nag_tsa_varma_forecast (g13djc) (PDF version)
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NAG Library Manual

NAG Library Function Document

nag_tsa_varma_forecast (g13djc)

+ Contents

    1  Purpose
    7  Accuracy

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 Zt = z1t,z2t,,zktT , for t=1,2,,n, denote a k-dimensional time series for which forecasts of Zn+1,Zn+2,,Zn+lmax are required. Let Wt = w1t,w2t,,wktT  be defined as follows:
wit=δiBzit*,  i=1,2,,k,
where δiB is the differencing operator applied to the ith series and where zit* is equal to either zit, zit or logezit 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 ith series is di, then the differencing operator for the ith series is defined by δiB=1-δi1B-δi2B2--δidiBdi where B is the backward shift operator; that is, BZt=Zt-1. The differencing parameters δij, for i=1,2,,k and j=1,2,,di, must be supplied by you. If the ith series does not require differencing, then di=0.
Wt is assumed to follow a multivariate ARMA model of the form:
Wt-μ=ϕ1Wt-1-μ+ϕ2Wt-2-μ++ϕpWt-p-μ+εt-θ1εt-1--θqεt-q, (1)
where εt = ε1t,ε2t,,εktT , for t=1,2,,n, is a vector of k residual series assumed to be Normally distributed with zero mean and positive definite covariance matrix Σ. The components of εt are assumed to be uncorrelated at non-simultaneous lags. The ϕi and θj are k by k matrices of parameters. The matrices ϕi, for i=1,2,,p, are the autoregressive (AR) parameter matrices, and the matrices θi, for i=1,2,,q, the moving average (MA) parameter matrices. The parameters in the model are thus the p (k by k) ϕ-matrices, the q (k by k) θ-matrices, the mean vector μ and the residual error covariance matrix Σ. 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 ϕB and θB are the autoregressive and moving average polynomials and δB denotes the k by k diagonal matrix whose ith diagonal elements is δiB and Zt* = z 1t * , z2t* zkt* T .
This may be rewritten as
Zt*=τ+ψ Bεt=τ+εt+ψ1εt- 1+ψ2εt- 2+
where ψB=δ-1Bϕ-1BθB and τ=δ-1Bμ is a vector of length k.
Forecasts are computed using a multivariate version of the procedure described in Box and Jenkins (1976). If Z^n*l denotes the forecast of Zn+l*, then Z^n*l is taken to be that linear function of Zn*,Zn-1*, which minimizes the elements of Eenlenl where enl=Zn+l*-Z^n*l is the forecast error. Z^n*l is referred to as the linear minimum mean square error forecast of Zn+l*.
The linear predictor which minimizes the mean square error may be expressed as
The forecast error at t for lead l is then
Let d=maxdi, for i=1,2,,k. Unless q=0 the function requires estimates of εt, for t=d+1,,n, which are obtainable from nag_tsa_varma_estimate (g13ddc). The terms εt are assumed to be zero, for t=n+1,,n+lmax. You may use nag_tsa_varma_update (g13dkc) to update these lmax forecasts should further observations, Zn+1,Zn+2,, become available. Note that when lmax or more further observations are available then nag_tsa_varma_forecast (g13djc) must be used to produce new forecasts for Zn+lmax+1,Zn+lmax+2,, 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, Zt; 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 knkmaxipiqparnparqq and v from nag_tsa_varma_estimate (g13ddc) are suitable for input to nag_tsa_varma_forecast (g13djc).
1:     kIntegerInput
On entry: k, the dimension of the multivariate time series.
Constraint: k1.
2:     nIntegerInput
On entry: n, the number of observations in the series, Zt, prior to differencing.
Constraint: n3.
The total number of observations must exceed the total number of parameters in the model; that is
  • if mean=Nag_MeanZero, n×k>ip+iq×k×k+k×k+1/2;
  • if mean=Nag_MeanInclude, n×k>ip+iq×k×k+k+k×k+1/2,
(see the arguments ip, iq and mean).
3:     z[kmax×n]const doubleInput
Note: the i,jth element of the matrix Z is stored in z[j-1×kmax+i-1].
On entry: z[t-1×kmax+i-1] must contain the ith series at time t, for t=1,2,,n and i=1,2,,k.
4:     kmaxIntegerInput
On entry: the stride separating row elements in the two-dimensional data stored in the arrays z, delta, qq, v, predz, sefz.
Constraint: kmaxk.
5:     tr[k]const IntegerInput
On entry: tr[i-1] indicates whether the ith series is to be transformed, for i=1,2,,k.
A square root transformation is used.
No transformation is used.
A log transformation is used.
Constraint: tr[i-1]=-1, 0 or 1, for i=1,2,,k.
6:     id[k]const IntegerInput
On entry: id[i-1] must specify, di, the order of differencing required for the ith series.
Constraint: 0id[i-1]<n-maxip,iq, for i=1,2,,k.
7:     delta[dim]const doubleInput
Note: the dimension, dim, of the array delta must be at least kmax×d, where d=maxid[i-1].
On entry: if id[i-1]>0, then delta[j-1×kmax+i-1] must be set equal to δij, for j=1,2,,di and i=1,2,,k.
If d=0, delta is not referenced.
8:     ipIntegerInput
On entry: p, the number of AR parameter matrices.
Constraint: ip0.
9:     iqIntegerInput
On entry: q, the number of MA parameter matrices.
Constraint: iq0.
10:   meanNag_IncludeMeanInput
On entry: mean=Nag_MeanInclude, if components of μ have been estimated and mean=Nag_MeanZero, if all elements of μ are to be taken as zero.
Constraint: mean=Nag_MeanInclude or Nag_MeanZero.
11:   par[lpar]const doubleInput
On entry: must contain the parameter estimates read in row by row in the order ϕ1,ϕ2,,ϕp, θ1,θ2,,θq, μ.
  • if ip>0, par[l-1×k×k+i-1×k+j-1] must be set equal to an estimate of the i,jth element of ϕl, for l=1,2,,p, i=1,2,,k and j=1,2,,k;
  • if iq>0, par[p×k×k+l-1×k×k+i-1×k+j-1] must be set equal to an estimate of the i,jth element of θl, for l=1,2,,q, i=1,2,,k and j=1,2,,k;
  • if mean=Nag_MeanInclude, par[p+q×k×k+i-1] must be set equal to an estimate of the ith component of μ, for i=1,2,,k.
Constraint: the first ip×k×k elements of par must satisfy the stationarity condition and the next iq×k×k elements of par must satisfy the invertibility condition.
12:   lparIntegerInput
On entry: the dimension of the array par.
  • if mean=Nag_MeanZero, lparmax1,ip+iq×k×k;
  • if mean=Nag_MeanInclude, lparip+iq×k×k+k.
13:   qq[kmax×k]doubleInput/Output
Note: the i,jth element of the matrix is stored in qq[j-1×kmax+i-1].
On entry: qq[j-1×kmax+i-1] must contain an estimate of the i,jth element of Σ. The lower triangle only is needed.
Constraint: qq must be positive definite.
On exit: if fail.code= NE_EIGENVALUESNE_G13D_ARNE_G13D_MANE_NEARLY_POS_DEFNE_NOT_POS_DEFNE_OVERFLOW_LIKELY or NE_TRANSFORMATION, then the upper triangle is set equal to the lower triangle.
14:   v[dim]const doubleInput
Note: the dimension, dim, of the array v must be at least kmax×(n-d), where d=maxid[i-1].
On entry: v[t-1×kmax+i-1] must contain an estimate of the ith component of εt+d, for i=1,2,,k and t=1,2,,n-d.
If q=0, v is not used.
15:   lmaxIntegerInput
On entry: the number, lmax, of forecasts required.
Constraint: lmax1.
16:   predz[kmax×lmax]doubleOutput
Note: the i,jth element of the matrix is stored in predz[j-1×kmax+i-1].
On exit: predz[l-1×kmax+i-1] contains the forecast of zi,n+l, for i=1,2,,k and l=1,2,,lmax.
17:   sefz[kmax×lmax]doubleOutput
Note: the i,jth element of the matrix is stored in sefz[j-1×kmax+i-1].
On exit: sefz[l-1×kmax+i-1] contains an estimate of the standard error of the forecast of zi,n+l, for i=1,2,,k and l=1,2,,lmax.
18:   ref[lref]doubleOutput
On exit: the reference vector which may be used to update forecasts using nag_tsa_varma_update (g13dkc). The first lmax-1×k×k elements contain the ψ weight matrices, ψ1,ψ2,,ψlmax-1. The next k×lmax elements contain the forecasts of the transformed series Z^n+1*,Z^n+2*,, Z^n+lmax* and the next k×lmax contain the variances of the forecasts of the transformed variables. The last k elements are used to store the transformations for the series.
19:   lrefIntegerInput
On entry: the dimension of the array ref.
Constraint: lreflmax-1×k×k+2×k×lmax+k.
20:   failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

