NAG Library Routine Document

G13DJF

+− 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

G13DJF computes forecasts of a multivariate time series. It is assumed that a vector ARMA model has already been fitted to the appropriately differenced/transformed time series using G13DDF. The standard deviations of the forecast errors are also returned. A reference vector is set up so that, should future series values become available, the forecasts and their standard errors may be updated by calling G13DKF.

2 Specification

SUBROUTINE G13DJF (

K, N, Z, KMAX, TR, ID, DELTA, IP, IQ, MEAN, PAR, LPAR, QQ, V, LMAX, PREDZ, SEFZ, REF, LREF, WORK, LWORK, IWORK, LIWORK, IFAIL)

INTEGER	K, N, KMAX, ID(K), IP, IQ, LPAR, LMAX, LREF, LWORK, IWORK(LIWORK), LIWORK, IFAIL
REAL (KIND=nag_wp)	Z(KMAX,N), DELTA(KMAX,), PAR(LPAR), QQ(KMAX,K), V(KMAX,), PREDZ(KMAX,LMAX), SEFZ(KMAX,LMAX), REF(LREF), WORK(LWORK)
CHARACTER(1)	TR(K), MEAN

3 Description

Let the vector

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

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

are required. Let

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

be defined as follows:

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

where

δ_{i} (B)

is the differencing operator applied to the

i

th series and where

z_{i t}^{*}

is equal to either

z_{i t}

\sqrt{z_{i t}}

\log_{e} z_{i t}

depending on whether or not a transformation was required to stabilize the variance before fitting the model.

If the order of differencing required for the

i

th series is

d_{i}

, then the differencing operator for the

i

th series 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}

. 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

W_{t}

is assumed to follow a multivariate ARMA model of the form:

W_{t} - μ = ϕ_{1} (W_{t - 1} - μ) + ϕ_{2} (W_{t - 2} - μ) + \dots + ϕ_{p} (W_{t - p} - μ) + ε_{t} - θ_{1} ε_{t - 1} - \dots - θ_{q} ε_{t - q},

(1)

where

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

, for

t = 1, 2, \dots, n

, is a vector of

k

residual series assumed to be Normally distributed with zero mean and positive definite covariance matrix

Σ

. The components of

ε_{t}

are assumed to be uncorrelated at non-simultaneous lags. The

ϕ_{i}

and

θ_{j}

are

k

k

matrices of parameters. The matrices

ϕ_{i}

, for

i = 1, 2, \dots, p

, are the autoregressive (AR) parameter matrices, and the matrices

θ_{i}

, for

i = 1, 2, \dots, q

, the moving average (MA) parameter matrices. The parameters in the model are thus the

p

(

k

k

)

ϕ

-matrices, the

q

(

k

k

)

θ

-matrices, the mean vector

μ

and the residual error covariance matrix

Σ

. The ARMA model (1) must be both stationary and invertible; see G13DXF for a method of checking these conditions.

The ARMA model (1) may be rewritten as

ϕ (B) (δ (B) Z_{t}^{*} - μ) = θ (B) ε_{t},

where

ϕ (B)

and

θ (B)

are the autoregressive and moving average polynomials and

δ (B)

denotes the

k

k

diagonal matrix whose

i

th diagonal elements is

δ_{i} (B)

and

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

This may be rewritten as

ϕ (B) δ (B) Z_{t}^{*} = ϕ (B) μ + θ (B) ε_{t}

Z_{t}^{*} = τ + ψ (B) ε_{t} = τ + ε_{t} + ψ_{1} ε_{t - 1} + ψ_{2} ε_{t - 2} + \dots

where

ψ (B) = δ^{- 1} (B) ϕ^{- 1} (B) θ (B)

and

τ = δ^{- 1} (B) μ

is a vector of length

k

Forecasts are computed using a multivariate version of the procedure described in Box and Jenkins (1976). If

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

denotes the forecast of

Z_{n + l}^{*}

, then

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

is taken to be that linear function of

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

which minimizes the elements of

E \{e_{n} (l) e_{n}^{'} (l)\}

where

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

is the forecast error.

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

is referred to as the linear minimum mean square error forecast of

Z_{n + l}^{*}

The linear predictor which minimizes the mean square error may be expressed as

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

The forecast error at

t

for lead

l

is then

e_{n} (l) = Z_{n + l}^{*} - {\hat{Z}}_{n}^{*} (l) = ε_{n + l} + ψ_{1} ε_{n + l - 1} + ψ_{2} ε_{n + l - 2} + \dots + ψ_{l - 1} ε_{n + 1} .

