NAG Library Function Document

nag_tsa_multi_part_regsn (g13dpc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

+− 10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

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

W_{t} = {(w_{1 t}, w_{2 t}, \dots, w_{k t})}^{T}

, for

t = 1, 2, \dots, n

, denote a vector of

k

time series. The partial autoregression matrix at lag

l

P_{l}

, is defined to be the last matrix coefficient when a vector autoregressive model of order

l

is fitted to the series.

P_{l}

has the property that if

W_{t}

follows a vector autoregressive model of order

p

then

P_{l} = 0

for

l > p

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

Q R

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

m

, which is usually taken to be at most

n / 4

The function also returns the asymptotic standard errors of the elements of

{\hat{P}}_{l}

and an estimate of the residual variance-covariance matrix

{\hat{Σ}}_{l}

, for

l = 1, 2, \dots, m

. If

S_{l}

denotes the residual sum of squares and cross-products matrix after fitting an

AR (l)

model to the series then under the null hypothesis

H_{0} : P_{l} = 0

the test statistic

X_{l} = - ((n - m - 1) - \frac{1}{2} - l k) \log (\frac{|S_{l}|}{|S_{l - 1}|})

is asymptotically distributed as

χ^{2}

with

k^{2}

degrees of freedom.

X_{l}

provides a useful diagnostic aid in determining the order of an autoregressive model. (Note that

{\hat{Σ}}_{l} = S_{l} / (n - l)

.) 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: k – IntegerInput: On entry: $k$ , the number of time series.
Constraint: $k \geq 1$ .
2: n – IntegerInput: On entry: $n$ , the number of observations in the time series.
Constraint: $n \geq 4$ .
3: z[ $k \times n$ ] – const doubleInput: On entry: $z [(t - 1) k + i - 1]$ must contain the value for the $i$ th series at time $t$ , for $i = 1, 2, \dots, k$ and $t = 1, 2, \dots, n$ .
4: m – IntegerInput: On entry: $m$ , the number of partial autoregression matrices to be computed. If in doubt set $m = 10$ .
Constraint: $m \geq 1$ and $n - m - (k \times m + 1) \geq k$ .
5: maxlag – 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 $fail . code =$ MATRIX_ILL_CONDITIONED on exit then maxlag will be less than m.
6: parlag[ $k \times k \times m$ ] – doubleOutput: On exit: $parlag [(l - 1) k^{2} + (j - 1) k + i - 1]$ contains an estimate of the $(i, j)$ th element of the partial autoregression matrix at lag $l$ , for $l = 1, 2, \dots, maxlag$ , $i = 1, 2, \dots, k$ and $j = 1, 2, \dots, k$ .
7: se[ $k \times k \times m$ ] – doubleOutput: On exit: $se [(l - 1) k^{2} + (j - 1) k + i - 1]$ contains an estimate of the standard error of the corresponding element in parlag.
8: qq[ $k \times k \times m$ ] – doubleOutput: On exit: $qq [(l - 1) k^{2} + (j - 1) k + i - 1]$ contains an estimate of the $(i, j)$ th element of the residual variance-covariance matrix, ${\hat{Σ}}_{l}$ , for $l = 1, 2, \dots, maxlag$ , $i = 1, 2, \dots, k$ and $j = 1, 2, \dots, k$ .
9: x[m] – doubleOutput: On exit: $x [l - 1]$ contains $X_{l}$ , the likelihood ratio statistic at lag $l$ , for $l = 1, 2, \dots, maxlag$ .
10: pvalue[m] – doubleOutput: On exit: $pvalue [l - 1]$ contains the significance level of the statistic in the corresponding element of x.
11: loglhd[m] – doubleOutput: On exit: $loglhd [l - 1]$ contains an estimate of the maximum of the log-likelihood function when an $AR (l - 1)$ model has been fitted to the series, for $l = 1, 2, \dots, maxlag$ .
12: fail – NagError *Input/Output: The NAG error argument (see Section 3.6 in the Essential Introduction).

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 $⟨value⟩$ .
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, $m = ⟨value⟩$ .
Constraint: $m \geq 1$ .
On entry, $n = ⟨value⟩$ .
Constraint: $n \geq 4$ .
NE_INT_3: On entry, $k = ⟨value⟩$ , $m = ⟨value⟩$ and $n = ⟨value⟩$ .
Constraint: $n - m - (k \times m + 1) \geq k$ .
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.

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 Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

The time taken is roughly proportional to

n m k

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

l

for which

- 2 \times loglhd [l] + 2 l k^{2}

is smallest. You should be warned that this does not always lead to the same choice of

l

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

48

up to lag

10

NAG Library Function Documentnag_tsa_multi_part_regsn (g13dpc)