NAG Library Routine Document

G13AHF

+− 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

G13AHF produces forecasts of a time series, given a time series model which has already been fitted to the time series using G13AEF or G13AFF. The original observations are not required, since G13AHF uses as input either the original state set produced by G13AEF or G13AFF or the state set updated by a series of new observations using G13AGF. Standard errors of the forecasts are also provided.

2 Specification

SUBROUTINE G13AHF (

ST, NST, MR, PAR, NPAR, C, RMS, NFV, FVA, FSD, WA, NWA, IFAIL)

INTEGER	NST, MR(7), NPAR, NFV, NWA, IFAIL
REAL (KIND=nag_wp)	ST(NST), PAR(NPAR), C, RMS, FVA(NFV), FSD(NFV), WA(NWA)

3 Description

The original time series is

x_{t}

, for

t = 1, 2, \dots, n

and parameters have been fitted to the model of this time series using G13AEF or G13AFF.

Forecasts of

x_{t}

, for

t = n + 1, \dots, n + L

, are calculated in five stages, as follows:

(i)	set $a_{t} = 0$ for $t = N + 1, N + 2, \dots, N + L$ , where $N = n - d - (D \times s)$ is the number of differenced values in the series;
(ii)	calculate the values of $e_{t}$ , for $t = N + 1, \dots, N + L$ , and $e_{t} = ϕ_{1} \times e_{t - 1} + \dots + ϕ_{p} \times e_{t - p} + a_{t} - θ_{1} \times a_{t - 1} - \dots - θ_{q} \times a_{t - q}$ ;
(iii)	calculate the values of $w_{t}$ , for $t = N + 1, \dots, N + L$ , where $w_{t} = Φ_{1} \times w_{t - s} + \dots + Φ_{P} \times w_{t - s \times P} + e_{t} - Θ_{1} \times e_{t - s} - \dots - Θ_{Q} \times e_{t - s \times Q}$ and $w_{t}$ for $t \leq N$ are the first $s \times P$ values in the state set, corrected for the constant;
(iv)	add the constant term $c$ to give the differenced series $\nabla^{d} \nabla_{s}^{D} x_{t} = w_{t} + c$ , for $t = N + 1, \dots, N + L$ ;
(v)	the differencing operations are reversed to reconstitute $x_{t}$ , for $t = n + 1, \dots, n + L$ .

The standard errors of these forecasts are given by

s_{t} = {[V \times (ψ_{0}^{2} + ψ_{1}^{2} + \dots + ψ_{t - n - 1}^{2})]}^{1 / 2}

, for

t = n + 1, \dots, n + L

, where

ψ_{0} = 1

V

is the residual variance of

a_{t}

, and

ψ_{j}

is the coefficient expressing the dependence of

x_{t}

a_{t - j}

To calculate

ψ_{j}

, for

j = 1, 2, \dots, (L - 1)

, the following device is used.

A copy of the state set is initialized to zero throughout and the calculations outlined above for the construction of forecasts are carried out with the settings

a_{N + 1} = 1

, and

a_{t} = 0

, for

t = N + 2, \dots, N + L

The resulting quantities corresponding to the sequence

x_{N + 1}, x_{N + 2}, \dots, x_{N + L}

are precisely

1

ψ_{1}, ψ_{2}, \dots, ψ_{L - 1}

The supplied time series model is used throughout these calculations, with the exception that the constant term

c

is taken to be zero.

4 References

None.

5 Parameters

1: ST(NST) – REAL (KIND=nag_wp) arrayInput

On entry: the state set derived from G13AEF or G13AFF originally, or as modified using earlier calls of G13AGF.

2: NST – INTEGERInput

On entry: the number of values in the state set array ST.

Constraint:

NST = P \times s + D \times s + d + q + \max (p, Q \times s)

. (As returned by G13AEF or G13AFF).

3: MR( $7$ ) – INTEGER arrayInput

On entry: the orders vector

(p, d, q, P, D, Q, s)

of the ARIMA model, in the usual notation.

Constraints:

$p, d, q, P, D, Q, s \geq 0$ ;
$p + q + P + Q > 0$ ;
$s \neq 1$ ;
if $s = 0$ , $P + D + Q = 0$ ;
if $s > 1$ , $P + D + Q > 0$ .

4: PAR(NPAR) – REAL (KIND=nag_wp) arrayInput

On entry: the estimates of the

p

values of the

ϕ

parameters, the

q

values of the

θ

parameters, the

P

values of the

Φ

parameters and the

Q

values of the

Θ

parameters which specify the model and which were output originally by G13AEF or G13AFF.

5: NPAR – INTEGERInput

On entry: the number of

ϕ

θ

Φ

and

Θ

parameters in the model.

Constraint:

NPAR = p + q + P + Q

6: C – REAL (KIND=nag_wp)Input

On entry:

c

, the value of the model constant. This will have been output by G13AEF or G13AFF.

7: RMS – REAL (KIND=nag_wp)Input

On entry:

V

, the residual variance associated with the model.

If G13AFF was used to estimate the model, RMS should be set to

S / NDF

, where S and NDF were output by G13AFF.

If G13AEF was used to estimate the model, RMS should be set to

S / ICOUNT (5)

, where S and

ICOUNT (5)

were output by G13AEF.

Constraint:

RMS \geq 0.0

8: NFV – INTEGERInput

On entry:

L

, the required number of forecasts.

Constraint:

NFV > 0

9: FVA(NFV) – REAL (KIND=nag_wp) arrayOutput

On exit: NFV forecast values relating to the original undifferenced series.

10: FSD(NFV) – REAL (KIND=nag_wp) arrayOutput

On exit: the standard errors associated with each of the NFV forecast values in FVA.

11: WA(NWA) – REAL (KIND=nag_wp) arrayWorkspace

12: NWA – INTEGERInput

On entry: the dimension of the array WA as declared in the (sub)program from which G13AHF is called.

Constraint:

NWA \geq (4 \times NPAR + 3 \times NST)

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,	$NPAR \neq p + q + P + Q$ ,
or	the orders vector MR is invalid (check the constraints given in Section 5).

$IFAIL = 2$

On entry,

NST \neq P \times s + D \times s + d + q + \max (Q \times s, p)

$IFAIL = 3$

On entry,

NFV \leq 0

$IFAIL = 4$

On entry,

NWA < 4 \times NPAR + 3 \times NST

$IFAIL = 5$

On entry,

RMS < 0.0

7 Accuracy

The computations are believed to be stable.

8 Further Comments

The time taken by G13AHF is approximately proportional to

NFV \times NPAR

9 Example

The following program is based on the data derived in the example used to illustrate G13AGF.

These consist of a set of orders indicating that there are two moving average parameters (one non-seasonal, and one seasonal with periodicity

12

The model constant is zero.

The state set contains

26

values.

In addition the residual mean-square derived when the model was originally fitted is given.

Twelve forecasts and their associated errors are obtained.

NAG Library Routine DocumentG13AHF

+− 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

G13AHF