Let

d = \max (d_{i})

, for

i = 1, 2, \dots, k

. Unless

q = 0

the routine requires estimates of

ε_{t}

, for

t = d + 1, \dots, n

, which are obtainable from G13DDF. The terms

ε_{t}

are assumed to be zero, for

t = n + 1, \dots, n + l_{\max}

. You may use G13DKF to update these

l_{\max}

forecasts should further observations,

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

, become available. Note that when

l_{\max}

or more further observations are available then G13DJF must be used to produce new forecasts for

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

, should they be required.

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 output quantities K, N, KMAX, IP, IQ, PAR, NPAR, QQ and V from G13DDF are suitable for input to G13DJF.

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,

Z_{t}

, prior to differencing.

Constraint:

N \geq 3

The total number of observations must exceed the total number of parameters in the model; that is

if $MEAN ='Z'$ , $N \times K > (IP + IQ) \times K \times K + K \times (K + 1) / 2$ ;
if $MEAN ='M'$ , $N \times K > (IP + IQ) \times K \times K + K + K \times (K + 1) / 2$ ,

(see the parameters IP, IQ and MEAN).

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, QQ, V, PREDZ and SEFZ as declared in the (sub)program from which G13DJF 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:

ID (i)

must specify,

d_{i}

, the order of differencing required for the

i

th series.

Constraint:

0 \leq ID (i) < N - \max (IP, IQ)

, 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

, DELTA is not referenced.

8: IP – INTEGERInput

On entry:

p

, the number of AR parameter matrices.

Constraint:

IP \geq 0

9: IQ – INTEGERInput

On entry:

q

, the number of MA parameter matrices.

Constraint:

IQ \geq 0

10: MEAN – CHARACTER(1)Input

On entry:

MEAN ='M'

, if components of

μ

have been estimated and

MEAN ='Z'

, if all elements of

μ

are to be taken as zero.

Constraint:

MEAN ='M'

'Z'

11: PAR(LPAR) – REAL (KIND=nag_wp) arrayInput

On entry: must contain the parameter estimates read in row by row in the order

ϕ_{1}, ϕ_{2}, \dots, ϕ_{p}

θ_{1}, θ_{2}, \dots, θ_{q}

μ

Thus,

if $IP > 0$ , $PAR ((l - 1) \times k \times k + (i - 1) \times k + j)$ must be set equal to an estimate of the $(i, j)$ th element of $ϕ_{l}$ , for $l = 1, 2, \dots, p$ , $i = 1, 2, \dots, k$ and $j = 1, 2, \dots, k$ ;
if $IQ > 0$ , $PAR (p \times k \times k + (l - 1) \times k \times k + (i - 1) \times k + j)$ must be set equal to an estimate of the $(i, j)$ th element of $θ_{l}$ , for $l = 1, 2, \dots, q$ , $i = 1, 2, \dots, k$ and $j = 1, 2, \dots, k$ ;
if $MEAN ='M'$ , $PAR ((p + q) \times k \times k + i)$ must be set equal to an estimate of the $i$ th component of $μ$ , for $i = 1, 2, \dots, k$ .

Constraint: the first

IP \times K \times K

elements of PAR must satisfy the stationarity condition and the next

IQ \times K \times K

elements of PAR must satisfy the invertibility condition.

12: LPAR – INTEGERInput

On entry: the dimension of the array PAR as declared in the (sub)program from which G13DJF is called.

Constraints:

if $MEAN ='Z'$ , $LPAR \geq \max (1, (IP + IQ) \times K \times K)$ ;
if $MEAN ='M'$ , $LPAR \geq (IP + IQ) \times K \times K + K$ .

13: QQ(KMAX,K) – REAL (KIND=nag_wp) arrayInput/Output

On entry:

QQ (i, j)

must contain an estimate of the

(i, j)

th element of

Σ

. The lower triangle only is needed.

Constraint:

QQ

must be positive definite.

On exit: if

IFAIL \neq 1

, then the upper triangle is set equal to the lower triangle.

14: V(KMAX, $*$ ) – REAL (KIND=nag_wp) arrayInput

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

\max (1, N - d)

, where

d = \max (ID (i))

On entry:

V (i, t)

must contain an estimate of the

i

th component of

ε_{t + d}

, for

i = 1, 2, \dots, k

and

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

q = 0

, V is not used.

15: LMAX – INTEGERInput

On entry: the number,

l_{\max}

, of forecasts required.

