NAG Library Routine Document

G13DBF

+− 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

G13DBF calculates the multivariate partial autocorrelation function of a multivariate time series.

2 Specification

SUBROUTINE G13DBF (

C0, C, LDC0, NS, NL, NK, P, V0, V, D, DB, W, WB, NVP, WA, IWA, IFAIL)

INTEGER	LDC0, NS, NL, NK, NVP, IWA, IFAIL
REAL (KIND=nag_wp)	C0(LDC0,NS), C(LDC0,LDC0,NL), P(NK), V0, V(NK), D(LDC0,LDC0,NK), DB(LDC0,NS), W(LDC0,LDC0,NK), WB(LDC0,LDC0,NK), WA(IWA)

3 Description

The input is a set of lagged autocovariance matrices

C_{0}, C_{1}, C_{2}, \dots, C_{m}

. These will generally be sample values such as are obtained from a multivariate time series using G13DMF.

The main calculation is the recursive determination of the coefficients in the finite lag (forward) prediction equation

x_{t} = Φ_{l, 1} x_{t - 1} + \dots + Φ_{l, l} x_{t - l} + e_{l, t}

and the associated backward prediction equation

x_{t - l - 1} = Ψ_{l, 1} x_{t - l} + \dots + Ψ_{l, l} x_{t - 1} + f_{l, t}

together with the covariance matrices

D_{l}

e_{l, t}

and

G_{l}

f_{l, t}

The recursive cycle, by which the order of the prediction equation is extended from

l

l + 1

, is to calculate

M_{l + 1} = C_{l + 1}^{T} - Φ_{l, 1} C_{l}^{T} - \dots - Φ_{l, l} C_{1}^{T}

(1)

then

Φ_{l + 1, l + 1} = M_{l + 1} D_{l}^{- 1}

Ψ_{l + 1, l + 1} = {M^{T}}_{l + 1} G_{l}^{- 1}

from which

Φ_{l + 1, j} = Φ_{l, j} - Φ_{l + 1, l + 1} Ψ_{l, l + 1 - j}, j = 1, 2, \dots, l

(2)

and

Ψ_{l + 1, j} = Ψ_{l, j} - Ψ_{l + 1, l + 1} Φ_{l, l + 1 - j}, j = 1, 2, \dots, l .

(3)

Finally,

D_{l + 1} = D_{l} - M_{l + 1} {Φ^{T}}_{l + 1, l + 1}

and

G_{l + 1} = G_{l} - {M^{T}}_{l + 1} {Ψ^{T}}_{l + 1, l + 1}

(Here

T

denotes the transpose of a matrix.)

The cycle is initialized by taking (for

l = 0

)

D_{0} = G_{0} = C_{0} .

In the step from

l = 0

1

, the above equations contain redundant terms and simplify. Thus (1) becomes

M_{1} = {C^{T}}_{1}

and neither (2) or (3) are needed.

Quantities useful in assessing the effectiveness of the prediction equation are generalized variance ratios

v_{l} = \det D_{l} / \det C_{0}, l = 1, 2, \dots

and multiple squared partial autocorrelations

p_{l}^{2} = 1 - v_{l} / v_{l - 1} .

4 References

Akaike H (1971) Autoregressive model fitting for control Ann. Inst. Statist. Math. 23 163–180

Whittle P (1963) On the fitting of multivariate autoregressions and the approximate canonical factorization of a spectral density matrix Biometrika 50 129–134

