g13bjc produces forecasts of a time series (the output series) which may depend on one or more other (input) series via a previously estimated multi-input model. The future values of any input series must be supplied. Standard errors of the forecasts are produced. If future values of some of the input series have been obtained as forecasts using ARIMA models for those series, this may be allowed for in the calculation of the standard errors.

2 Specification

#include <nag.h>

void

g13bjc (Nag_ArimaOrder *arimav, Integer nseries, Nag_TransfOrder *transfv, double para[], Integer npara, Integer nev, Integer nfv, const double xxy[], Integer tdxxy, double rmsxy[], const Integer mrx[], Integer tdmrx, const double parx[], Integer ldparx, Integer tdparx, double fva[], double fsd[], Nag_G13_Opt *options, NagError *fail)

The function may be called by the names: g13bjc, nag_tsa_multi_inputmod_forecast or nag_tsa_multi_inp_model_forecast.

3 Description

g13bjc has two stages. The first stage is essentially the same as a call to the model estimation function g13bec, with zero iterations. In particular, all the arguments remain unchanged in the supplied input series transfer function models and output noise series ARIMA model except that a further iteration takes place for any

ω

corresponding to a simple input. The internal nuisance arguments associated with the pre-observation period effects of the input series are estimated where requested, and so are any backforecasts of the output noise series. The output components

z_{t}

and

n_{t}

, and residuals

a_{t}

are calculated exactly as described in the Section 3 of g13bec.

In the second stage, the forecasts of the output series

y_{t}

are calculated for

t = n + 1, n + 2, \dots, n + L

where

n

is the latest time point of the observations and

L

is the maximum lead time of the forecasts.

First the new values,

x_{t}

for any input series are used to form the input components

z_{t}

for

t = n + 1, n + 2, \dots, n + L

using the transfer function models:

(a) $z_{t} = δ_{1} z_{t - 1} + δ_{2} z_{t - 2} + \dots + δ_{p} z_{t - p} + ω_{0} x_{t - b} - ω_{1} x_{t - b - 1} - \dots - ω_{q} x_{t - b - q}$ .
The output noise component $n_{t}$ for $t = n + 1, n + 2, \dots, n + L$ is then forecast by setting $a_{t} = 0$ for $t = n + 1, n + 2, \dots, n + L$ and using the ARIMA model equations:
(b) $e_{t} = ϕ_{1} e_{t - 1} + ϕ_{2} e_{t - 2} + \dots + ϕ_{p} e_{t - p} + a_{t} - θ_{1} a_{t - 1} - θ_{2} a_{t - 2} - \dots - θ_{q} a_{t - q}$ .
(c) $w_{t} = Φ_{1} w_{t - s} + Φ_{2} w_{t - 2 \times s} + \dots + Φ_{P} w_{t - P \times s} + e_{t} - Θ_{1} e_{t - s} - Θ_{2} e_{t - 2 \times s} - \dots - Θ_{Q} e_{t - Q \times s}$ .
(d) $n_{t} = {(\nabla^{d} \nabla_{s}^{D})}^{−1} (w_{t} + c)$ .

This last step of ‘integration’ reverses the process of differencing. Finally the output forecasts are calculated as

y_{t} = z_{1, t} + z_{2, t} + \dots + z_{m, t} + n_{t} .

The forecast error variance of

y_{t + l}

(i.e., at lead time

l

) is

S_{l}^{2}

, which is the sum of parts which arise from the various input series, and the output noise component. That part due to the output noise is

{s n}_{l}^{2} = V_{n} \times (ψ_{0}^{2} + ψ_{1}^{2} + \dots + ψ_{l - 1}^{2})

V_{n}

is the estimated residual variance of the output noise ARIMA model, and

ψ_{0}, ψ_{1}, \dots

, are the ‘psi-weights’ of this model as defined in Box and Jenkins (1976). They are calculated by applying the equations (b), (c) and (d) above for

t = 0, 1, \dots, L

, but with artificial values for the various series and with the constant

c

set to

0

. Thus all values of

a_{t}

e_{t}

w_{t}

and

n_{t}

are taken as zero for

t < 0

;

a_{t}

is taken to be 1 for

t = 0

and 0 for

t > 0

. The resulting values of

n_{t}

for

t = 0, 1, \dots, L

are precisely

ψ_{0}, ψ_{1}, \dots, ψ_{L}

as required.

