1. Multivariate Time Series Analysis

2. Let {xt : t  T} be a Multivariate time series.
Definition:
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. Definitions: i)
is called the cross spectrum of
Note: since sij(k) ≠ sij(-k) then fij(l) is complex.
ii)
If fij(l) = cij(l) - i qij(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. Lemma:
Assume that
Then F(l) is:
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. Corrollary:
The fact that F(l) is positive semidefinite also means that all square submatrices along the diagonal have a positive determinant
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 Filters with 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. Recall that the periodogram
has asymptotic expectation 4pfxx(l).
Similarly the asymptotic expectation of
is 4pfxy(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 quadrature spectrum

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

51. Recall

52. Thus their sample counter parts can be defined in a similar manner
Thus their sample counter parts can be defined in a similar manner. Namely

53. Consistent Estimation of the Cross-spectrum fxy(l)

54. Daniell Estimator

55. = The Daniell Estimator of the Cospectrum
= The Daniell Estimator of the quadrature spectrum

56. Weighted Covariance Estimator

58. Again once the Cospectrum and Quadrature Spectrum have been estimated,
The Cross spectrum, Amplitude Spectrum, Phase Spectrum and Coherency can be estimated generally as follows using either the
a) Daniell Estimator or
b) the weighted covariance estimator
of cxy(l) and qxy(l):

59. Namely