NAG Library Routine Document

G13DSF

G13DSF is a diagnostic checking routine suitable for use after fitting a vector ARMA model to a multivariate time series using G13DDF. The residual cross-correlation matrices are returned along with an estimate of their asymptotic standard errors and correlations. Also, G13DSF calculates the modified Li–McLeod portmanteau statistic and its significance level for testing model adequacy.

2 Specification

SUBROUTINE G13DSF (

K, N, V, KMAX, IP, IQ, M, PAR, PARHLD, QQ, ISHOW, R0, R, RCM, LDRCM, CHI, IDF, SIGLEV, IW, LIW, WORK, LWORK, IFAIL)

INTEGER	K, N, KMAX, IP, IQ, M, ISHOW, LDRCM, IDF, IW(LIW), LIW, LWORK, IFAIL
REAL (KIND=nag_wp)	V(KMAX,N), PAR((IP+IQ)KK), QQ(KMAX,K), R0(KMAX,K), R(KMAX,KMAX,M), RCM(LDRCM,MKK), CHI, SIGLEV, WORK(LWORK)
LOGICAL	PARHLD((IP+IQ)KK)

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 which is assumed to follow a multivariate ARMA model of the form

\begin{array}{l} W_{t} - μ = & ϕ_{1} (W_{t - 1} - μ) + ϕ_{2} (W_{t - 2} - μ) + \dots + ϕ_{p} (W_{t - p} - μ) \\ + ε_{t} - θ_{1} ε_{t - 1} - θ_{2} ε_{t - 2} - \dots - θ_{q} ε_{t - q}, \end{array}

(1)

where

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

, for

t = 1, 2, \dots, n

, is a vector of

k

residual series assumed to be Normally distributed with zero mean and positive definite covariance matrix

Σ

. The components of

ε_{t}

are assumed to be uncorrelated at non-simultaneous lags. The

ϕ_{i}

and

θ_{j}

are

k

k

matrices of parameters.

\{ϕ_{i}\}

, for

i = 1, 2, \dots, p

, are called the autoregressive (AR) parameter matrices, and

\{θ_{i}\}

, for

i = 1, 2, \dots, q

, the moving average (MA) parameter matrices. The parameters in the model are thus the

p

(

k

k

)

ϕ

-matrices, the

q

(

k

k

)

θ

-matrices, the mean vector

μ

and the residual error covariance matrix

Σ

. Let

A (ϕ) = {[\begin{array}{l} ϕ_{1} & I & 0 & . & . & . & 0 \\ ϕ_{2} & 0 & I & 0 & . & . & 0 \\ . & . \\ . & . \\ . & . \\ ϕ_{p - 1} & 0 & . & . & . & 0 & I \\ ϕ_{p} & 0 & . & . & . & 0 & 0 \end{array}]}_{p k \times p k} and B (θ) = {[\begin{array}{l} θ_{1} & I & 0 & . & . & . & 0 \\ θ_{2} & 0 & I & 0 & . & . & 0 \\ . & . \\ . & . \\ . & . \\ θ_{q - 1} & 0 & . & . & . & I \\ θ_{q} & 0 & . & . & . & . & 0 \end{array}]}_{q k \times q k}

where

I

denotes the

k

k

identity matrix.

The ARMA model (1) is said to be stationary if the eigenvalues of

A (ϕ)

lie inside the unit circle, and invertible if the eigenvalues of

B (θ)

lie inside the unit circle. The ARMA model is assumed to be both stationary and invertible. Note that some of the elements of the

ϕ

- and/or

θ

-matrices may have been fixed at pre-specified values (for example by calling G13DDF).

The estimated residual cross-correlation matrix at lag

l

is defined to the

k

k

matrix

{\hat{R}}_{l}

whose

(i, j)

th element is computed as

{\hat{r}}_{i j} (l) = \frac{\sum_{t = l + 1}^{n} ({\hat{ε}}_{i t - l} - {\bar{ε}}_{i}) ({\hat{ε}}_{j t} - {\bar{ε}}_{j})}{\sqrt{\sum_{t = 1}^{n} {({\hat{ε}}_{i t} - {\bar{ε}}_{i})}^{2} \sum_{t = 1}^{n} {({\hat{ε}}_{j t} - {\bar{ε}}_{j})}^{2}}}, l = 0, 1, \dots, i ​ and ​ j = 1, 2, \dots, k,

where

{\hat{ε}}_{i t}

denotes an estimate of the

t

th residual for the

i

th series

ε_{i t}

and

{\bar{ε}}_{i} = \sum_{t = 1}^{n} {\hat{ε}}_{i t} / n

. (Note that

{\hat{R}}_{l}

is an estimate of

E (ε_{t - l} ε_{t}^{T})

, where

E

is the expected value.)