Further contributions to

S_{l}^{2}

come only from those input series, for which future values are forecasts which have been obtained by applying input series ARIMA models. For such a series the contribution is

{s z}_{l}^{2} = V_{x} \times (ν_{0}^{2} + ν_{1}^{2} + \dots + ν_{l - 1}^{2})

V_{x}

is the estimated residual variance of the input series ARIMA model. The coefficients

ν_{0}, ν_{1}, \dots

are calculated by applying the transfer function model equation (a) above for

t = 0, 1, \dots, L

, but again with artificial values of the series. Thus all values of

z_{t}

and

x_{t}

for

t < 0

are taken to be zero, and

x_{0}, x_{1}, \dots

are taken to be the psi-weight sequence

ψ_{0}, ψ_{1}, \dots

for the input series ARIMA model. The resulting values of

z_{t}

for

t = 0, 1, \dots, L

are precisely

ν_{0}, ν_{1}, \dots, ν_{L}

as required.

In adding such contributions

{s z}_{l}^{2}

{s n}_{l}^{2}

to make up the total forecast error variance

S_{l}^{2}

, it is assumed that the various input series with which these contributions are associated, are statistically independent of each other.

When using the function in practice an ARIMA model is required for all the input series. In the case of those inputs for which no such ARIMA model is available (or its effects are to be excluded), the corresponding orders and arguments and the estimated residual variance should be set to zero.

4 References

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

5 Arguments

1: $arimav$ – Nag_ArimaOrder *

Pointer to structure of type Nag_ArimaOrder with the following members:

p – Integer

d – IntegerInput

q – IntegerInput

bigp – IntegerInput

bigd – IntegerInput

bigq – IntegerInput

s – IntegerInput

On entry: these seven members of arimav must specify the orders vector

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

, respectively, of the ARIMA model for the output noise component.

p

q

P

and

Q

refer, respectively, to the number of autoregressive

(ϕ)

, moving average

(θ)

, seasonal autoregressive

(Φ)

and seasonal moving average

(Θ)

arguments.

d

D

and

s

refer, respectively, to the order of non-seasonal differencing, the order of seasonal differencing and the seasonal period.

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 $s > 1$ , $P + D + Q > 0$ ;
$d + s \times (P + D) \leq n$ ;
$p + d - q + s \times (P + D - Q) \leq n$ .

2: $nseries$ – Integer Input

On entry: the number of input and output series. There may be any number of input series (including none), but only one output series.

Constraint:

nseries > 1

if there are no arguments in the model (that is

p = q = P = Q = 0

and

options . cfixed = Nag_TRUE

nseries \geq 1

otherwise.

3: $transfv$ – Nag_TransfOrder *

Pointer to structure of type Nag_TransfOrder with the following members:

b – Integer *Input
q – Integer *Input
p – Integer *
r – Integer *Input: On entry: before use these member pointers must be allocated memory by calling g13byc which allocates $nseries - 1$ elements to each pointer. The memory allocated to these pointers must be given the transfer function model orders $b$ , $q$ and $p$ of each of the input series. The order arguments for input series $i$ are held in the $i$ th element of the allocated memory for each pointer. $transfv \to b [i - 1]$ holds the value $b_{i}$ , $transfv \to q [i - 1]$ holds the value $q_{i}$ and $transfv \to p [i - 1]$ holds the value $p_{i}$ .
For a simple input, $b_{i} = q_{i} = p_{i} = 0$ .

$transfv \to r [i - 1]$ holds the value $r_{i}$ , where $r_{i} = 1$ for a simple input, and $r_{i} = 2$ or 3 for a transfer function input.

The choice $r_{i} = 3$ leads to estimation of the pre-period input effects as nuisance arguments, and $r_{i} = 2$ suppresses this estimation. This choice may affect the returned forecasts.

