NAG Library Function Document
nag_tsa_varma_forecast (g13djc)
1
Purpose
nag_tsa_varma_forecast (g13djc) 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
nag_tsa_varma_estimate (g13ddc).
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
nag_tsa_varma_update (g13dkc).
2
Specification
#include <nag.h> |
#include <nagg13.h> |
void |
nag_tsa_varma_forecast (Integer k,
Integer n,
const double z[],
Integer kmax,
const Integer tr[],
const Integer id[],
const double delta[],
Integer ip,
Integer iq,
Nag_IncludeMean mean,
const double par[],
Integer lpar,
double qq[],
const double v[],
Integer lmax,
double predz[],
double sefz[],
double ref[],
Integer lref,
NagError *fail) |
|
3
Description
Let the vector
, for
, denote a
-dimensional time series for which forecasts of
are required. Let
be defined as follows:
where
is the differencing operator applied to the
th series and where
is equal to either
,
or
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 th series is , then the differencing operator for the th series is defined by where is the backward shift operator; that is, . The differencing parameters , for and , must be supplied by you. If the th series does not require differencing, then .
is assumed to follow a multivariate ARMA model of the form:
where
, for
, is a vector of
residual series assumed to be Normally distributed with zero mean and positive definite covariance matrix
. The components of
are assumed to be uncorrelated at non-simultaneous lags. The
and
are
by
matrices of parameters. The matrices
, for
, are the autoregressive (AR) parameter matrices, and the matrices
, for
, the moving average (MA) parameter matrices. The parameters in the model are thus the
(
by
)
-matrices, the
(
by
)
-matrices, the mean vector
and the residual error covariance matrix
. The ARMA model
(1) must be both stationary and invertible; see
nag_tsa_arma_roots (g13dxc) for a method of checking these conditions.
The ARMA model
(1) may be rewritten as
where
and
are the autoregressive and moving average polynomials and
denotes the
by
diagonal matrix whose
th diagonal elements is
and
.
This may be rewritten as
or
where
and
is a vector of length
.
Forecasts are computed using a multivariate version of the procedure described in
Box and Jenkins (1976). If
denotes the forecast of
, then
is taken to be that linear function of
which minimizes the elements of
where
is the forecast error.
is referred to as the linear minimum mean square error forecast of
.
The linear predictor which minimizes the mean square error may be expressed as
The forecast error at
for lead
is then
Let
, for
. Unless
the function requires estimates of
, for
, which are obtainable from
nag_tsa_varma_estimate (g13ddc).
The terms
are assumed to be zero, for
. You may use
nag_tsa_varma_update (g13dkc) to update these
forecasts should further observations,
, become available. Note that when
or more further observations are available then
nag_tsa_varma_forecast (g13djc) must be used to produce new forecasts for
, 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,
; 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,
n,
kmax,
ip,
iq,
par,
npar,
qq and
v from
nag_tsa_varma_estimate (g13ddc)
are suitable for input to
nag_tsa_varma_forecast (g13djc).
- 1:
– IntegerInput
-
On entry: , the dimension of the multivariate time series.
Constraint:
.
- 2:
– IntegerInput
-
On entry: , the number of observations in the series, , prior to differencing.
Constraint:
.
The total number of observations must exceed the total number of parameters in the model; that is
- if , ;
- if , ,
(see the arguments
ip,
iq and
mean).
- 3:
– const doubleInput
-
On entry: must contain the th series at time , for and .
- 4:
– IntegerInput
-
On entry: the stride separating row elements in the two-dimensional data stored in the arrays
z,
delta,
qq,
v,
predz,
sefz.
Constraint:
.
- 5:
– const IntegerInput
-
On entry:
indicates whether the
th series is to be transformed, for
.
- A square root transformation is used.
- No transformation is used.
- A log transformation is used.
Constraint:
, or , for .
- 6:
– const IntegerInput
-
On entry: must specify, , the order of differencing required for the th series.
Constraint:
, for .
- 7:
– const doubleInput
-
Note: the dimension,
dim, of the array
delta
must be at least
, where
.
On entry: if
, then
must be set equal to
, for
and
.
If
,
delta is not referenced.
- 8:
– IntegerInput
-
On entry: , the number of AR parameter matrices.
Constraint:
.
- 9:
– IntegerInput
-
On entry: , the number of MA parameter matrices.
Constraint:
.
- 10:
– Nag_IncludeMeanInput
-
On entry: , if components of have been estimated and , if all elements of are to be taken as zero.
