NAG Library Routine Document

G13DPF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

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

G13DPF 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

SUBROUTINE G13DPF (

K, N, Z, KMAX, M, MAXLAG, PARLAG, SE, QQ, X, PVALUE, LOGLHD, WORK, LWORK, IWORK, IFAIL)

INTEGER	K, N, KMAX, M, MAXLAG, LWORK, IWORK(K*M), IFAIL
REAL (KIND=nag_wp)	Z(KMAX,N), PARLAG(KMAX,KMAX,M), SE(KMAX,KMAX,M), QQ(KMAX,KMAX,M), X(M), PVALUE(M), LOGLHD(M), WORK(LWORK)

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 routines G02DCF and G02DFF. They are calculated up to lag

m

, which is usually taken to be at most

n / 4

The routine 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 routine 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 Parameters

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(KMAX,N) – REAL (KIND=nag_wp) arrayInput: On entry: $Z (i, t)$ must contain the observation $w_{i t}$ , for $i = 1, 2, \dots, k$ and $t = 1, 2, \dots, n$ .
4: KMAX – INTEGERInput: On entry: the first dimension of the arrays Z, PARLAG, SE and QQ and the second dimension of the arrays PARLAG, SE and QQ as declared in the (sub)program from which G13DPF is called.
Constraint: $KMAX \geq K$ .
5: 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 $N - M - (K \times M + 1) \geq K$ .
6: MAXLAG – INTEGEROutput: 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 $IFAIL = 2$ on exit then MAXLAG will be less than M.
7: PARLAG(KMAX,KMAX,M) – REAL (KIND=nag_wp) arrayOutput: On exit: $PARLAG (i, j, l)$ contains an estimate of the $(i, j)$ th element of the partial autoregression matrix at lag $l$ , ${\hat{P}}_{l} (i j)$ , for $l = 1, 2, \dots, MAXLAG$ , $i = 1, 2, \dots, k$ and $j = 1, 2, \dots, k$ .
8: SE(KMAX,KMAX,M) – REAL (KIND=nag_wp) arrayOutput: On exit: $SE (i, j, l)$ contains an estimate of the standard error of the corresponding element in the array PARLAG.
9: QQ(KMAX,KMAX,M) – REAL (KIND=nag_wp) arrayOutput: On exit: $QQ (i, j, l)$ contains an estimate of the $(i, j)$ th element of the corresponding variance-covariance matrix ${\hat{Σ}}_{l}$ , for $l = 1, 2, \dots, MAXLAG$ , $i = 1, 2, \dots, k$ and $j = 1, 2, \dots, k$ .
10: X(M) – REAL (KIND=nag_wp) arrayOutput: On exit: $X (l)$ contains $X_{l}$ , the likelihood ratio statistic at lag $l$ , for $l = 1, 2, \dots, MAXLAG$ .
11: PVALUE(M) – REAL (KIND=nag_wp) arrayOutput: On exit: $PVALUE (l)$ contains the significance level of the statistic in the corresponding element of X.
12: LOGLHD(M) – REAL (KIND=nag_wp) arrayOutput: On exit: $LOGLHD (l)$ contains an estimate of the maximum of the log-likelihood function when an $AR (l)$ model has been fitted to the series, for $l = 1, 2, \dots, MAXLAG$ .
13: WORK(LWORK) – REAL (KIND=nag_wp) arrayWorkspace
14: LWORK – INTEGERInput: On entry: the dimension of the array WORK as declared in the (sub)program from which G13DPF is called.
Constraint: $LWORK \geq (k + 1) k + l (4 + k) + 2 l^{2}$ , where $l = m k + 1$ .
15: IWORK( $K \times M$ ) – INTEGER arrayWorkspace
16: IFAIL – INTEGERInput/Output: On entry: IFAIL must be set to $0$ , $- 1 or 1$ . If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $- 1 or 1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is $0$ . When the value $- 1 or 1$ is used it is essential to test the value of IFAIL on exit.

On exit: $IFAIL = 0$ unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

IFAIL = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by X04AAF).

Errors or warnings detected by the routine:

$IFAIL = 1$

On entry,	$K < 1$ ,
or	$N < 4$ ,
or	$KMAX < K$ ,
or	$M < 1$ ,
or	$N - M - (K \times M + 1) < K$ ,
or	LWORK is too small.

$IFAIL = 2$: The recursive equations used to compute the sample partial autoregression matrices have broken down at lag $MAXLAG + 1$ . This exit could occur if the regression model is overparameterised. For your settings of $k$ and $n$ the value returned by MAXLAG is the largest permissible value of $m$ for which the model is not overparameterised. All output quantities in the arrays PARLAG, SE, QQ, X, PVALUE and LOGLHD up to and including lag MAXLAG will be correct.

7 Accuracy

The computations are believed to be stable.

8 Further Comments

The time taken is roughly proportional to

n m k

For each order of autoregressive model that has been estimated, G13DPF 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.

9 Example

This example computes the sample partial autoregression matrices of two time series of length

48

up to lag

10

NAG Library Routine DocumentG13DPF