A modified portmanteau statistic,

Q_{(m)}^{*}

, is calculated from the formula (see Li and McLeod (1981))

Q_{(m)}^{*} = \frac{k^{2} m (m + 1)}{2 n} + n \sum_{l = 1}^{m} \hat{r} {(l)}^{T} ({\hat{R}}_{0}^{- 1} \otimes {\hat{R}}_{0}^{- 1}) \hat{r} (l),

where

\otimes

denotes Kronecker product,

{\hat{R}}_{0}

is the estimated residual cross-correlation matrix at lag zero and

\hat{r} (l) = vec ({\hat{R}}_{l}^{T})

, where

vec

of a

k

k

matrix is a vector with the

(i, j)

th element in position

(i - 1) k + j

m

denotes the number of residual cross-correlation matrices computed. (Advice on the choice of

m

is given in Section 8.2.) Let

l_{C}

denote the total number of ‘free’ parameters in the ARMA model excluding the mean,

μ

, and the residual error covariance matrix

Σ

. Then, under the hypothesis of model adequacy,

Q_{(m)}^{*}

, has an asymptotic

χ^{2}

-distribution on

m k^{2} - l_{C}

degrees of freedom.

Let

\underset{̲}{\hat{r}} = (vec (R_{1}^{T}), vec (R_{2}^{T}), \dots, vec (R_{m}^{T}))

then the covariance matrix of

\underset{̲}{\hat{r}}

is given by

Var (\hat{\underset{̲}{r}}) = [Y - X {(X^{T} G G^{T} X)}^{- 1} X^{T}] / n,

where

Y = I_{m} \otimes (Δ \otimes Δ)

and

G = I_{m} (G G^{T})

Δ

is the dispersion matrix

Σ

in correlation form and

G

a nonsingular

k

k

matrix such that

G G^{T} = Δ^{- 1}

and

G Δ G^{T} = I_{k}

. The construction of the matrix

X

is discussed in Li and McLeod (1981). (Note that the mean,

μ

, plays no part in calculating

Var (\hat{\underset{̲}{r}})

and therefore is not required as input to G13DSF.)

4 References

Li W K and McLeod A I (1981) Distribution of the residual autocorrelations in multivariate ARMA time series models J. Roy. Statist. Soc. Ser. B 43 231–239

5 Parameters

The output quantities K, N, V, KMAX, IP, IQ, PAR, PARHLD and QQ from G13DDF are suitable for input to G13DSF.

1: K – INTEGERInput

On entry:

k

, the number of residual time series.

Constraint:

K \geq 1

2: N – INTEGERInput

On entry:

n

, the number of observations in each residual series.

3: V(KMAX,N) – REAL (KIND=nag_wp) arrayInput

On entry:

V (i, t)

must contain an estimate of the

i

th component of

ε_{t}

, for

i = 1, 2, \dots, k

and

t = 1, 2, \dots, n

Constraints:

no two rows of $V$ may be identical;
in each row there must be at least two distinct elements.

4: KMAX – INTEGERInput

On entry: the first dimension of the arrays V, QQ and R0 and

Constraint:

KMAX \geq K

5: IP – INTEGERInput

On entry:

p

, the number of AR parameter matrices.

Constraint:

IP \geq 0

6: IQ – INTEGERInput

On entry:

q

, the number of MA parameter matrices.

Constraint:

IQ \geq 0

Note:

IP = IQ = 0

is not permitted.

7: M – INTEGERInput

On entry: the value of

m

, the number of residual cross-correlation matrices to be computed. See Section 8.2 for advice on the choice of M.

Constraint:

IP + IQ < M < N

8: PAR( $(IP + IQ) \times K \times K$ ) – REAL (KIND=nag_wp) arrayInput

On entry: the parameter estimates read in row by row in the order

ϕ_{1}, ϕ_{2}, \dots, ϕ_{p}

θ_{1}, θ_{2}, \dots, θ_{q}

Thus,

if $IP > 0$ , $PAR ((l - 1) \times k \times k + (i - 1) \times k + j)$ must be set equal to an estimate of the $(i, j)$ th element of $ϕ_{l}$ , for $l = 1, 2, \dots, p$ and $i = 1, 2, \dots, k$ ;
if $IQ \geq 0$ , $PAR (p \times k \times k + (l - 1) \times k \times k + (i - 1) \times k + j)$ must be set equal to an estimate of the $(i, j)$ th element of $θ_{l}$ , for $l = 1, 2, \dots, q$ and $i = 1, 2, \dots, k$ .

The first

p \times k \times k

elements of PAR must satisfy the stationarity condition and the next

q \times k \times k

elements of PAR must satisfy the invertibility condition.

