NAG Library Routine Document
G13DMF
1 Purpose
G13DMF calculates the sample crosscorrelation (or crosscovariance) matrices of a multivariate time series.
2 Specification
INTEGER 
K, N, M, KMAX, IFAIL 
REAL (KIND=nag_wp) 
W(KMAX,N), WMEAN(K), R0(KMAX,K), R(KMAX,KMAX,M) 
CHARACTER(1) 
MATRIX 

3 Description
Let
${W}_{t}={\left({w}_{1t},{w}_{2t},\dots ,{w}_{kt}\right)}^{\mathrm{T}}$, for
$t=1,2,\dots ,n$, denote
$n$ observations of a vector of
$k$ time series. The sample crosscovariance matrix at lag
$l$ is defined to be the
$k$ by
$k$ matrix
$\hat{C}\left(l\right)$, whose (
$i,j$)th element is given by
where
${\stackrel{}{w}}_{i}$ and
${\stackrel{}{w}}_{j}$ denote the sample means for the
$i$th and
$j$th series respectively. The sample crosscorrelation matrix at lag
$l$ is defined to be the
$k$ by
$k$ matrix
$\hat{R}\left(l\right)$, whose
$\left(i,j\right)$th element is given by
The number of lags, $m$, is usually taken to be at most $n/4$.
If
${W}_{t}$ follows a vector moving average model of order
$q$, then it can be shown that the theoretical crosscorrelation matrices
$\left(R\left(l\right)\right)$ are zero beyond lag
$q$. In order to help spot a possible cutoff point, the elements of
$\hat{R}\left(l\right)$ are usually compared to their approximate standard error of 1/
$\sqrt{n}$. For further details see, for example,
Wei (1990).
The routine uses a single pass through the data to compute the means and the crosscovariance matrix at lag zero. The crosscovariance matrices at further lags are then computed on a second pass through the data.
4 References
Wei W W S (1990) Time Series Analysis: Univariate and Multivariate Methods Addison–Wesley
West D H D (1979) Updating mean and variance estimates: An improved method Comm. ACM 22 532–555
5 Parameters
 1: MATRIX – CHARACTER(1)Input
On entry: indicates whether the crosscovariance or crosscorrelation matrices are to be computed.
 ${\mathbf{MATRIX}}=\text{'V'}$
 The crosscovariance matrices are computed.
 ${\mathbf{MATRIX}}=\text{'R'}$
 The crosscorrelation matrices are computed.
Constraint:
${\mathbf{MATRIX}}=\text{'V'}$ or $\text{'R'}$.
 2: K – INTEGERInput
On entry: $k$, the dimension of the multivariate time series.
Constraint:
${\mathbf{K}}\ge 1$.
 3: N – INTEGERInput
On entry: $n$, the number of observations in the series.
Constraint:
${\mathbf{N}}\ge 2$.
 4: M – INTEGERInput
On entry:
$m$, the number of crosscorrelation (or crosscovariance) matrices to be computed. If in doubt set
${\mathbf{M}}=10$. However it should be noted that
M is usually taken to be at most
${\mathbf{N}}/4$.
Constraint:
$1\le {\mathbf{M}}<{\mathbf{N}}$.
 5: W(KMAX,N) – REAL (KIND=nag_wp) arrayInput
On entry: ${\mathbf{W}}\left(\mathit{i},\mathit{t}\right)$ must contain the observation ${w}_{\mathit{i}\mathit{t}}$, for $\mathit{i}=1,2,\dots ,k$ and $\mathit{t}=1,2,\dots ,n$.
 6: KMAX – INTEGERInput
On entry: the first dimension of the arrays
W,
R0 and
R and the second dimension of the array
R as declared in the (sub)program from which G13DMF is called.
Constraint:
${\mathbf{KMAX}}\ge {\mathbf{K}}$.
 7: WMEAN(K) – REAL (KIND=nag_wp) arrayOutput
On exit: the means,
${\stackrel{}{w}}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,k$.