Constraint:
or .
- 11:
– const doubleInput
-
On entry: must contain the parameter estimates read in row by row in the order
,
,
.
Thus,
- if ,
must be set equal to an estimate of the th element of , for , and ;
- if ,
must be set equal to an estimate of the th element of , for , and ;
- if ,
must be set equal to an estimate of the th component of , for .
Constraint:
the first
elements of
par must satisfy the stationarity condition and the next
elements of
par must satisfy the invertibility condition.
- 12:
– IntegerInput
-
On entry: the dimension of the array
par.
Constraints:
- if , ;
- if , .
- 13:
– doubleInput/Output
-
On entry: must contain an estimate of the th element of . The lower triangle only is needed.
Constraint:
must be positive definite.
- 14:
– const doubleInput
-
Note: the dimension,
dim, of the array
v
must be at least
, where
.
On entry:
must contain an estimate of the
th component of
, for
and
.
- 15:
– IntegerInput
-
On entry: the number, , of forecasts required.
Constraint:
.
- 16:
– doubleOutput
-
On exit: contains the forecast of , for and .
- 17:
– doubleOutput
-
On exit: contains an estimate of the standard error of the forecast of , for and .
- 18:
– doubleOutput
-
On exit: the reference vector which may be used to update forecasts using
nag_tsa_varma_update (g13dkc). The first
elements contain the
weight matrices,
. The next
elements contain the forecasts of the transformed series
and the next
contain the variances of the forecasts of the transformed variables. The last
k elements are used to store the transformations for the series.
- 19:
– IntegerInput
-
On entry: the dimension of the array
ref.
Constraint:
.
- 20:
– NagError *Input/Output
-
The NAG error argument (see
Section 3.7 in How to Use the NAG Library and its Documentation).
6
Error Indicators and Warnings
- NE_ALLOC_FAIL
-
Dynamic memory allocation failed.
See
Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
- NE_BAD_PARAM
-
On entry, argument had an illegal value.
- NE_EIGENVALUES
-
An excessive number of iterations were needed by
nag_tsa_arma_roots (g13dxc) to evaluate the eigenvalues of the matrices used to test for stationarity and invertibility.
- NE_G13D_AR
-
On entry, the AR parameter matrices are outside the stationarity region.
- NE_G13D_MA
-
On entry, the MA parameter matrices are outside the invertibility region.
- NE_INT
-
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry,
lpar is too small:
but must be at least
.
On entry,
lref is too small:
but must be at least
.
On entry, .
Constraint: .
- NE_INT_2
-
On entry, and .
Constraint: .
- NE_INT_ARRAY
-
On entry, and .
Constraint: .
On entry, .
Constraint: , or .
- 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.
See
Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
- NE_NEARLY_POS_DEF
-
The covariance matrix may be nearly non-positive definite.
- NE_NO_LICENCE
-
Your licence key may have expired or may not have been installed correctly.
See
Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
- NE_NOT_POS_DEF
-
On entry, the covariance matrix
qq is not positive definite.
- NE_OBSERV_LT_P
-
On entry, the total number of observations is less than the total number of parameters (including the covariance matrix). Number of observations and number of parameters .
- NE_OVERFLOW_LIKELY
-
The 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
Parallelism and Performance
nag_tsa_varma_forecast (g13djc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_tsa_varma_forecast (g13djc) makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the
x06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the
Users' Note for your implementation for any additional implementation-specific information.
The same differencing operator does not have to be applied to all the series. For example, suppose we have
, and wish to apply the second order differencing operator
to the first series and the first-order differencing operator
to the second series:
Then
,
, and
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 function to produce the forecasts.
nag_tsa_varma_forecast (g13djc) should not be used when the moving average parameters lie close to the boundary of the invertibility region. The function does test for both invertibility and stationarity but if in doubt, you may use
nag_tsa_arma_roots (g13dxc), before calling this function, to check that the VARMA model being used is invertible.
On a successful exit, the quantities
k,
lmax,
kmax,
ref and
lref will be suitable for input to
nag_tsa_varma_update (g13dkc).
10
Example
This example computes forecasts of the next five values in two series each of length
. No transformation is to be used and no differencing is to be applied to either of the series.
nag_tsa_varma_estimate (g13ddc)
is first called to fit an AR(1) model to the series. The mean vector
is to be estimated and
constrained to be zero.
10.1
Program Text
Program Text (g13djce.c)
10.2
Program Data
Program Data (g13djce.d)
10.3
Program Results
Program Results (g13djce.r)