When $r_{i} = 1$ , any nonzero contents of the $i$ th element of the memory of $transfv \to b$ , $transfv \to q$ and $transfv \to p$ are ignored.

Constraint: $transfv \to r [i - 1] = 1$ , $2$ or $3$ , for $i = 1, 2, \dots, nseries - 1$
The memory allocated to the members of transfv must be freed by a call to g13bzc

4: $para [npara]$ – double Input/Output

On entry: estimates of the multi-input model arguments. These are in order firstly the ARIMA model arguments:

p

values of

ϕ

arguments,

q

values of

θ

arguments,

P

values of

Φ

arguments,

Q

values of

Θ

arguments. These are followed by the transfer function model argument values

ω_{0}, ω_{1}, \dots, ω_{q_{1}}

, and

δ_{1}, δ_{2}, \dots, δ_{p_{1}}

for the first of any input series and similarly for each subsequent input series. The final component of para is the value of the constant

c

On exit: the input estimates are unaltered except that any

ω

estimates corresponding to a simple input will be undated by a single iteration.

5: $npara$ – Integer Input

On entry: the exact number of

ϕ

θ

Φ

Θ

ω

δ

c

arguments, so that

npara = p + q + P + Q + nseries + \sum (p_{i} + q_{i})

, the summation being over all the input series. (

c

must be included whether its value was previously estimated or was set fixed.)

6: $nev$ – Integer Input

On entry: the number of original (undifferenced) values in each of the input and output time-series.

7: $nfv$ – Integer Input

On entry: the number of forecast values of the output series required.

Constraint:

nfv > 0

8: $xxy [(nev + nfv) \times tdxxy]$ – const double Input

Note: the

(i, j)

th element of the matrix is stored in

xxy [(i - 1) \times tdxxy + j - 1]

On entry: the columns of xxy must contain in the first nev places, the past values of each of the input and output series, in that order. In the next nfv places, the columns relating to the input series (i.e., columns 0 to

nseries - 2

) contain the future values of the input series which are necessary for construction of the forecasts of the output series

y

9: $tdxxy$ – Integer Input

On entry: the stride separating matrix column elements in the array xxy.

Constraint:

tdxxy \geq nseries

10: $rmsxy [nseries]$ – double Input/Output

On entry: elements of

rmsxy [0]

rmsxy [nseries - 2]

must contain the estimated residual variance of the input series ARIMA models. In the case of those inputs for which no ARIMA model is available or its effects are to be excluded in the calculation of forecast standard errors, the corresponding entry of rmsxy should be set to 0.

On exit:

rmsxy [nseries - 1]

contains the estimated residual variance of the output noise ARIMA model which is calculated from the supplied series. Otherwise rmsxy is unchanged.

11: $mrx [7 \times tdmrx]$ – const Integer Input

On entry: the orders array for each of the input series ARIMA models. Thus, column

i - 1

contains values of

p

d

q

P

D

Q

s

for input series

i

. In the case of those inputs for which no ARIMA model is available, the corresponding orders should be set to 0.

If there are no input series then NULL may be supplied in place of mrx.

12: $tdmrx$ – Integer Input

On entry: the stride separating matrix column elements in the array mrx.

Constraint:

tdmrx \geq nseries - 1

13: $parx [ldparx \times tdparx]$ – const double Input

Note: the

(i, j)

th element of the matrix is stored in

parx [(i - 1) \times tdparx + j - 1]

On entry: values of the arguments (

ϕ

θ

Φ

, and

Θ

) for each of the input series ARIMA models. Thus column

i

contains

mrx [i]

values of

ϕ

mrx [(2) \times tdmrx + i]

values of

θ

mrx [(3) \times tdmrx + i]

values of

Φ

and

mrx [(5) \times tdmrx + i]

values of

Θ

– in that order.

Values in the columns relating to those input series for which no ARIMA model is available are ignored.

If there are no input series then NULL may be supplied in place of parx.

14: $ldparx$ – Integer Input

On entry: the maximum number of arguments in any of the input series ARIMA models. If there are no input series then ldparx is not referenced.