 8: R0(KMAX,K) – REAL (KIND=nag_wp) arrayOutput
On exit: if
$i\ne j$, then
${\mathbf{R0}}\left(i,j\right)$ contains an estimate of the
$\left(i,j\right)$th element of the crosscorrelation (or crosscovariance) matrix at lag zero,
${\hat{R}}_{ij}\left(0\right)$; if
$i=j$, then if
${\mathbf{MATRIX}}=\text{'V'}$,
${\mathbf{R0}}\left(i,i\right)$ contains the variance of the
$i$th series,
${\hat{C}}_{ii}\left(0\right)$, and if
${\mathbf{MATRIX}}=\text{'R'}$,
${\mathbf{R0}}\left(i,i\right)$ contains the standard deviation of the
$i$th series,
$\sqrt{{\hat{C}}_{ii}\left(0\right)}$.
If
${\mathbf{IFAIL}}={\mathbf{2}}$ and
${\mathbf{MATRIX}}=\text{'R'}$, then on exit all the elements in
R0 whose computation involves the zero variance are set to zero.
 9: R(KMAX,KMAX,M) – REAL (KIND=nag_wp) arrayOutput
On exit:
${\mathbf{R}}\left(\mathit{i},\mathit{j},\mathit{l}\right)$ contains an estimate of the (
$\mathit{i},\mathit{j}$)th element of the crosscorrelation (or crosscovariance) at lag
$\mathit{l}$,
${\hat{R}}_{\mathit{i}\mathit{j}}\left(\mathit{l}\right)$, for
$\mathit{l}=1,2,\dots ,m$,
$\mathit{i}=1,2,\dots ,k$ and
$\mathit{j}=1,2,\dots ,k$.
If
${\mathbf{IFAIL}}={\mathbf{2}}$ and
${\mathbf{MATRIX}}=\text{'R'}$, then on exit all the elements in
R whose computation involves the zero variance are set to zero.
 10: IFAIL – INTEGERInput/Output

On entry:
IFAIL must be set to
$0$,
$1\text{ or}1$. If you are unfamiliar with this parameter you should refer to
Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value
$1\text{ or}1$ is recommended. If the output of error messages is undesirable, then the value
$1$ is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is
$0$.
When the value $\mathbf{1}\text{ or}\mathbf{1}$ is used it is essential to test the value of IFAIL on exit.
On exit:
${\mathbf{IFAIL}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see
Section 6).
6 Error Indicators and Warnings
If on entry
${\mathbf{IFAIL}}={\mathbf{0}}$ or
${{\mathbf{1}}}$, explanatory error messages are output on the current error message unit (as defined by
X04AAF).
Errors or warnings detected by the routine:
 ${\mathbf{IFAIL}}=1$
On entry,  ${\mathbf{MATRIX}}\ne \text{'V'}$ or $\text{'R'}$, 
or  ${\mathbf{K}}<1$, 
or  ${\mathbf{N}}<2$, 
or  ${\mathbf{M}}<1$, 
or  ${\mathbf{M}}\ge {\mathbf{N}}$, 
or  ${\mathbf{KMAX}}<{\mathbf{K}}$. 
 ${\mathbf{IFAIL}}=2$
On entry, at least one of the
$k$ series is such that all its elements are practically equal giving zero (or near zero) variance. In this case if
${\mathbf{MATRIX}}=\text{'R'}$ all the correlations in
R0 and
R involving this variance are set to zero.
7 Accuracy
For a discussion of the accuracy of the onepass algorithm used to compute the sample crosscovariances at lag zero see
West (1979). For the other lags a twopass algorithm is used to compute the crosscovariances; the accuracy of this algorithm is also discussed in
West (1979). The accuracy of the crosscorrelations will depend on the accuracy of the computed crosscovariances.
The time taken is roughly proportional to $mn{k}^{2}$.
9 Example
This program computes the sample crosscorrelation matrices of two time series of length $48$, up to lag $10$. It also prints the crosscorrelation matrices together with plots of symbols indicating which elements of the correlation matrices are significant. Three * represent significance at the $0.5$% level, two * represent significance at the 1% level and a single * represents significance at the 5% level. The * are plotted above or below the line depending on whether the elements are significant in the positive or negative direction.
9.1 Program Text
Program Text (g13dmfe.f90)
9.2 Program Data
Program Data (g13dmfe.d)
9.3 Program Results
Program Results (g13dmfe.r)