NAG Library Routine Document

G13DLF

+− 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

G13DLF differences and/or transforms a multivariate time series. It is intended to be used prior to G13DDF to fit a vector autoregressive moving average (VARMA) model to the differenced/transformed series.

2 Specification

SUBROUTINE G13DLF (

K, N, Z, KMAX, TR, ID, DELTA, W, ND, WORK, IFAIL)

INTEGER	K, N, KMAX, ID(K), ND, IFAIL
REAL (KIND=nag_wp)	Z(KMAX,N), DELTA(KMAX,), W(KMAX,), WORK(K*N)
CHARACTER(1)	TR(K)

3 Description

For certain time series it may first be necessary to difference the original data to obtain a stationary series before calculating autocorrelations, etc. This routine also allows you to apply either a square root or a log transformation to the original time series to stabilize the variance if required.

If the order of differencing required for the

i

th series is

d_{i}

, then the differencing operator is defined by

δ_{i} (B) = 1 - δ_{i 1} B - δ_{i 2} B^{2} - \dots - δ_{i d_{i}} B^{d_{i}}

, where

B

is the backward shift operator; that is,

B Z_{t} = Z_{t - 1}

. Let

d

denote the maximum of the orders of differencing,

d_{i}

, over the

k

series. The routine computes values of the differenced/transformed series

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

, for

t = d + 1, \dots, n

, as follows:

w_{i t} = δ_{i} (B) z_{i t}^{*}, i = 1, 2, \dots, k

where

z_{i t}^{*}

are the transformed values of the original

k

-dimensional time series

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

The differencing parameters

δ_{i j}

, for

i = 1, 2, \dots, k

and

j = 1, 2, \dots, d_{i}

, must be supplied by you. If the

i

th series does not require differencing, then

d_{i} = 0

4 References

Box G E P and Jenkins G M (1976) Time Series Analysis: Forecasting and Control (Revised Edition) Holden–Day

Wei W W S (1990) Time Series Analysis: Univariate and Multivariate Methods Addison–Wesley

5 Parameters

1: K – INTEGERInput

On entry:

k

, the dimension of the multivariate time series.

Constraint:

K \geq 1

2: N – INTEGERInput

On entry:

n

, the number of observations in the series, prior to differencing.

Constraint:

N \geq 1

3: Z(KMAX,N) – REAL (KIND=nag_wp) arrayInput

On entry:

Z (i, t)

must contain,

z_{i t}

, the

i

th component of

Z_{t}

, for

i = 1, 2, \dots, k

and

t = 1, 2, \dots, n

Constraints:

if $TR (i) ='L'$ , $Z (i, t) > 0.0$ ;
if $TR (i) ='S'$ , $Z (i, t) \geq 0.0$ , for $i = 1, 2, \dots, k$ and $t = 1, 2, \dots, n$ .

4: KMAX – INTEGERInput

On entry: the first dimension of the arrays Z, DELTA and W as declared in the (sub)program from which G13DLF is called.

Constraint:

KMAX \geq K

5: TR(K) – CHARACTER(1) arrayInput

On entry:

TR (i)

indicates whether the

i

th time series is to be transformed, for

i = 1, 2, \dots, k

$TR (i) ='N'$: No transformation is used.
$TR (i) ='L'$: A log transformation is used.
$TR (i) ='S'$: A square root transformation is used.

Constraint:

TR (i) ='N'

'L'

'S'

, for

i = 1, 2, \dots, k

6: ID(K) – INTEGER arrayInput

On entry: the order of differencing for each series,

d_{1}, d_{2}, \dots, d_{k}

Constraint:

0 \leq ID (i) < N

, for

i = 1, 2, \dots, K

7: DELTA(KMAX, $*$ ) – REAL (KIND=nag_wp) arrayInput

Note: the second dimension of the array DELTA must be at least

\max (1, d)

, where

d = \max (ID (i))

On entry: if

ID (i) > 0

, then

DELTA (i, j)

must be set equal to

δ_{i j}

, for

j = 1, 2, \dots, d_{i}

and

i = 1, 2, \dots, k

d = 0

, then DELTA is not referenced.

8: W(KMAX, $*$ ) – REAL (KIND=nag_wp) arrayOutput

Note: the second dimension of the array W must be at least

N - d

, where

d = \max (ID (i))

On exit:

W (i, t)

contains the value of

w_{i, t + d}

, for

i = 1, 2, \dots, k

and

t = 1, 2, \dots, n - d

9: ND – INTEGEROutput

On exit: the number of differenced values,

n - d

, in the series, where

d = \max (ID (i))

10: WORK( $K \times N$ ) – REAL (KIND=nag_wp) arrayWorkspace

11: 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,	$K < 1$ ,
or	$N < 1$ ,
or	$KMAX < K$ .

$IFAIL = 2$

On entry,	$ID (i) < 0$ , for some $i = 1, 2, \dots, k$ ,
or	$ID (i) \geq N$ , for some $i = 1, 2, \dots, k$ .

$IFAIL = 3$

On entry,

at least one of the first

k

elements of TR is not equal to 'N', 'L' or 'S'.

$IFAIL = 4$: On entry, one or more of the elements of Z is invalid, for the transformation requested; that is, you may be trying to log or square root a series, some of whose values are negative.

7 Accuracy

The computations are believed to be stable.

8 Further Comments

The same differencing operator does not have to be applied to all the series. For example, suppose we have

k = 2

, and wish to apply the second-order differencing operator

\nabla^{2}

to the first series and the first-order differencing operator

\nabla

to the second series:

\begin{array}{l} w_{1 t} & = \nabla^{2} z_{1 t} = {(1 - B)}^{2} z_{1 t} = (1 - 2 B + B^{2}) z_{1 t},   and \\ w_{2 t} & = \nabla z_{2 t} = (1 - B) z_{2 t} . \end{array}

Then

d_{1} = 2, d_{2} = 1

d = \max (d_{1}, d_{2}) = 2

, and

DELTA = [\begin{array}{r} δ_{11} & δ_{12} \\ δ_{21} \end{array}] = [\begin{array}{r} 2 & - 1 \\ 1 \end{array}] .

9 Example

A program to difference (non-seasonally) each of two time series of length

48

. No transformation is to be applied to either of the series.

NAG Library Routine DocumentG13DLF

+− 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

G13DLF