NAG Library Function Document
nag_tsa_multi_part_regsn (g13dpc)
1
Purpose
nag_tsa_multi_part_regsn (g13dpc) calculates the sample partial autoregression matrices of a multivariate time series. A set of likelihood ratio statistics and their significance levels are also returned. These quantities are useful for determining whether the series follows an autoregressive model and, if so, of what order.
2
Specification
#include <nag.h> |
#include <nagg13.h> |
void |
nag_tsa_multi_part_regsn (Integer k,
Integer n,
const double z[],
Integer m,
Integer *maxlag,
double parlag[],
double se[],
double qq[],
double x[],
double pvalue[],
double loglhd[],
NagError *fail) |
|
3
Description
Let , for , denote a vector of time series. The partial autoregression matrix at lag , , is defined to be the last matrix coefficient when a vector autoregressive model of order is fitted to the series. has the property that if follows a vector autoregressive model of order then for .
Sample estimates of the partial autoregression matrices may be obtained by fitting autoregressive models of successively higher orders by multivariate least squares; see
Tiao and Box (1981) and
Wei (1990). These models are fitted using a
algorithm based on the functions
nag_regsn_mult_linear_addrem_obs (g02dcc) and
nag_regsn_mult_linear_delete_var (g02dfc). They are calculated up to lag
, which is usually taken to be at most
.
The function also returns the asymptotic standard errors of the elements of
and an estimate of the residual variance-covariance matrix
, for
. If
denotes the residual sum of squares and cross-products matrix after fitting an
model to the series then under the null hypothesis
the test statistic
is asymptotically distributed as
with
degrees of freedom.
provides a useful diagnostic aid in determining the order of an autoregressive model. (Note that
.) The function also returns an estimate of the maximum of the log-likelihood function for each AR model that has been fitted.
4
References
Tiao G C and Box G E P (1981) Modelling multiple time series with applications J. Am. Stat. Assoc. 76 802–816
Wei W W S (1990) Time Series Analysis: Univariate and Multivariate Methods Addison–Wesley
5
Arguments
- 1:
– IntegerInput
-
On entry: , the number of time series.
Constraint:
.
- 2:
– IntegerInput
-
On entry: , the number of observations in the time series.
Constraint:
.
- 3:
– const doubleInput
-
On entry: must contain the value for the th series at time , for and .
- 4:
– IntegerInput
-
On entry: , the number of partial autoregression matrices to be computed. If in doubt set .
Constraint:
and .
- 5:
– Integer *Output
-
On exit: the maximum lag up to which partial autoregression matrices (along with their likelihood ratio statistics and their significance levels) have been successfully computed. On a successful exit
maxlag will equal
m. If
MATRIX_ILL_CONDITIONED on exit then
maxlag will be less than
m.
- 6:
– doubleOutput
-
On exit: contains an estimate of the th element of the partial autoregression matrix at lag , for , and .
- 7:
– doubleOutput
-
On exit:
contains an estimate of the standard error of the corresponding element in
parlag.
- 8:
– doubleOutput
-
On exit: contains an estimate of the th element of the residual variance-covariance matrix, , for , and .
- 9:
– doubleOutput
-
On exit: contains , the likelihood ratio statistic at lag , for .
- 10:
– doubleOutput
-
On exit:
contains the significance level of the statistic in the corresponding element of
x.
- 11:
– doubleOutput
-
On exit: contains an estimate of the maximum of the log-likelihood function when an model has been fitted to the series, for .
- 12:
– 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
- MATRIX_ILL_CONDITIONED
-
The recursive equations used to compute the partial autoregression matrices are ill-conditioned. They have been computed up to lag .
- 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_INT
-
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: .
- NE_INT_3
-
On entry, , and .
Constraint: .
- 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_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.
7
Accuracy
The computations are believed to be stable.
8
Parallelism and Performance
nag_tsa_multi_part_regsn (g13dpc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_tsa_multi_part_regsn (g13dpc) 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 time taken is roughly proportional to .
For each order of autoregressive model that has been estimated, nag_tsa_multi_part_regsn (g13dpc) returns the maximum of the log-likelihood function. An alternative means of choosing the order of a vector AR process is to choose the order for which Akaike's information criterion is smallest. That is, choose the value of for which is smallest. You should be warned that this does not always lead to the same choice of as indicated by the sample partial autoregression matrices and the likelihood ratio statistics.
10
Example
This example computes the sample partial autoregression matrices of two time series of length up to lag .
10.1
Program Text
Program Text (g13dpce.c)
10.2
Program Data
Program Data (g13dpce.d)
10.3
Program Results
Program Results (g13dpce.r)