Constraint:

ldparx \geq n c e = \max (1, (mrx [i] + mrx [(2) \times tdmrx + i] + mrx [(3) \times tdmrx + i] + mrx [(5) \times tdmrx + i]))

, for

i = 0, 1, \dots, nseries - 1

15: $tdparx$ – Integer Input

On entry: the stride separating matrix column elements in the array parx.

Constraint:

tdparx \geq nseries - 1

16: $fva [nfv]$ – double Output

On exit: the required forecast values for the output series.

17: $fsd [nfv]$ – double Output

On exit: the standard errors for each of the forecast values.

18: $options$ – Nag_G13_Opt * Input/Output

On entry/exit: a pointer to a structure of type Nag_G13_Opt whose members are optional parameters for g13bjc. If the optional parameters are not required, then the null pointer, G13_DEFAULT, can be used in the function call tog13bjc. Details of the optional parameters and their types are given below in Section 11.

19: $fail$ – NagError * Input/Output

The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_2_INT_ARG_LT: On entry, $ldparx = ⟨ value ⟩$ while $n c e = ⟨ value ⟩$ . These arguments must satisfy $ldparx \geq n c e$ . (See the expression for $n c e$ in Section 5 where ldparx is described).

On entry, $tdmrx = ⟨ value ⟩$ while $nseries - 1 = ⟨ value ⟩$ . These arguments must satisfy $tdmrx \geq nseries - 1$ .

On entry, $tdparx = ⟨ value ⟩$ while $nseries - 1 = ⟨ value ⟩$ . These arguments must satisfy $tdparx \geq nseries - 1$ .

On entry, $tdxxy = ⟨ value ⟩$ while $nseries = ⟨ value ⟩$ . These arguments must satisfy $tdxxy \geq nseries$ .
NE_ALLOC_FAIL: Dynamic memory allocation failed.
NE_ARIMA_TEST_FAILED: On entry, or during execution, one or more sets of the ARIMA ( $ϕ$ , $θ$ , $Φ$ or $Θ$ ) arguments do not satisfy the stationarity or invertibility test conditions.
NE_BAD_PARAM: On entry, argument $options . cfixed$ had an illegal value.
NE_CONSTRAINT: General constraint: $⟨ value ⟩$ .
NE_DELTA_TEST_FAILED: On entry, or during execution, one or more sets of $δ$ arguments do not satisfy the stationarity or invertibility test conditions.
NE_G13_OPTIONS_NOT_INIT: On entry, the option structure, options, has not been initialized using g13bxc.
NE_G13_ORDERS_NOT_INIT: On entry, the orders array structure transfv in function g13byc has not been initialized.
NE_INT_ARG_LE: On entry, $nfv = ⟨ value ⟩$ .
Constraint: $nfv > 0$ .
NE_INT_ARG_LT: On entry, $nseries = ⟨ value ⟩$ .
Constraint: $nseries \geq 1$ .
NE_INT_ARRAY_2: Value $⟨ value ⟩$ given to $transfv \to r [⟨ value ⟩]$ not valid. Correct range for elements if $transfv \to r$ is $1 \leq transfv \to r [i] \leq 3$ .
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_INVALID_NSER: On entry, $nseries = 1$ and there are no arguments in the model, i.e., ( $p = q = P = Q = 0$ and $options . cfixed = Nag_TRUE$ ).
NE_MAT_NOT_POS_DEF: Attempt to invert the second derivative matrix needed in the calculation of the covariance matrix of the parameter estimates has failed. The matrix is not positive definite, possibly due to rounding errors.
NE_NPARA_MR_MT_INCONSIST: On entry, there is inconsistency between npara on the one hand and the elements in the orders structures, arimav and transfv on the other.
NE_NSER_INCONSIST: Value of nseries passed to g13byc was $⟨ value ⟩$ which is not equal to the value $⟨ value ⟩$ passed in this function.
NE_SOLUTION_FAIL_CONV: Iterative refinement has failed to improve the solution of the equations giving the latest estimates of the arguments. This occurred because the matrix of the set of equations is too ill-conditioned.

7 Accuracy

The computation used is believed to be stable.

