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Multivariate Time Series Analysis

Multivariate Time Series Analysis. Definition :. Let { x t : t  T } be a Multivariate time series. m ( t ) = mean value function of { x t : t  T } = E [ x t ] for t  T .

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Multivariate Time Series Analysis

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  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)

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

  31. Also

  32. continuing

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

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

  35. Estimation of the Cross Spectrum

  36. 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.

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

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

  39. The Periodogram & the Cross-Periodogram

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

  41. Finally

  42. Note: and

  43. Also and

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

  45. 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.

  46. The sample cross spectrum

  47. The sample cospectrum

  48. The sample quadraturespectrum

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

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