NAG Library Function Document

nag_tsa_multi_cross_corr (g13dmc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

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

nag_tsa_multi_cross_corr (g13dmc) calculates the sample cross-correlation (or cross-covariance) matrices of a multivariate time series.

2 Specification

#include <nag.h>

#include <nagg13.h>

void	nag_tsa_multi_cross_corr (Nag_CovOrCorr matrix, Integer k, Integer n, Integer m, const double w[], double wmean[], double r0[], double r[], NagError *fail)

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 function 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 Arguments

1: matrix – Nag_CovOrCorrInput

On entry: indicates whether the cross-covariance or cross-correlation matrices are to be computed.

$matrix = Nag_AutoCov$: The cross-covariance matrices are computed.
$matrix = Nag_AutoCorr$: The cross-correlation matrices are computed.

Constraint:

matrix = Nag_AutoCov

Nag_AutoCorr

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[ $k \times n$ ] – const doubleInput

On entry:

w [(t - 1) k + i - 1]

must contain the value for series

i

at time

t

, for

i = 1, 2, \dots, k

and

t = 1, 2, \dots, n

6: wmean[k] – doubleOutput

On exit: the means,

{\bar{w}}_{i}

, for

i = 1, 2, \dots, k

7: r0[ $k \times k$ ] – doubleOutput

On exit: if

matrix = Nag_AutoCov

r0 [(j - 1) k + i - 1]

contains the

(i, j)

th element of the sample cross-covariance matrix.

matrix = Nag_AutoCorr

r0 [(j - 1) k + i - 1]

i \neq j

contains the

(i, j)

th element of the sample cross-correlation matrix and

r0 [(i - 1) k + i - 1]

contains the standard deviation of the

i

th series.

8: r[ $k \times k \times m$ ] – doubleOutput

On exit: if

matrix = Nag_AutoCov

r [(l - 1) k^{2} + (j - 1) k + i - 1]

contains the

(i, j)

th element of the sample cross-covariance matrix at lag

l

matrix = Nag_AutoCorr

, then it contains the

(i, j)

th element of the sample cross-correlation matrix lag

l

, for

l = 1, 2, \dots, m

i = 1, 2, \dots, k

and

j = 1, 2, \dots, k

9: fail – NagError *Input/Output

The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_BAD_PARAM: On entry, argument $〈value〉$ had an illegal value.
NE_INT: On entry, $k = 〈value〉$ .
Constraint: $k \geq 1$ .; On entry, $n = 〈value〉$ .
Constraint: $n \geq 2$ .
NE_INT_2: On entry, $m = 〈value〉$ and $n = 〈value〉$ .
Constraint: $m \geq 1$ and $m < n$ .
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
NE_ZERO_VARIANCE: On entry, at least one of the series is such that all its elements are practically identical giving zero (or near zero) variance.

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 Function Documentnag_tsa_multi_cross_corr (g13dmc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

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 Function Document

nag_tsa_multi_cross_corr (g13dmc)