8 Parallelism and Performance

g13bjc is not threaded in any implementation.

9 Further Comments

The time taken by g13bjc is approximately proportional to the product of the length of each series and the square of the number of arguments in the multi-input model.

10 Example

The data in this example relate to 40 observations of an output time series and 5 input time series. This example differs from g13bec in g13bec in that there are now 4 simple input series. The output series has one autoregressive

(ϕ)

argument and one seasonal moving average

(Θ)

argument. The seasonal period is

4

. The transfer function input (the fifth in the set) is defined by orders

b_{5} = 1

q_{5} = 0

p_{5} = 1

r_{5} = 3

, so that it allows for pre-observation period effects. The initial values of the specified model are:

\begin{matrix} ϕ = 0.495, ​ Θ = 0.238, ω_{1} = - 0.367 ω_{2} = - 3.876 ω_{3} = 4.516 \\ ω_{4} = 2.474 ω_{5} = 8.629 δ_{1} = 0.688, ​ c = - 82.858 . \end{matrix}

A further eight values of the input series are supplied, and it is assumed that the values for the fifth series have themselves been forecast from an ARIMA model with orders 2 0 2 0 1 1 4, in which

ϕ_{1} = 1.6743

ϕ_{2} = - 0.9505

θ_{1} = 1.4605

θ_{2} = - 0.4862

and

Θ_{1} = 0.8993

, and for which the residual mean square is 0.1720.

The following are computed and printed out: the estimated residual variance for the output noise series, the eight forecast values and their standard errors, and the values of the components

z_{t}

and the output noise component

n_{t}

11 Optional Parameters

A number of optional input and output arguments to g13bjc are available through the structure argument options of type Nag_G13_Opt. An argument may be selected by assigning an appropriate value to the relevant structure member and those arguments not selected will be assigned default values. If no use is to be made of any of the optional parameters you should use the null pointer, G13_DEFAULT, in place of options when calling g13bjc; the default settings will then be used for all arguments.

Before assigning values to options the structure must be initialized by a call to the function g13bxc. Values may then be assigned directly to the structure members in the normal C manner.

Options selected by direct assignment are checked within g13bjc for being within the required range, if outside the range, an error message is generated.

When all calls to g13bjc have been completed and the results contained in the options structure are no longer required; then g13xzc should be called to free the NAG allocated memory from options.

11.1 Optional Parameters Checklist and Default Values

For easy reference, the following list shows the input and output members of options which are valid for g13bjc together with their default values where relevant.

Boolean list	Nag_TRUE
Boolean cfixed	Nag_FALSE
double *zt
double *noise

11.2 Description of the Optional Parameters

List – Nag_Boolean

Default

= Nag_TRUE

On entry: if

options . List = Nag_TRUE

then the argument settings which are used in the call to g13bjc will be printed.

cfixed – Nag_Boolean

Default

= Nag_FALSE

On entry:

options . cfixed

must be set to Nag_FALSE if the constant was estimated when the model was fitted, and Nag_TRUE if it was held at a fixed value. This only affects the degrees of freedom used in calculating the estimated residual variance.

zt – double *

Default memory

= (nev + nfv) \times (nseries - 1)

On exit: this pointer is allocated memory internally with

(nev + nfv) \times (nseries - 1)

elements corresponding to

(nev + nfv)

rows by

nseries - 1

columns. The columns of

options . zt

hold the values of the input component series

z_{t}

noise – double *

Default memory

= nev + nfv

On exit: this pointer is allocated memory internally with

nev + nfv

elements. It holds the output noise component

n_{t}

NAG Library Manual, Mark 27.2

Interfaces: FL CL CPP AD

NAG CL Interface Introduction

G13 (Tsa) Chapter Contents

G13 (Tsa) Chapter Introduction

g13bj: FL CL CPP AD

NAG CL Interfaceg13bjc (multi_​inputmod_​forecast)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

11 Optional Parameters

11.1 Optional Parameters Checklist and Default Values

11.2 Description of the Optional Parameters

NAG CL Interface
g13bjc (multi_inputmod_forecast)