9: PARHLD( $(IP + IQ) \times K \times K$ ) – LOGICAL arrayInput

On entry:

PARHLD (i)

must be set to .TRUE. if

PAR (i)

has been held constant at a pre-specified value and .FALSE. if

PAR (i)

is a free parameter, for

i = 1, 2, \dots, (p + q) \times k \times k

10: QQ(KMAX,K) – REAL (KIND=nag_wp) arrayInput/Output

On entry:

QQ (i, j)

is an efficient estimate of the

(i, j)

th element of

Σ

. The lower triangle only is needed.

Constraint:

QQ

must be positive definite.

On exit: if

IFAIL \neq 1

, then the upper triangle is set equal to the lower triangle.

11: ISHOW – INTEGERInput

On entry: must be nonzero if the residual cross-correlation matrices

\{{\hat{r}}_{i j} (l)\}

and their standard errors

\{se ({\hat{r}}_{i j} (l))\}

, the modified portmanteau statistic with its significance and a summary table are to be printed. The summary table indicates which elements of the residual correlation matrices are significant at the

5 %

level in either a positive or negative direction; i.e., if

{\hat{r}}_{i j} (l) > 1.96 \times se ({\hat{r}}_{i j} (l))

then a ‘

+

’ is printed, if

{\hat{r}}_{i j} (l) < - 1.96 \times se ({\hat{r}}_{i j} (l))

then a ‘

-

’ is printed, otherwise a fullstop (.) is printed. The summary table is only printed if

k \leq 6

on entry.

The residual cross-correlation matrices, their standard errors and the modified portmanteau statistic with its significance are available also as output variables in R, RCM, CHI, IDF and SIGLEV.

12: R0(KMAX,K) – REAL (KIND=nag_wp) arrayOutput

On exit: if

i \neq j

, then

R0 (i, j)

contains an estimate of the

(i, j)

th element of the residual cross-correlation matrix at lag zero,

{\hat{R}}_{0}

. When

i = j

R0 (i, j)

contains the standard deviation of the

i

th residual series. If

IFAIL = 3

on exit then the first K rows and columns of R0 are set to zero.

13: R(KMAX,KMAX,M) – REAL (KIND=nag_wp) arrayOutput

On exit:

R (l, i, j)

is an estimate of the

(i, j)

th element of the residual cross-correlation matrix at lag

l

, for

i = 1, 2, \dots, k

j = 1, 2, \dots, k

and

l = 1, 2, \dots, m

. If

IFAIL = 3

on exit then all elements of R are set to zero.

14: RCM(LDRCM, $M \times K \times K$ ) – REAL (KIND=nag_wp) arrayOutput

On exit: the estimated standard errors and correlations of the elements in the array R. The correlation between

R (l, i, j)

and

R (l_{2}, i_{2}, j_{2})

is returned as

RCM (s, t)

where

s = (l - 1) \times k \times k + (j - 1) \times k + i

and

t = (l_{2} - 1) \times k \times k + (j_{2} - 1) \times k + i_{2}

except that if

s = t

, then

RCM (s, t)

contains the standard error of

R (l, i, j)

. If on exit,

IFAIL \geq 5

, then all off-diagonal elements of RCM are set to zero and all diagonal elements are set to

1 / \sqrt{n}

15: LDRCM – INTEGERInput

On entry: the first dimension of the array RCM as declared in the (sub)program from which G13DSF is called.

Constraint:

LDRCM \geq M \times K \times K

16: CHI – REAL (KIND=nag_wp)Output

On exit: the value of the modified portmanteau statistic,

Q_{(m)}^{*}

. If

IFAIL = 3

on exit then CHI is returned as zero.

17: IDF – INTEGEROutput

On exit: the number of degrees of freedom of CHI.

18: SIGLEV – REAL (KIND=nag_wp)Output

On exit: the significance level of CHI based on IDF degrees of freedom. If

IFAIL = 3

on exit, SIGLEV is returned as one.

19: IW(LIW) – INTEGER arrayWorkspace

20: LIW – INTEGERInput

On entry: the dimension of the array IW as declared in the (sub)program from which G13DSF is called.

Constraint:

LIW \geq K \times \max (IP, IQ)

21: WORK(LWORK) – REAL (KIND=nag_wp) arrayWorkspace

22: LWORK – INTEGERInput

On entry: the dimension of the array WORK as declared in the (sub)program from which G13DSF is called.

Constraint: if

LWORK \geq k (n + KMAX + 2) + m k^{2} (NPAR + m k^{2} + 1) + 3 k^{2} + (NPAR + 1) NPAR

NPAR = (p + q) k^{2}

23: 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, because for this routine the values of the output parameters may be useful even if

IFAIL \neq 0

on exit, the recommended value is

- 1

. 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).

Note: G13DSF may return useful information for one or more of the following detected errors or warnings.

