nag_tsa_multi_inp_model_forecast (g13bjc)

1 **Purpose**

2 **Specification**

3 **Description**

4 **References**

5 **Parameters**

6 **Error Indicators and Warnings**

7 **Accuracy**

8 **Further Comments**

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9 **Example**

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9.1 **Example 1**

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10 **Optional Parameters**

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11 **Example 2**
## 1 Purpose

## 2 Specification

## 3 Description

## 4 References

## 5 Parameters

## 6 Error Indicators and Warnings

## 7 Accuracy

## 8 Further Comments

## 9 Example

### 9.1 Example 1

#### 9.1.1 Program Text

#### 9.1.2 Program Data

#### 9.1.3 Program Results

## 10 Optional Parameters

### 10.1 Optional Parameters Checklist and Default Values

### 10.2 Description of Optional Parameters

*On entry*: if ${\mathbf{list}}=\mathbf{TRUE}$ then the parameter settings which are used in the call to nag_tsa_multi_inp_model_forecast (g13bjc) will be printed.

*On entry*: cfixed must be set to **FALSE** if the constant was estimated when the model was fitted, and **TRUE** if it was held at a fixed value. This only affects the degrees of freedom used in calculating the estimated residual variance.

*On exit*: this pointer is allocated memory internally with $\left({\mathbf{nev}}+{\mathbf{nfv}}\right)\times \left({\mathbf{nseries}}-1\right)$ elements corresponding to $\left({\mathbf{nev}}+{\mathbf{nfv}}\right)$ rows by ${\mathbf{nseries}}-1$ columns. The columns of zt hold the values of the input component series ${z}_{t}$.

*On exit*: this pointer is allocated memory internally with ${\mathbf{nev}}+{\mathbf{nfv}}$ elements. It holds the output noise component ${n}_{t}$.
## 11 Example 2

### 11.1 Program Text

### 11.2 Program Data

### 11.3 Program Results

© The Numerical Algorithms Group Ltd, Oxford, UK. 2004

nag_tsa_multi_inp_model_forecast (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.

#include <nag.h> #include <nagg13.h> | |

void nag_tsa_multi_inp_model_forecast | (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) |

nag_tsa_multi_inp_model_forecast (g13bjc) has two stages. The first stage is essentially the same as a call to the model estimation function nag_tsa_multi_inp_model_estim (g13bec), with zero iterations. In particular, all the parameters 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 $\omega $ corresponding to a simple input. The internal nuisance parameters 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 nag_tsa_multi_inp_model_estim (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:

- ${z}_{t}={\delta}_{1}{z}_{t-1}+{\delta}_{2}{z}_{t-2}+\cdots +{\delta}_{p}{z}_{t-p}+{\omega}_{0}{x}_{t-b}-{\omega}_{1}{x}_{t-b-1}-\cdots -{\omega}_{q}{x}_{t-b-q}$.
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:
- ${e}_{t}={\varphi}_{1}{e}_{t-1}+{\varphi}_{2}{e}_{t-2}+\cdots +{\varphi}_{p}{e}_{t-p}+{a}_{t}-{\theta}_{1}{a}_{t-1}-{\theta}_{2}{a}_{t-2}-\cdots -{\theta}_{q}{a}_{t-q}$.
- ${w}_{t}={\Phi}_{1}{w}_{t-s}+{\Phi}_{2}{w}_{t-2\times s}+\cdots +{\Phi}_{P}{w}_{t-P\times s}+{e}_{t}-{\Theta}_{1}{e}_{t-s}-{\Theta}_{2}{e}_{t-2\times s}-\cdots -{\Theta}_{Q}{e}_{t-Q\times s}$.
- ${n}_{t}={\left({\nabla}^{d}{\nabla}_{s}^{D}\right)}^{-1}\left({w}_{t}+c\right)$.

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

$${y}_{t}={z}_{1,t}+{z}_{2,t}+\cdots +{z}_{m,t}+{n}_{t}\text{.}$$ |

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

$${sn}_{l}^{2}={V}_{n}\times \left({\psi}_{0}^{2}+{\psi}_{1}^{2}+\cdots +{\psi}_{l-1}^{2}\right)$$ |

${V}_{n}$ is the estimated residual variance of the output noise ARIMA model, and ${\psi}_{0},{\psi}_{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 ${\psi}_{0},{\psi}_{1},\dots ,{\psi}_{L}$ 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

$${sz}_{l}^{2}={V}_{x}\times \left({\nu}_{0}^{2}+{\nu}_{1}^{2}+\cdots +{\nu}_{l-1}^{2}\right)$$ |

${V}_{x}$ is the estimated residual variance of the input series ARIMA model. The coefficients ${\nu}_{0},{\nu}_{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 ${\psi}_{0},{\psi}_{1},\dots $ for the input series ARIMA model. The resulting values of ${z}_{t}$ for $t=0,1,\dots ,L$ are precisely ${\nu}_{0},{\nu}_{1},\dots ,{\nu}_{L}$ as required.