Dynamic memory allocation failed.
On entry, argument value had an illegal value.
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.
On entry, the AR parameter matrices are outside the stationarity region.
On entry, the MA parameter matrices are outside the invertibility region.
On entry, ip=value.
Constraint: ip0.
On entry, iq=value.
Constraint: iq0.
On entry, k=value.
Constraint: k1.
On entry, lmax=value.
Constraint: lmax1.
On entry, lpar is too small: lpar=value but must be at least value.
On entry, lref is too small: lref=value but must be at least value.
On entry, n=value.
Constraint: n3.
On entry, kmax=value and k=value.
Constraint: kmaxk.
On entry, id[value]=value and n-maxip,iq=value.
Constraint: 0id[i-1]<n-maxip,iq.
On entry, tr[i-1]=value and k=value.
Constraint: tr[i-1]=-1, 0 or 1, for i=1,2,,k.
On entry, tr[value]=value. Constraint: tr[i]=-1, 0 or 1.
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.
The covariance matrix may be nearly non positive definite.
On entry, the covariance matrix qq is not positive definite.
On entry, the total number of observations is less than the total number of parameters (including the covariance matrix). Number of observations =value and number of parameters =value.
The forecasts will overflow if computed.
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 Users' Note for your implementation for any additional implementation-specific information.

9  Further Comments

The same differencing operator does not have to be applied to all the series. For example, suppose we have k=2, and wish to apply the second order differencing operator 2 to the first series and the first-order differencing operator  to the second series:
w1t=2z1t= 1-B 2z1t=1-2B+B2Z1t,   and w2t=z2t=1-Bz2t.
Then d1=2,d2=1, d=maxd1,d2=2, and
delta= δ11 δ12 δ21 = 2 -1 1 .
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 48. 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 μ is to be estimated and ϕ12,1 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)

nag_tsa_varma_forecast (g13djc) (PDF version)
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NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2014