NAG Library Routine Document

G13ACF

+− 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

G13ACF calculates partial autocorrelation coefficients given a set of autocorrelation coefficients. It also calculates the predictor error variance ratios for increasing order of finite lag autoregressive predictor, and the autoregressive parameters associated with the predictor of maximum order.

2 Specification

SUBROUTINE G13ACF (

R, NK, NL, P, V, AR, NVL, IFAIL)

INTEGER	NK, NL, NVL, IFAIL
REAL (KIND=nag_wp)	R(NK), P(NL), V(NL), AR(NL)

3 Description

The data consist of values of autocorrelation coefficients

r_{1}, r_{2}, \dots, r_{K}

, relating to lags

1, 2, \dots, K

. These will generally (but not necessarily) be sample values such as may be obtained from a time series

x_{t}

using G13ABF.

The partial autocorrelation coefficient at lag

l

may be identified with the parameter

p_{l, l}

in the autoregression

x_{t} = c_{l} + p_{l, 1} x_{t - 1} + p_{l, 2} x_{t - 2} + \dots + p_{l, l} x_{t - l} + e_{l, t}

where

e_{l, t}

is the predictor error.

The first subscript

l

p_{l, l}

and

e_{l, t}

emphasizes the fact that the parameters will in general alter as further terms are introduced into the equation (i.e., as

l

is increased).

The parameters are determined from the autocorrelation coefficients by the Yule–Walker equations

r_{i} = p_{l, 1} r_{i - 1} + p_{l, 2} r_{i - 2} + \dots + p_{l, l} r_{i - l}, i = 1, 2, \dots, l

taking

r_{j} = r_{|j|}

when

j < 0

, and

r_{0} = 1

The predictor error variance ratio

v_{l} = var (e_{l, t}) / var (x_{t})

is defined by

v_{l} = 1 - p_{l, 1} r_{1} - p_{l, 2} r_{2} - \dots - p_{l, l} r_{l} .

The above sets of equations are solved by a recursive method (the Durbin–Levinson algorithm). The recursive cycle applied for

l = 1, 2, \dots, (L - 1)

, where

L

is the number of partial autocorrelation coefficients required, is initialized by setting

p_{1, 1} = r_{1}

and

v_{1} = 1 - r_{1}^{2}

Then

\begin{array}{lcl} p_{l + 1, l + 1} & = & (r_{l + 1} - p_{l, 1} r_{l} - p_{l, 2} r_{l - 1} - \dots - p_{l, l} r_{1}) / v_{l} \\ p_{l + 1, j} & = & p_{l, j} - p_{l + 1, l + 1} p_{l, l + 1 - j}, j = 1, 2, \dots, l \\ v_{l + 1} & = & v_{l} (1 - p_{l + 1, l + 1}) (1 + p_{l + 1, l + 1}) . \end{array}

If the condition

|p_{l, l}| \geq 1

occurs, say when

l = l_{0}

, it indicates that the supplied autocorrelation coefficients do not form a positive definite sequence (see Hannan (1960)), and the recursion is not continued. The autoregressive parameters are overwritten at each recursive step, so that upon completion the only available values are

p_{L j}

, for

j = 1, 2, \dots, L

, or

p_{l_{0} - 1, j}

if the recursion has been prematurely halted.

4 References

Box G E P and Jenkins G M (1976) Time Series Analysis: Forecasting and Control (Revised Edition) Holden–Day

Durbin J (1960) The fitting of time series models Rev. Inst. Internat. Stat. 28 233

Hannan E J (1960) Time Series Analysis Methuen

5 Parameters

1: R(NK) – REAL (KIND=nag_wp) arrayInput: On entry: the autocorrelation coefficient relating to lag $k$ , for $k = 1, 2, \dots, K$ .
2: NK – INTEGERInput: On entry: $K$ , the number of lags. The lags range from $1$ to $K$ and do not include zero.
Constraint: $NK > 0$ .
3: NL – INTEGERInput: On entry: $L$ , the number of partial autocorrelation coefficients required.
Constraint: $0 < NL \leq NK$ .
4: P(NL) – REAL (KIND=nag_wp) arrayOutput: On exit: $P (l)$ contains the partial autocorrelation coefficient at lag $l$ , $p_{l, l}$ , for $l = 1, 2, \dots, NVL$ .
5: V(NL) – REAL (KIND=nag_wp) arrayOutput: On exit: $V (l)$ contains the predictor error variance ratio $v_{l}$ , for $l = 1, 2, \dots, NVL$ .
6: AR(NL) – REAL (KIND=nag_wp) arrayOutput: On exit: the autoregressive parameters of maximum order, i.e., $p_{L j}$ if $IFAIL = 0$ , or $p_{l_{0} - 1, j}$ if $IFAIL = 3$ , for $j = 1, 2, \dots, NVL$ .
7: NVL – INTEGEROutput: On exit: the number of valid values in each of P, V and AR. Thus in the case of premature termination at iteration $l_{0}$ (see Section 3), NVL is returned as $l_{0} - 1$ .
8: 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,	$NK \leq 0$ ,
or	$NL \leq 0$ ,
or	$NK < NL$ .

$IFAIL = 2$: On entry, the autocorrelation coefficient of lag $1$ has an absolute value greater than or equal to $1.0$ ; no recursions could be performed.

$IFAIL = 3$: Recursion has been prematurely terminated; the supplied autocorrelation coefficients do not form a positive definite sequence (see Section 3). Parameter NVL returns the number of valid values computed.

7 Accuracy

The computations are believed to be stable.

8 Further Comments

The time taken by G13ACF is proportional to

{(NVL)}^{2}

9 Example

This example uses an input series of

10

sample autocorrelation coefficients derived from the original series of sunspot numbers generated by the G13ABF example program. The results show five values of each of the three output arrays: partial autocorrelation coefficients, predictor error variance ratios and autoregressive parameters. All of these were valid.

NAG Library Routine DocumentG13ACF