Multivariate Time Series Analysis

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

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.

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.

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.

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}

The Cross Correlation Function and the Cross Spectrum

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

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

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.

Definition: is called the Spectral Matrix

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

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) = .

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

Definition: = Squared Coherency function Note:

Definition:

Applications and Examples of Multivariate Spectral Analysis

Example I - Linear Filters

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

continuing

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

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.

Also

continuing

Thus cross spectrum of the bivariate time series is:

Comment B: = Squared Coherency function.

Example II - Linear Filters with additive noise at the output

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)

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

Also

continuing

Thus cross spectrum of the bivariate time series is:

Thus = Squared Coherency function. Noise to Signal Ratio

Estimation of the Cross Spectrum

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.

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

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

The Periodogram & the Cross-Periodogram

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

Finally

Note: and

Also and

The sample cross-spectrum, cospectrum & quadrature spectrum

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.

The sample cross spectrum

The sample cospectrum

The sample quadrature spectrum

The sample Cross amplitude spectrum, Phase spectrum & Squared Coherency

Recall

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

Consistent Estimation of the Cross-spectrum fxy(l)

Daniell Estimator

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

Weighted Covariance Estimator

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):

Namely