Multivariate Time Series Analysis

1. Multivariate Time Series Analysis

2. Definition: Let {xt : tT} be a Multivariate time series. m(t) = mean value function of {xt : tT} = E[xt] for tT. S(t,s) = Lagged covariance matrix of {xt : tT} = E{[ xt - m(t)][ xs - m(s)]'} for t,sT

3. Definition: The time series {xt : tT} is stationary if the joint distribution of is the same as the joint distribution of for all finite subsets t1, t2, ... , tk of T and all choices of h.

4. In this case then for tT. and S(t,s) = E{[ xt - m][ xs - m]'} = E{[ xt+h - m][ xs+h - m]'} = E{[ xt-s - m][ x0 - m]'} = S(t - s) for t,sT.

5. Definition: The time series {xt : tT} is weakly stationary if : for tT. and S(t,s) = S(t - s) for t, sT.

6. In this case S(h) = E{[ xt+h - m][ xs - m]'} = Cov(xt+h,xt ) is called the Lagged covariance matrix of the process {xt : tT}

7. The Cross Correlation Function and the Cross Spectrum

8. Note:sij(h) = (i,j)th element of S(h), and is called the cross covariance function of is called the cross correlation function of

9. i) is called the cross spectrum of Definitions: Note: since sij(k) ≠ sij(-k) then fij(l) is complex. ii) If fij(l) = cij(l) - iqij(l) then cij(l) is called the Cospectrum (Coincident spectral density) and qij(l) is called the quadrature spectrum

10. iii) If fij(l) = Aij(l) exp{ifij(l)} then Aij(l) is called the Cross Amplitude Spectrum and fij(l) is called the Phase Spectrum.

11. Definition: is called the Spectral Matrix

12. The Multivariate Wiener-Khinchin Relations (p-variate) and

13. Assume that Then F(l) is: Lemma: i) Positive semidefinite: a*F(l)a ≥ 0 if a*a ≥ 0, where a is any complex vector. ii) Hermitian:F(l) = F*(l) = the Adjoint of F(l) = the complex conjugate transpose of F(l). i.e.fij(l) = .

14. The fact that F(l) is positive semidefinite also means that all square submatrices along the diagonal have a positive determinant Corrollary: Hence and or

15. Definition: = Squared Coherency function Note:

16. Definition:

17. Applications and Examples of Multivariate Spectral Analysis

18. Example I - Linear Filters

19. Let denote a bivariate time series with zero mean. Suppose that the time series {yt : tT} is constructed as follows: t = ..., -2, -1, 0, 1, 2, ...

20. The time series {yt : t T} is said to be constructed from {xt : tT} by means of a Linear Filter.

21. continuing

22. continuing Thus the spectral density of the time series {yt : tT} is:

23. Comment A: is called the Transfer function of the linear filter. is called the Gain of the filter while is called the Phase Shift of the filter.

24. Also

25. continuing

26. Thus cross spectrum of the bivariate time series is:

27. Comment B: = Squared Coherency function.

28. Example II - Linear Filterswith additive noise at the output

29. Let denote a bivariate time series with zero mean. Suppose that the time series {yt : tT} is constructed as follows: t = ..., -2, -1, 0, 1, 2, ... The noise {vt : tT} is independent of the series {xt : tT} (may be white)

31. continuing Thus the spectral density of the time series {yt : tT} is:

32. Also

33. continuing

34. Thus cross spectrum of the bivariate time series is:

35. Thus = Squared Coherency function. Noise to Signal Ratio

36. Estimation of the Cross Spectrum

37. Let denote T observations on a bivariate time series with zero mean. If the series has non-zero mean one uses in place of Again assume that T = 2m +1 is odd.

38. Then define: and with lk = 2pk/T and k = 0, 1, 2, ... , m.

39. Also and for k = 0, 1, 2, ... , m.

40. The Periodogram & the Cross-Periodogram

41. Also and for k = 0, 1, 2, ... , m.

42. Finally

43. Note: and

44. Also and

45. The sample cross-spectrum, cospectrum & quadrature spectrum

46. Similarly the asymptotic expectation of is 4pfxy(l). Recall that the periodogram has asymptotic expectation 4pfxx(l). An asymptotic unbiased estimator of fxy(l) can be obtained by dividing by 4p.

47. The sample cross spectrum

48. The sample cospectrum

49. The sample quadraturespectrum

50. The sample Cross amplitude spectrum, Phase spectrum & Squared Coherency