Errors or warnings detected by the routine:

$IFAIL = 1$

On entry,	$K < 1$ ,
or	$KMAX < K$ ,
or	$IP < 0$ ,
or	$IQ < 0$ ,
or	$IP = IQ = 0$ ,
or	$M \leq IP + IQ$ ,
or	$M \geq N$ ,
or	$LDRCM < M \times K \times K$ ,
or	LIW is too small,
or	LWORK is too small.

$IFAIL = 2$: On entry, either QQ is not positive definite or the autoregressive parameter matrices are extremely close to or outside the stationarity region, or the moving average parameter matrices are extremely close to or outside the invertibility region. To proceed, you must supply different parameter estimates in the arrays PAR and QQ.

$IFAIL = 3$: On entry, at least one of the $k$ residual series is such that all its elements are practically identical giving zero (or near zero) variance or at least two of the residual series are identical. In this case CHI is set to zero, SIGLEV to one and all the elements of R0 and R set to zero.

$IFAIL = 4$: This is an unlikely exit brought about by an excessive number of iterations being needed to evaluate the zeros of the determinantal polynomials $\det (A (ϕ))$ and $\det (B (θ))$ . All output parameters are undefined.

$IFAIL = 5$: On entry, either the eigenvalues and eigenvectors of $Δ$ (the matrix QQ in correlation form) could not be computed or the determinantal polynomials $\det (A (ϕ))$ and $\det (B (θ))$ have a factor in common. To proceed, you must either supply different parameter estimates in the array QQ or delete this common factor from the model. In this case, the off-diagonal elements of RCM are returned as zero and the diagonal elements set to $1 / \sqrt{n}$ . All other output quantities will be correct.

$IFAIL = 6$: This is an unlikely exit. At least one of the diagonal elements of RCM was found to be either negative or zero. In this case all off-diagonal elements of RCM are returned as zero and all diagonal elements of RCM set to $1 / \sqrt{n}$ .

7 Accuracy

The computations are believed to be stable.

8 Further Comments

8.1 Timing

The time taken by G13DSF depends upon the number of residual cross-correlation matrices to be computed,

m

, and the number of time series,

k

8.2 Choice of $m$

The number of residual cross-correlation matrices to be computed,

m

, should be chosen to ensure that when the ARMA model (1) is written as either an infinite order autoregressive process, i.e.,

W_{t} - μ = \sum_{j = 1}^{\infty} π_{j} (W_{t - j} - μ) + ε_{t}

or as an infinite order moving average process, i.e.,

W_{t} - μ = \sum_{j = 1}^{\infty} ψ_{j} ε_{t - j} + ε_{t}

then the two sequences of

k

k

matrices

\{π_{1}, π_{2}, \dots\}

and

\{ψ_{1}, ψ_{2}, \dots\}

are such that

π_{j}

and

ψ_{j}

are approximately zero for

j > m

. An overestimate of

m

is therefore preferable to an under-estimate of

m

. In many instances the choice

m = 10

will suffice. In practice, to be on the safe side, you should try setting

m = 20

8.3 Checking a ‘White Noise’ Model

If you have fitted the ‘white noise’ model

W_{t} - μ = ε_{t}

then G13DSF should be entered with

p = 1

q = 0

, and the first

k^{2}

elements of PAR and PARHLD set to zero and .TRUE. respectively.

8.4 Approximate Standard Errors

When

IFAIL = 5

6

all the standard errors in RCM are set to

1 / \sqrt{n}

. This is the asymptotic standard error of

{\hat{r}}_{i j} (l)

when all the autoregressive and moving average parameters are assumed to be known rather than estimated.

8.5 Alternative Tests

{\hat{R}}_{0}

is useful in testing for instantaneous causality. If you wish to carry out a likelihood ratio test then the covariance matrix at lag zero

({\hat{C}}_{0})

can be used. It can be recovered from

{\hat{R}}_{0}

by setting

\begin{array}{l} {\hat{C}}_{0} (i, j) & = {\hat{R}}_{0} (i, j) \times {\hat{R}}_{0} (i, i) \times {\hat{R}}_{0} (j, j), & for ​ i \neq j \\ = {\hat{R}}_{0} (i, j) \times {\hat{R}}_{0} (i, j), & for ​ i = j \end{array}

9 Example

This example fits a bivariate AR(1) model to two series each of length

48

μ

has been estimated but

ϕ_{1} (2, 1)

has been constrained to be zero. Ten residual cross-correlation matrices are to be computed.

NAG Library Routine DocumentG13DSF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

8.1 Timing

8.2 Choice of m

8.3 Checking a ‘White Noise’ Model

8.4 Approximate Standard Errors

8.5 Alternative Tests

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine Document

G13DSF

8.2 Choice of $m$