NAG Library Function Document

nag_tsa_varma_update (g13dkc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

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

nag_tsa_varma_update (g13dkc) 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 nag_tsa_varma_forecast (g13djc) must be made prior to calling this function 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 nag_tsa_varma_update (g13dkc) the reference vector is updated so that should future series values become available these forecasts may be updated by recalling nag_tsa_varma_update (g13dkc).

2 Specification

#include <nag.h>

#include <nagg13.h>

void	nag_tsa_varma_update (Integer k, Integer lmax, Integer m, Integer mlast, const double z[], Integer kmax, double ref[], Integer lref, double v[], double predz[], double sefz[], NagError fail)

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 nag_tsa_varma_forecast (g13djc). Given

m

further observations

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

, where

m < l_{\max}

, nag_tsa_varma_update (g13dkc) updates the forecasts of

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

and their corresponding standard errors.

nag_tsa_varma_update (g13dkc) uses a multivariate version of the procedure described in Box and Jenkins (1976). The forecasts are updated using the

ψ

weights, computed in nag_tsa_varma_forecast (g13djc). 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 nag_tsa_varma_update (g13dkc) 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 Arguments

The quantities k, lmax, kmax, ref and lref from nag_tsa_varma_forecast (g13djc) are suitable for input to nag_tsa_varma_update (g13dkc).

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 nag_tsa_varma_forecast (g13djc).
Constraint: $lmax \geq 2$ .
3: m – IntegerInput: On entry: $m$ , the number of new observations available since the last call to either nag_tsa_varma_forecast (g13djc) or nag_tsa_varma_update (g13dkc). The number of new observations since the last call to nag_tsa_varma_forecast (g13djc) is then $m + mlast$ .
Constraint: $0 < m < lmax - mlast$ .
4: mlast – Integer *Input/Output: On entry: on the first call to nag_tsa_varma_update (g13dkc), since calling
nag_tsa_varma_forecast (g13djc), 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 nag_tsa_varma_update (g13dkc).

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 nag_tsa_varma_forecast (g13djc).
mlast must not be changed between calls to nag_tsa_varma_update (g13dkc), unless a call to nag_tsa_varma_forecast (g13djc) has been made between the calls in which case mlast should be reset to $0$ .

Constraint: $0 \leq mlast < lmax - m$ .
5: z[ $kmax \times m$ ] – const doubleInput: On entry: $z [kmax \times (j - 1) + i - 1]$ 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 nag_tsa_varma_forecast (g13djc).
Constraint: if the transformation defined in tr in nag_tsa_varma_forecast (g13djc) for the $i$ th series is the log transformation, then $z [kmax \times (j - 1) + i - 1] > 0.0$ , and if it is the square-root transformation, then $z [kmax \times (j - 1) + i - 1] \geq 0.0$ , 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.
Constraint: $kmax \geq k$ .
7: ref[lref] – doubleInput/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 nag_tsa_varma_forecast (g13djc) (or a previous call to nag_tsa_varma_update (g13dkc)).

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 nag_tsa_varma_forecast (g13djc). The last k elements are not updated.
8: lref – IntegerInput: On entry: the dimension of the array ref.
Constraint: $lref \geq (lmax - 1) \times k \times k + 2 \times k \times lmax + k$ .
9: v[ $kmax \times m$ ] – doubleOutput: On exit: $v [kmax \times (j - 1) + i - 1]$ 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[ $kmax \times lmax$ ] – doubleInput/Output: On entry: nonupdated values are kept intact.

On exit: $predz [kmax \times (j - 1) + i - 1]$ 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 nag_tsa_varma_forecast (g13djc) or nag_tsa_varma_update (g13dkc) are set equal to the corresponding columns of z.
11: sefz[ $kmax \times lmax$ ] – doubleInput/Output: On entry: nonupdated values are kept intact.

On exit: $sefz [kmax \times (j - 1) + i - 1]$ 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 nag_tsa_varma_forecast (g13djc) or nag_tsa_varma_update (g13dkc) are set equal to zero.
12: fail – NagError *Input/Output: The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
NE_BAD_PARAM: On entry, argument $〈value〉$ had an illegal value.
NE_INT: On entry, $k = 〈value〉$ .
Constraint: $k \geq 1$ .; On entry, $lmax = 〈value〉$ .
Constraint: $lmax \geq 2$ .; On entry, lref is too small: $lref = 〈value〉$ but must be at least $〈value〉$ .; On entry, $m = 〈value〉$ .
Constraint: $m > 0$ .; On entry, $mlast = 〈value〉$ .
Constraint: $mlast \geq 0$ .
NE_INT_2: On entry, $kmax = 〈value〉$ and $k = 〈value〉$ .
Constraint: $kmax \geq k$ .
NE_INT_3: On entry, $lmax = 〈value〉$ , $m = 〈value〉$ and $mlast = 〈value〉$ .
Constraint: $0 < m < lmax - mlast$ .; On entry, $lmax = 〈value〉$ , $m = 〈value〉$ and $mlast = 〈value〉$ .
Constraint: $0 \leq mlast < lmax - m$ .
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_REF_VEC: On entry, some of the elements of the array ref have been corrupted.
NE_RESULT_OVERFLOW: The updated forecasts will overflow if computed.
NE_TRANSFORMATION: On entry, one (or more) of the transformations requested is invalid.

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 nag_tsa_varma_update (g13dkc) 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. nag_tsa_varma_estimate (g13ddc) 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 nag_tsa_varma_forecast (g13djc) 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 Function Documentnag_tsa_varma_update (g13dkc)