In adding such contributions ${sz}_{l}^{2}$ to ${sn}_{l}^{2}$ 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 parameters and the estimated residual variance should be set to zero.

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

- 1:
**arimav**– Nag_ArimaOrder * -
Pointer to structure of type
**Nag_ArimaOrder**with the following members:**p**– Integer**d**– Integer Input**q**– Integer Input**bigp**– Integer Input**bigd**– Integer Input**bigq**– Integer Input**s**– Integer Input-
*On entry*: these seven members of arimav must specify the orders vector $\left(p,d,q,P,D,Q,s\right)$, respectively, of the ARIMA model for the output noise component.$p$, $q$, $P$ and $Q$ refer, respectively, to the number of autoregressive $\left(\varphi \right)$, moving average $\left(\theta \right)$, seasonal autoregressive $\left(\Phi \right)$ and seasonal moving average $\left(\Theta \right)$ parameters.$d$, $D$ and $s$ refer, respectively, to the order of non-seasonal differencing, the order of seasonal differencing and the seasonal period.

- 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*: ${\mathbf{nseries}}>1$ if there are no parameters in the model (that is $p=q=P=Q=0$ and ${\mathbf{options.cfixed}}=\mathbf{TRUE}$), ${\mathbf{nseries}}\ge 1$ 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 nag_tsa_transf_orders (g13byc) which allocates ${\mathbf{nseries}}-1$ 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 parameters for input series $i$ are held in the $i$th element of the allocated memory for each pointer. ${\mathbf{b}}\left[i-1\right]$ holds the value ${b}_{i}$, ${\mathbf{q}}\left[i-1\right]$ holds the value ${q}_{i}$ and ${\mathbf{p}}\left[i-1\right]$ holds the value ${p}_{i}$.For a simple input, ${b}_{i}={q}_{i}={p}_{i}=0$.${\mathbf{r}}\left[i-1\right]$ 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 parameters, and ${r}_{i}=2$ suppresses this estimation. This choice may affect the returned forecasts.*Constraint*: ${\mathbf{r}}\left[i-1\right]$ $=$ $1$, $2$ or $3$, for $\mathit{i}=1,2,\dots ,{\mathbf{nseries}}-1$The memory allocated to the members of transfv must be freed by a call to nag_tsa_trans_free (g13bzc)

