NAG Library Routine Document

G13DMF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

G13DMF calculates the sample cross-correlation (or cross-covariance) matrices of a multivariate time series.

2 Specification

SUBROUTINE G13DMF (

MATRIX, K, N, M, W, KMAX, WMEAN, R0, R, IFAIL)

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} = {(w_{1 t}, w_{2 t}, \dots, w_{k t})}^{T}

, for

t = 1, 2, \dots, n

, denote

n

observations of a vector of

k

time series. The sample cross-covariance matrix at lag

l

is defined to be the

k

k

matrix

\hat{C} (l)

, whose (

i, j

)th element is given by

{\hat{C}}_{i j} (l) = \frac{1}{n} \sum_{t = l + 1}^{n} (w_{i (t - l)} - {\bar{w}}_{i}) (w_{j t} - {\bar{w}}_{j}), l = 0, 1, 2, \dots, m, ​ i = 1, 2, \dots, k ​ and ​ j = 1, 2, \dots, k,

where

{\bar{w}}_{i}

and

{\bar{w}}_{j}

denote the sample means for the

i

th and

j

th series respectively. The sample cross-correlation matrix at lag

l

is defined to be the

k

k

matrix

\hat{R} (l)

, whose

(i, j)

th element is given by

{\hat{R}}_{i j} (l) = \frac{{\hat{C}}_{i j} (l)}{\sqrt{{\hat{C}}_{i i} (0) {\hat{C}}_{j j} (0)}}, l = 0, 1, 2, \dots, m, ​ i = 1, 2, \dots, k ​ and ​ j = 1, 2, \dots, k .

The number of lags,

m

, is usually taken to be at most

n / 4

W_{t}

follows a vector moving average model of order

q

, then it can be shown that the theoretical cross-correlation matrices

(R (l))

are zero beyond lag

q

. In order to help spot a possible cut-off point, the elements of

\hat{R} (l)

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 cross-covariance matrix at lag zero. The cross-covariance 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 cross-covariance or cross-correlation matrices are to be computed.

$MATRIX ='V'$: The cross-covariance matrices are computed.
$MATRIX ='R'$: The cross-correlation matrices are computed.

Constraint:

MATRIX ='V'

'R'

2: K – INTEGERInput

On entry:

k

, the dimension of the multivariate time series.

Constraint:

K \geq 1

3: N – INTEGERInput

On entry:

n

, the number of observations in the series.

Constraint:

N \geq 2

4: M – INTEGERInput

On entry:

m

, the number of cross-correlation (or cross-covariance) matrices to be computed. If in doubt set

M = 10

. However it should be noted that M is usually taken to be at most

N / 4

Constraint:

1 \leq M < N

5: W(KMAX,N) – REAL (KIND=nag_wp) arrayInput

On entry:

W (i, t)

must contain the observation

w_{i t}

, for

i = 1, 2, \dots, k

and

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:

KMAX \geq K

7: WMEAN(K) – REAL (KIND=nag_wp) arrayOutput

On exit: the means,

{\bar{w}}_{i}

, for

i = 1, 2, \dots, k

8: R0(KMAX,K) – REAL (KIND=nag_wp) arrayOutput

On exit: if

i \neq j

, then

R0 (i, j)

contains an estimate of the

(i, j)

th element of the cross-correlation (or cross-covariance) matrix at lag zero,

{\hat{R}}_{i j} (0)

; if

i = j

, then if

MATRIX ='V'

R0 (i, i)

contains the variance of the

i

th series,

{\hat{C}}_{i i} (0)

, and if

MATRIX ='R'

R0 (i, i)

contains the standard deviation of the

i

th series,

\sqrt{{\hat{C}}_{i i} (0)}

IFAIL = 2

and

MATRIX ='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:

R (i, j, l)

contains an estimate of the (

i, j

)th element of the cross-correlation (or cross-covariance) at lag

l

{\hat{R}}_{i j} (l)

, for

l = 1, 2, \dots, m

i = 1, 2, \dots, k

and

j = 1, 2, \dots, k

IFAIL = 2

and

MATRIX ='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 ​ 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 ​ 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 $- 1 or 1$ is used it is essential to test the value of IFAIL on exit.

On exit:

IFAIL = 0

unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

IFAIL = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by X04AAF).

Errors or warnings detected by the routine:

$IFAIL = 1$

On entry,	$MATRIX \neq'V'$ or $'R'$ ,
or	$K < 1$ ,
or	$N < 2$ ,
or	$M < 1$ ,
or	$M \geq N$ ,
or	$KMAX < K$ .

$IFAIL = 2$: 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 $MATRIX ='R'$ all the correlations in R0 and R involving this variance are set to zero.

7 Accuracy

For a discussion of the accuracy of the one-pass algorithm used to compute the sample cross-covariances at lag zero see West (1979). For the other lags a two-pass algorithm is used to compute the cross-covariances; the accuracy of this algorithm is also discussed in West (1979). The accuracy of the cross-correlations will depend on the accuracy of the computed cross-covariances.

8 Further Comments

The time taken is roughly proportional to

m n k^{2}

9 Example

This program computes the sample cross-correlation matrices of two time series of length

48

, up to lag

10

. It also prints the cross-correlation 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.

NAG Library Routine DocumentG13DMF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine Document

G13DMF