NAG Library Routine Document

G13DKF

+− 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

G13DKF accepts a sequence of new observations in a multivariate time series and updates both the forecasts and the standard deviations of the forecast errors. A call to G13DJF must be made prior to calling this routine in order to calculate the elements of a reference vector together with a set of forecasts and their standard errors. On a successful exit from G13DKF the reference vector is updated so that should future series values become available these forecasts may be updated by recalling G13DKF.

2 Specification

SUBROUTINE G13DKF (

K, LMAX, M, MLAST, Z, KMAX, REF, LREF, V, PREDZ, SEFZ, WORK, IFAIL)

INTEGER	K, LMAX, M, MLAST, KMAX, LREF, IFAIL
REAL (KIND=nag_wp)	Z(KMAX,M), REF(LREF), V(KMAX,M), PREDZ(KMAX,LMAX), SEFZ(KMAX,LMAX), WORK(K*M)

3 Description

Let

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

, for

t = 1, 2, \dots, n

, denote a

k

-dimensional time series for which forecasts of

{\hat{Z}}_{n + 1}, {\hat{Z}}_{n + 2}, \dots, {\hat{Z}}_{n + l_{\max}}

have been computed using G13DJF. Given

m

further observations

Z_{n + 1}, Z_{n + 2}, \dots, Z_{n + m}

, where

m < l_{\max}

, G13DKF updates the forecasts of

Z_{n + m + 1}, Z_{n + m + 2}, \dots, Z_{n + l_{\max}}

and their corresponding standard errors.

G13DKF uses a multivariate version of the procedure described in Box and Jenkins (1976). The forecasts are updated using the

ψ

weights, computed in G13DJF. If

Z_{t}^{*}

denotes the transformed value of

Z_{t}

and

{\hat{Z}}_{t}^{*} (l)

denotes the forecast of

Z_{t + l}^{*}

from time

t

with a lead of

l

(that is the forecast of

Z_{t + l}^{*}

given observations

Z_{t}^{*}, Z_{t - 1}^{*}, \dots

), then

{\hat{Z}}_{t + 1}^{*} (l) = τ + ψ_{l} ε_{t + 1} + ψ_{l + 1} ε_{t} + ψ_{l + 2} ε_{t - 1} + \dots

and

{\hat{Z}}_{t}^{*} (l + 1) = τ + ψ_{l + 1} ε_{t} + ψ_{l + 2} ε_{t - 1} + \dots

where

τ

is a constant vector of length

k

involving the differencing parameters and the mean vector

μ

. By subtraction we obtain

{\hat{Z}}_{t + 1}^{*} (l) = {\hat{Z}}_{t}^{*} (l + 1) + ψ_{l} ε_{t + 1} .

Estimates of the residuals corresponding to the new observations are also computed as

ε_{n + l} = Z_{n + l}^{*} - {\hat{Z}}_{n}^{*} (l)

, for

l = 1, 2, \dots, m

. These may be of use in checking that the new observations conform to the previously fitted model.

On a successful exit, the reference array is updated so that G13DKF may be called again should future series values become available, see Section 8.

When a transformation has been used the forecasts and their standard errors are suitably modified to give results in terms of the original series

Z_{t}

; see Granger and Newbold (1976).

4 References

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

Granger C W J and Newbold P (1976) Forecasting transformed series J. Roy. Statist. Soc. Ser. B 38 189–203

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

5 Parameters

The quantities K, LMAX, KMAX, REF and LREF from G13DJF are suitable for input to G13DKF.

1: K – INTEGERInput: On entry: $k$ , the dimension of the multivariate time series.

Constraint: $K \geq 1$ .
2: LMAX – INTEGERInput: On entry: the number, $l_{\max}$ , of forecasts requested in the call to G13DJF.
Constraint: $LMAX \geq 2$ .
3: M – INTEGERInput: On entry: $m$ , the number of new observations available since the last call to either G13DJF or G13DKF. The number of new observations since the last call to G13DJF is then $M + MLAST$ .
Constraint: $0 < M < LMAX - MLAST$ .
4: MLAST – INTEGERInput/Output: On entry: on the first call to G13DKF, since calling G13DJF, MLAST must be set to $0$ to indicate that no new observations have yet been used to update the forecasts; on subsequent calls MLAST must contain the value of MLAST as output on the previous call to G13DKF.

On exit: is incremented by $m$ to indicate that $MLAST + M$ observations have now been used to update the forecasts since the last call to G13DJF.
MLAST must not be changed between calls to G13DKF, unless a call to G13DJF has been made between the calls in which case MLAST should be reset to $0$ .