- 4:
**para**[npara] – double Input/Output -
*On entry*: estimates of the multi-input model parameters. These are in order firstly the ARIMA model parameters: $p$ values of $\varphi $ parameters, $q$ values of $\theta $ parameters, $P$ values of $\Phi $ parameters, $Q$ values of $\Theta $ parameters. These are followed by the transfer function model parameter values ${\omega}_{0},{\omega}_{1},\dots ,{\omega}_{{q}_{1}}$, and ${\delta}_{1},{\delta}_{2},\dots ,{\delta}_{{p}_{1}}$ 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 $\omega $ estimates corresponding to a simple input will be undated by a single iteration. - 5:
**npara**– Integer Input -
*On entry*: the exact number of $\varphi $, $\theta $, $\Phi $, $\Theta $, $\omega $, $\delta $, $c$ parameters, so that ${\mathbf{npara}}=p+q+P+Q+{\mathbf{nseries}}+\sum \left({p}_{i}+{q}_{i}\right)$, 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*: ${\mathbf{nfv}}>0$. - 8:
**xxy**[${\mathbf{nev}}+{\mathbf{nfv}}$][tdxxy] – const double Input -
*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 ${\mathbf{nseries}}-2$) 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 second dimension of the array xxy as declared in the subroutine from which nag_tsa_multi_inp_model_forecast (g13bjc) is called.*Constraint*: ${\mathbf{tdxxy}}\ge {\mathbf{nseries}}$. - 10:
**rmsxy**[nseries] – double Input/Output -
*On entry*: elements of ${\mathbf{rmsxy}}\left[0\right]$ to ${\mathbf{rmsxy}}\left[{\mathbf{nseries}}-2\right]$ 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*: ${\mathbf{rmsxy}}\left[{\mathbf{nseries}}-1\right]$ contains the estimated residual variance of the output noise ARIMA model which is calculated from the supplied series. Otherwise rmsxy is unchanged. - 11:
**mrx**[$7$][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 the null pointer (Integer *)0 may be supplied in place of mrx. - 12:
**tdmrx**– Integer Input -
*On entry*: the second dimension of the array mrx as declared in the subroutine from which nag_tsa_multi_inp_model_forecast (g13bjc) is called.*Constraint*: ${\mathbf{tdmrx}}\ge {\mathbf{nseries}}-1$. - 13:
**parx**[ldparx][tdparx] – const double Input -
*On entry*: values of the parameters ($\varphi $, $\theta $, $\Phi $, and $\Theta $) for each of the input series ARIMA models. Thus column $i$ contains ${\mathbf{mrx}}\left[0\right]\left[i\right]$ values of $\varphi $, ${\mathbf{mrx}}\left[2\right]\left[i\right]$ values of $\theta $, ${\mathbf{mrx}}\left[3\right]\left[i\right]$ values of $\Phi $ and ${\mathbf{mrx}}\left[5\right]\left[i\right]$ values of $\Theta $ – 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 the null pointer (double *)0 may be supplied in place of parx. - 14:
**ldparx**– Integer Input -
*On entry*: the maximum number of parameters in any of the input series ARIMA models. If there are no input series then ldparx is not referenced.*Constraint*: ${\mathbf{ldparx}}\ge nce=\mathrm{max}\left(1,\left({\mathbf{mrx}}\left[0\right]\left[i\right]+{\mathbf{mrx}}\left[2\right]\left[i\right]+{\mathbf{mrx}}\left[3\right]\left[i\right]+{\mathbf{mrx}}\left[5\right]\left[i\right]\right)\right)$, for $\mathit{i}=0,1,\dots ,{\mathbf{nseries}}-1$. - 15:
**tdparx**– Integer Input -
*On entry*: the second dimension of the array parx as declared in the subroutine from which nag_tsa_multi_inp_model_forecast (g13bjc) is called.*Constraint*: ${\mathbf{tdparx}}\ge {\mathbf{nseries}}-1$. - 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 nag_tsa_multi_inp_model_forecast (g13bjc). If the optional parameters are not required, then the null pointer, G13_DEFAULT, can be used in the function call tonag_tsa_multi_inp_model_forecast (g13bjc). Details of the optional parameters and their types are given below in Section 10. - 19:
**fail**– NagError *Input/Output

**NE_G13_OPTIONS_NOT_INIT**-
On entry, the option structure, options, has not been initialised using nag_tsa_options_init (g13bxc).
**NE_G13_ORDERS_NOT_INIT**-
On entry, the orders array structure transfv in function nag_tsa_transf_orders (g13byc) has not been initialised.
**NE_INT_ARRAY_2**-
Value $\u2329\mathit{\text{value}}\u232a$ given to ${\mathbf{transfv}}.{\mathbf{r}}\left[\u2329\mathit{\text{value}}\u232a\right]$ not valid. Correct range for elements if transfv.r is $1\le {\mathbf{r}}\left[i\right]\le 3$.
**NE_BAD_PARAM**-
On entry, parameter options.cfixed had an illegal value.
**NE_INT_ARG_LT****NE_INT_ARG_LE****NE_2_INT_ARG_LT**-
On entry, ${\mathbf{tdxxy}}=\u2329\mathit{\text{value}}\u232a$ while ${\mathbf{nseries}}=\u2329\mathit{\text{value}}\u232a$. These parameters must satisfy ${\mathbf{tdxxy}}\ge {\mathbf{nseries}}$.

On entry, ${\mathbf{tdmrx}}=\u2329\mathit{\text{value}}\u232a$ while ${\mathbf{nseries}}-1=\u2329\mathit{\text{value}}\u232a$. These parameters must satisfy ${\mathbf{tdmrx}}\ge {\mathbf{nseries}}-1$.

On entry, ${\mathbf{ldparx}}=\u2329\mathit{\text{value}}\u232a$ while $nce=\u2329\mathit{\text{value}}\u232a$. These parameters must satisfy ${\mathbf{ldparx}}\ge nce$. (See the expression for $nce$ in Section 5 where ldparx is described).