5 Parameters

1: C0(LDC0,NS) – REAL (KIND=nag_wp) arrayInput: On entry: contains the zero lag cross-covariances between the NS series as returned by G13DMF. (C0 is assumed to be symmetric, upper triangle only is used.)
2: C(LDC0,LDC0,NL) – REAL (KIND=nag_wp) arrayInput: On entry: contains the cross-covariances at lags $1$ to NL. $C (i, j, k)$ must contain the cross-covariance, $c_{i j k}$ , of series $i$ and series $j$ at lag $k$ . Series $j$ leads series $i$ .
3: LDC0 – INTEGERInput: On entry: the first dimension of the arrays C0, C, D, DB, W and WB and the second dimension of the arrays C, D, W and WB as declared in the (sub)program from which G13DBF is called.
Constraint: $LDC0 \geq \max (NS, 1)$ .
4: NS – INTEGERInput: On entry: $k$ , the number of time series whose cross-covariances are supplied in C and C0.
Constraint: $NS \geq 1$ .
5: NL – INTEGERInput: On entry: $m$ , the maximum lag for which cross-covariances are supplied in C.
Constraint: $NL \geq 1$ .
6: NK – INTEGERInput: On entry: the number of lags to which partial auto-correlations are to be calculated.
Constraint: $1 \leq NK \leq NL$ .
7: P(NK) – REAL (KIND=nag_wp) arrayOutput: On exit: the multiple squared partial autocorrelations from lags $1$ to NVP; that is, $P (l)$ contains $p_{l}^{2}$ , for $l = 1, 2, \dots, NVP$ . For lags $NVP + 1$ to NK the elements of P are set to zero.
8: V0 – REAL (KIND=nag_wp)Output: On exit: the lag zero prediction error variance (equal to the determinant of C0).
9: V(NK) – REAL (KIND=nag_wp) arrayOutput: On exit: the prediction error variance ratios from lags $1$ to NVP; that is, $V (l)$ contains $v_{l}$ , for $l = 1, 2, \dots, NVP$ . For lags $NVP + 1$ to NK the elements of V are set to zero.
10: D(LDC0,LDC0,NK) – REAL (KIND=nag_wp) arrayOutput: On exit: the prediction error variance matrices at lags $1$ to NVP.
Element $(i, j, k)$ of D contains the prediction error covariance of series $i$ and series $j$ at lag $k$ , for $k = 1, 2, \dots, NVP$ . Series $j$ leads series $i$ ; that is, the $(i, j)$ th element of $D_{k}$ . For lags $NVP + 1$ to NK the elements of D are set to zero.
11: DB(LDC0,NS) – REAL (KIND=nag_wp) arrayOutput: On exit: the backward prediction error variance matrix at lag NVP.
$DB (i, j)$ contains the backward prediction error covariance of series $i$ and series $j$ ; that is, the $(i, j)$ th element of the $G_{k}$ , where $k = NVP$ .
12: W(LDC0,LDC0,NK) – REAL (KIND=nag_wp) arrayOutput: On exit: the prediction coefficient matrices at lags $1$ to NVP.
$W (i, j, l)$ contains the $j$ th prediction coefficient of series $i$ at lag $l$ ; that is, the $(i, j)$ th element of $Φ_{k l}$ , where $k = NVP$ , for $l = 1, 2, \dots, NVP$ . For lags $NVP + 1$ to NK the elements of W are set to zero.
13: WB(LDC0,LDC0,NK) – REAL (KIND=nag_wp) arrayOutput: On exit: the backward prediction coefficient matrices at lags $1$ to NVP.
$WB (i, j, l)$ contains the $j$ th backward prediction coefficient of series $i$ at lag $l$ ; that is, the $(i, j)$ th element of $Ψ_{k l}$ , where $k = NVP$ , for $l = 1, 2, \dots, NVP$ . For lags $NVP + 1$ to NK the elements of WB are set to zero.
14: NVP – INTEGEROutput: On exit: the maximum lag, $L$ , for which calculation of P, V, D, DB, W and WB was successful. If the routine completes successfully NVP will equal NK.
15: WA(IWA) – REAL (KIND=nag_wp) arrayWorkspace
16: IWA – INTEGERInput: On entry: the dimension of the array WA as declared in the (sub)program from which G13DBF is called.
Constraint: $IWA \geq (2 \times NS + 1) \times NS$ .
17: 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,	$LDC0 < 1$ ,
or	$NS < 1$ ,
or	$NS > LDC0$ ,
or	$NL < 1$ ,
or	$NK < 1$ ,
or	$NK > NL$ ,
or	$IWA < (2 \times NS + 1) \times NS$ .

$IFAIL = 2$: C0 is not positive definite.

V0, V, P, D, DB, W, WB and NVP are set to zero.

$IFAIL = 3$: At lag $k = NVP + 1 \leq NK$ , $D_{k}$ was found not to be positive definite. Up to lag NVP, V0, V, P, D, W and WB contain the values calculated so far and from lag $NVP + 1$ to lag NK the matrices contain zero. DB contains the backward prediction coefficients for lag NVP.

7 Accuracy

The conditioning of the problem depends on the prediction error variance ratios. Very small values of these may indicate loss of accuracy in the computations.

8 Further Comments

The time taken by G13DBF is roughly proportional to

{NK}^{2} \times {NS}^{3}

If sample autocorrelation matrices are used as input, then the output will be relevant to the original series scaled by their standard deviations. If these autocorrelation matrices are produced by G13DMF, you must replace the diagonal elements of

C_{0}

(otherwise used to hold the series variances) by

1

9 Example

This example reads the autocovariance matrices for four series from lag

0

5

. It calls G13DBF to calculate the multivariate partial autocorrelation function and other related matrices of statistics up to lag

3

. It prints the results.

NAG Library Routine DocumentG13DBF