Constraint: $0 \leq MLAST < LMAX - M$ .
5: Z(KMAX,M) – REAL (KIND=nag_wp) arrayInput: On entry: $Z (i, j)$ must contain the value of $z_{i, n + MLAST + j}$ , for $i = 1, 2, \dots, k$ and $j = 1, 2, \dots, m$ , and where $n$ is the number of observations in the time series in the last call made to G13DJF.
Constraint: if the transformation defined in TR in G13DJF for the $i$ th series is the log transformation, then $Z (i, j) > 0.0$ , and if it is the square-root transformation, then $Z (i, j) \geq 0.0$ , for $j = 1, 2, \dots, m$ and $i = 1, 2, \dots, k$ .
6: KMAX – INTEGERInput: On entry: the first dimension of the arrays Z, PREDZ, SEFZ and V as declared in the (sub)program from which G13DKF is called.
Constraint: $KMAX \geq K$ .
7: REF(LREF) – REAL (KIND=nag_wp) arrayInput/Output: On entry: must contain the first $(LMAX - 1) \times K \times K + 2 \times K \times LMAX + K$ elements of the reference vector as returned on a successful exit from G13DJF (or a previous call to G13DKF).

On exit: the elements of REF are updated. The first $(LMAX - 1) \times K \times K$ elements store the $ψ$ weights $ψ_{1}, ψ_{2}, \dots, ψ_{l_{\max} - 1}$ . The next $K \times LMAX$ elements contain the forecasts of the transformed series and the next $K \times LMAX$ elements contain the variances of the forecasts of the transformed variables; see G13DJF. The last K elements are not updated.
8: LREF – INTEGERInput: On entry: the dimension of the array REF as declared in the (sub)program from which G13DKF is called.
Constraint: $LREF \geq (LMAX - 1) \times K \times K + 2 \times K \times LMAX + K$ .
9: V(KMAX,M) – REAL (KIND=nag_wp) arrayOutput: On exit: $V (i, j)$ contains an estimate of the $i$ th component of $ε_{n + MLAST + j}$ , for $i = 1, 2, \dots, k$ and $j = 1, 2, \dots, m$ .
10: PREDZ(KMAX,LMAX) – REAL (KIND=nag_wp) arrayInput/Output: On entry: nonupdated values are kept intact.

On exit: $PREDZ (i, j)$ contains the updated forecast of $z_{i, n + j}$ , for $i = 1, 2, \dots, k$ and $j = MLAST + M + 1, \dots, l_{\max}$ .
The columns of PREDZ corresponding to the new observations since the last call to either G13DJF or G13DKF are set equal to the corresponding columns of Z.
11: SEFZ(KMAX,LMAX) – REAL (KIND=nag_wp) arrayInput/Output: On entry: nonupdated values are kept intact.

On exit: $SEFZ (i, j)$ contains an estimate of the standard error of the corresponding element of PREDZ, for $i = 1, 2, \dots, k$ and $j = MLAST + M + 1, \dots, l_{\max}$ .
The columns of SEFZ corresponding to the new observations since the last call to either G13DJF or G13DKF are set equal to zero.
12: WORK( $K \times M$ ) – REAL (KIND=nag_wp) arrayWorkspace
13: 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	$LMAX < 2$ ,
or	$M \leq 0$ ,
or	$MLAST + M \geq LMAX$ ,
or	$MLAST < 0$ ,
or	$KMAX < K$ ,
or	$LREF < (LMAX - 1) \times K \times K + 2 \times K \times LMAX + K$ .

$IFAIL = 2$: On entry, some of the elements of the reference vector, REF, have been corrupted since the most recent call to G13DJF (or G13DKF).

$IFAIL = 3$: On entry, one or more of the elements of Z is invalid, for the transformation being used; that is you may be trying to log or square root a series, some of whose values are negative.

$IFAIL = 4$: This is an unlikely exit. For one of the series, overflow will occur if the forecasts are updated. You should check whether the elements of REF have been corrupted.

7 Accuracy

The matrix computations are believed to be stable.

8 Further Comments

If a further

m^{*}

observations,

Z_{n + MLAST + 1}, Z_{n + MLAST + 2}, \dots, Z_{n + MLAST + m^{*}}

, become available, then forecasts of

Z_{n + MLAST + m^{*} + 1}, Z_{n + MLAST + m^{*} + 2}, \dots, Z_{n + l_{\max}}

may be updated by recalling G13DKF with

M = m^{*}

. Note that M and the contents of the array Z are the only quantities which need updating; MLAST is updated on exit from the previous call. On a successful exit, V contains estimates of

ε_{n + MLAST + 1}, ε_{n + MLAST + 2}, \dots, ε_{n + MLAST + m^{*}}

; columns

MLAST + 1, MLAST + 2, \dots, MLAST + m^{*}

of PREDZ contain the new observed values

Z_{n + MLAST + 1}, Z_{n + MLAST + 2}, \dots, Z_{n + MLAST + m^{*}}

and columns

MLAST + 1, MLAST + 2, \dots, MLAST + m^{*}

of SEFZ are set to zero.

9 Example

This example shows how to update the forecasts of two series each of length

48

. No transformation has been used and no differencing applied to either of the series. G13DDF is first called to fit an AR(1) model to the series.

μ

is to be estimated and

ϕ_{1} (2, 1)

constrained to be zero. A call to G13DJF is then made in order to compute forecasts of the next five series values. After one new observation becomes available the four forecasts are updated. A further observation becomes available and the three forecasts are updated.

NAG Library Routine DocumentG13DKF