Constraint:

LMAX \geq 1

16: PREDZ(KMAX,LMAX) – REAL (KIND=nag_wp) arrayOutput

On exit:

PREDZ (i, l)

contains the forecast of

z_{i, n + l}

, for

i = 1, 2, \dots, k

and

l = 1, 2, \dots, l_{\max}

17: SEFZ(KMAX,LMAX) – REAL (KIND=nag_wp) arrayOutput

On exit:

SEFZ (i, l)

contains an estimate of the standard error of the forecast of

z_{i, n + l}

, for

i = 1, 2, \dots, k

and

l = 1, 2, \dots, l_{\max}

18: REF(LREF) – REAL (KIND=nag_wp) arrayOutput

On exit: the reference vector which may be used to update forecasts using G13DKF. The first

(LMAX - 1) \times K \times K

elements contain the

ψ

weight matrices,

ψ_{1}, ψ_{2}, \dots, ψ_{l_{\max} - 1}

. The next

K \times LMAX

elements contain the forecasts of the transformed series

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

and the next

K \times LMAX

contain the variances of the forecasts of the transformed variables. The last K elements are used to store the transformations for the series.

19: LREF – INTEGERInput

On entry: the dimension of the array REF as declared in the (sub)program from which G13DJF is called.

Constraint:

LREF \geq (LMAX - 1) \times K \times K + 2 \times K \times LMAX + K

20: WORK(LWORK) – REAL (KIND=nag_wp) arrayWorkspace

21: LWORK – INTEGERInput

On entry: the dimension of the array WORK as declared in the (sub)program from which G13DJF is called.

Constraint: if

r = \max (IP, IQ)

and

d = \max (ID (i))

, for

i = 1, 2, \dots, k

LWORK \geq \max \{K r (K r + 2), (IP + d + 2) K^{2} + (N + LMAX) K\}

22: IWORK(LIWORK) – INTEGER arrayWorkspace

23: LIWORK – INTEGERInput

On entry: the dimension of the array IWORK as declared in the (sub)program from which G13DJF is called.

Constraint:

LIWORK \geq K \times \max (IP, IQ)

24: 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	$N < 3$ ,
or	$KMAX < K$ ,
or	$ID (i) < 0$ for some $i = 1, 2, \dots, k$ ,
or	$ID (i) \geq N - \max (IP, IQ)$ for some $i = 1, 2, \dots, k$ ,
or	$IP < 0$ ,
or	$IQ < 0$ ,
or	$MEAN \neq'M'$ or $'Z'$ ,
or	$LPAR < (IP + IQ) \times K \times K + K$ , and $MEAN ='M'$ ,
or	$LPAR < (IP + IQ) \times K \times K$ and $MEAN ='Z'$ ,
or	$N \times K \leq (IP + IQ) \times K \times K + K + K (K + 1) / 2$ , and $MEAN ='M'$ ,
or	$N \times K \leq (IP + IQ) \times K \times K + K (K + 1) / 2$ and $MEAN ='Z'$ ,
or	$LMAX < 1$ ,
or	$LREF < (LMAX - 1) \times K \times K + 2 \times K \times LMAX + K$ ,
or	LWORK is too small,
or	LIWORK is too small.

$IFAIL = 2$

On entry,

at least one of the first

k

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

$IFAIL = 3$: On entry, one or more of the transformations requested cannot be computed; that is, you may be trying to log or square-root a series, some of whose values are negative.

$IFAIL = 4$: On entry, either QQ is not positive definite or the autoregressive parameter matrices are extremely close to or outside the stationarity region, or the moving average parameter matrices are extremely close to or outside the invertibility region. To proceed, you must supply different parameter estimates in the arrays PAR and QQ.

$IFAIL = 5$: This is an unlikely exit brought about by an excessive number of iterations being needed to evaluate the eigenvalues of the matrices required to check for stationarity and invertibility; see G13DXF. All output parameters are undefined.

$IFAIL = 6$: This is an unlikely exit which could occur if QQ is nearly non positive definite. In this case the standard deviations of the forecast errors may be non-positive. To proceed, you must supply different parameter estimates in the array QQ.

$IFAIL = 7$: This is an unlikely exit. For one of the series, overflow will occur if the forecasts are computed. You should check whether the transformations requested in the array TR are sensible. All output parameters are undefined.

7 Accuracy

The matrix 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{matrix} 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{matrix}

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}] .

Note: although differencing may already have been applied prior to the model fitting stage, the differencing parameters supplied in DELTA are part of the model definition and are still required by this routine to produce the forecasts.

G13DJF should not be used when the moving average parameters lie close to the boundary of the invertibility region. The routine does test for both invertibility and stationarity but if in doubt, you may use G13DXF, before calling this routine, to check that the VARMA model being used is invertible.

On a successful exit, the output quantities K, LMAX, KMAX, REF and LREF will be suitable for input to G13DKF.

9 Example

This example computes forecasts of the next five values in two series each of length

48

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

μ

is to be estimated and

ϕ_{1} (2, 1)

constrained to be zero.

NAG Library Routine DocumentG13DJF

+− 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

G13DJF