57 e mple cro-pecrum, copecrum & qudrure pecrum

58 Recll e perodogrm ) co( ) n( I mpoc epecon 4 (). I Smlrl e mpoc epecon o 4 (). An mpoc unbed emor o () cn be obned b dvdng b 4. I

59 e mple cro pecrum ) ep( ˆ C I

60 e mple copecrum cˆ ˆ Re C co( )

61 e mple qudrure pecrum qˆ C Im ˆ n( )

62 e mple Cro mplude pecrum, Pe pecrum & Squred Coerenc

63 A Recll Cro Amplude Specrum c q Pe Specrum n q c K Squred Coerenc uncon c q

64 u er mple couner pr cn be dened n mlr mnner. Nmel ˆ A mple Cro Amplude Specrum ˆ mple Pe Specrum n cˆ q ˆ qˆ cˆ \ Kˆ ˆ ˆ ˆ cˆ ˆ qˆ ˆ mple Squred Coerenc uncon

65 Conen Emon o e Cro-pecrum ()

66 Dnell Emor

67 = e Dnell Emor o e Copecrum d d r r d c d c ˆ ˆ, = e Dnell Emor o e qudrure pecrum d d r r d q d q ˆ ˆ,

68 Weged Covrnce Emor

69 ˆ,, c q were w m wm C co ˆ,, w m wm C n : 0,,, w m weg uc w 0 ) 0 w ) w m m m w m re equence o ) w m 0 or m.

70 Agn once e Copecrum nd Qudrure Specrum ve been emed, e Cro pecrum, Amplude Specrum, Pe Specrum nd Coerenc cn be emed generll ollow ung eer e ) Dnell Emor or b) e weged covrnce emor o c () nd q ():

71 Nmel Aˆ cˆ qˆ ˆ n ˆ Kˆ cˆ qˆ qˆ cˆ ˆ cˆ ˆ ˆ ˆ qˆ ˆ