On entry, ${\mathbf{tdparx}}=\u2329\mathit{\text{value}}\u232a$ while ${\mathbf{nseries}}-1=\u2329\mathit{\text{value}}\u232a$.These parameters must satisfy ${\mathbf{tdparx}}\ge {\mathbf{nseries}}-1$. **NE_ALLOC_FAIL**-
Memory allocation failed.
**NE_INVALID_NSER**-
On entry, ${\mathbf{nseries}}=1$ and there are no parameters in the model, i.e., ($p=q=P=Q=0$ and ${\mathbf{options.cfixed}}=\mathbf{TRUE}$).
**NE_NSER_INCONSIST**-
Value of nseries passed to nag_tsa_transf_orders (g13byc) was $\u2329\mathit{\text{value}}\u232a$ which is not equal to the value $\u2329\mathit{\text{value}}\u232a$ passed in this function.
**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_DELTA_TEST_FAILED**-
On entry, or during execution, one or more sets of $\delta $ parameters do not satisfy the stationarity or invertibility test conditions.
**NE_SOLUTION_FAIL_CONV**-
Iterative refinement has failed to improve the solution of the equations giving the latest estimates of the parameters. This occurred because the matrix of the set of equations is too ill-conditioned.
**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_ARIMA_TEST_FAILED**-
On entry, or during execution, one or more sets of the ARIMA ($\varphi $, $\theta $, $\Phi $ or $\Theta $) parameters do not satisfy the stationarity or invertibility test conditions.
**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 consult NAG for assistance.

The computation used is believed to be stable.

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

This example illustrates the use of the default option G13_DEFAULT in a call to nag_tsa_multi_inp_model_forecast (g13bjc). An example showing the use of optional parameters is given in Section 11. There is one example program file, the main program of which calls both examples. The main program is given below.

This example illustrates the use of the default option G13_DEFAULT in a call to nag_tsa_multi_inp_model_forecast (g13bjc).

The data in the example relate to 40 observations of an output time series and 5 input time series. This example differs from Example 1 in nag_tsa_multi_inp_model_estim (g13bec) in that there are now 4 simple input series. The output series has one autoregressive $\left(\varphi \right)$ parameter and one seasonal moving average $\left(\Theta \right)$ parameter. 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:

$$\begin{array}{c}\varphi =0.495\text{,}\Theta =0.238\text{,}{\omega}_{1}=-0.367\text{\hspace{1em}}\omega 2=-3.876\text{\hspace{1em}}{\omega}_{3}=4.516\\ {\omega}_{4}=2.474\text{\hspace{1em}}{\omega}_{5,1}=8.629\text{\hspace{1em}}{\delta}_{5,1}=0.688\text{,}c=-82.858\text{.}\end{array}$$ |

A further 8 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 ${\varphi}_{1}=1.6743$, ${\varphi}_{2}=-0.9505$, ${\theta}_{1}=1.4605$, ${\theta}_{2}=-0.4862$ and ${\Theta}_{1}=0.8993$, 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 8 forecast values and their standard errors.

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

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

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

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

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

Boolean list | TRUE |

Boolean cfixed | FALSE |

double *zt | |

double *noise |

list –
Boolean | Input | Default $\text{}=\mathbf{TRUE}$ |

cfixed –
Boolean | Input | Default $\text{}=\mathbf{FALSE}$ |

zt –
double * | Output | Default memory $\text{}=\left({\mathbf{nev}}+{\mathbf{nfv}}\right)\times \left({\mathbf{nseries}}-1\right)$ |

noise –
double * | Output | Default memory $\text{}={\mathbf{nev}}+{\mathbf{nfv}}$ |

This example illustrates the use of the options parameter in a call to nag_tsa_multi_inp_model_forecast (g13bjc).

The data in the example relate to the same 40 obervations of an output time series and 5 input time series as in Example 1. This example differs from Example 2 in nag_tsa_multi_inp_model_estim (g13bec) in that there are now 4 simple input series. The output series has one autoregressive $\left(\varphi \right)$ parameter and one seasonal moving average $\left(\Theta \right)$ parameter. 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:

$$\begin{array}{c}\varphi =0.495\text{,}\Theta =0.238\text{,}{\omega}_{1}=-0.367\text{\hspace{1em}}\omega 2=-3.876\text{\hspace{1em}}{\omega}_{3}=4.516\\ {\omega}_{4}=2.474\text{\hspace{1em}}{\omega}_{5,1}=8.629\text{\hspace{1em}}{\delta}_{5,1}=0.688\text{,}c=-82.858\text{.}\end{array}$$ |

A further 8 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 ${\varphi}_{1}=1.6743$, ${\varphi}_{2}=-0.9505$, ${\theta}_{1}=1.4605$, ${\theta}_{2}=-0.4862$ and ${\Theta}_{1}=0.8993$, 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 8 forecast values and their standard errors, and the values of the components ${z}_{t}$ and the output noise component ${n}_{t}$.

© The Numerical Algorithms Group Ltd, Oxford, UK. 2004