NAG Library Routine Document
G13ASF
1 Purpose
G13ASF is a diagnostic checking routine suitable for use after fitting a Box–Jenkins ARMA model to a univariate time series using
G13AEF or
G13AFF.
The residual autocorrelation function is returned along with an estimate of its asymptotic standard errors and correlations. Also, G13ASF calculates the Box–Ljung portmanteau statistic and its significance level for testing model adequacy.
2 Specification
SUBROUTINE G13ASF ( 
N, V, MR, M, PAR, NPAR, ISHOW, R, RCM, LDRCM, CHI, IDF, SIGLEV, IW, LIW, WORK, LWORK, IFAIL) 
INTEGER 
N, MR(7), M, NPAR, ISHOW, LDRCM, IDF, IW(LIW), LIW, LWORK, IFAIL 
REAL (KIND=nag_wp) 
V(N), PAR(NPAR), R(M), RCM(LDRCM,M), CHI, SIGLEV, WORK(LWORK) 

3 Description
Consider the univariate multiplicative autoregressivemoving average model
where
${W}_{t}$, for
$\mathit{t}=1,2,\dots ,n$, denotes a time series and
${\epsilon}_{t}$, for
$\mathit{t}=1,2,\dots ,n$, is a residual series assumed to be normally distributed with zero mean and variance
${\sigma}^{2}$ (
$\text{}>0$). The
${\epsilon}_{t}$'s are also assumed to be uncorrelated. Here
$\mu $ is the overall mean term,
$s$ is the seasonal period and
$B$ is the backward shift operator such that
${B}^{r}{W}_{t}={W}_{tr}$. The polynomials in
(1) are defined as follows:
is the nonseasonal autoregressive (AR) operator;
is the nonseasonal moving average (MA) operator;
is the seasonal AR operator; and
is the seasonal MA operator. The model
(1) is assumed to be stationary, that is the zeros of
$\varphi \left(B\right)$ and
$\Phi \left({B}^{s}\right)$ are assumed to lie outside the unit circle. The model
(1) is also assumed to be invertible, that is the zeros of
$\theta \left(B\right)$ and
$\Theta \left({B}^{s}\right)$ are assumed to lie outside the unit circle. When both
$\Phi \left({B}^{s}\right)$ and
$\Theta \left({B}^{s}\right)$ are absent from the model, that is when
$P=Q=0$, then the model is said to be nonseasonal.
The estimated residual autocorrelation coefficient at lag
$l$,
${\hat{r}}_{l}$, is computed as:
where
${\hat{\epsilon}}_{t}$ denotes an estimate of the
$t$th residual,
${\epsilon}_{t}$, and
$\stackrel{}{\epsilon}={\displaystyle \sum _{t=1}^{n}}{\hat{\epsilon}}_{t}/n$. A portmanteau statistic,
${Q}_{\left(m\right)}$, is calculated from the formula (see
Box and Ljung (1978)):
where
$m$ denotes the number of residual autocorrelations computed. (Advice on the choice of
$m$ is given in
Section 8.2.) Under the hypothesis of model adequacy,
${Q}_{\left(m\right)}$ has an asymptotic
${\chi}^{2}$distribution on
$mpqPQ$ degrees of freedom. Let
${\hat{r}}^{\mathrm{T}}=\left({\hat{r}}_{1},{\hat{r}}_{2},\dots ,{\hat{r}}_{m}\right)$ then the variancecovariance matrix of
$\hat{r}$ is given by:
The construction of the matrix
$X$ is discussed in
McLeod (1978). (Note that the mean,
$\mu $, and the residual variance,
${\sigma}^{2}$, play no part in calculating
$\mathrm{Var}\left(\hat{r}\right)$ and therefore are not required as input to G13ASF.)
Note: for additive models with fixed parameter values (i.e., fitted by
G13DDF)
G13DSF should be used instead of G13ASF.
4 References
Box G E P and Ljung G M (1978) On a measure of lack of fit in time series models Biometrika 65 297–303
McLeod A I (1978) On the distribution of the residual autocorrelations in Box–Jenkins models J. Roy. Statist. Soc. Ser. B 40 296–302
5 Parameters
 1: N – INTEGERInput
On entry:
$n$, the number of observations in the residual series.
If G13ASF is used following a call to
G13AEF, then
N must be the value
${\mathbf{ICOUNT}}\left(2\right)$ returned by
G13AEF.
If G13ASF is used following a call to
G13AFF, then
N must be the value
NRES returned by
G13AFF.
Constraint:
${\mathbf{N}}\ge 3$.
 2: V(N) – REAL (KIND=nag_wp) arrayInput
On entry:
${\mathbf{V}}\left(\mathit{t}\right)$ must contain an estimate of
${\epsilon}_{\mathit{t}}$, for
$\mathit{t}=1,2,\dots ,n$.
If G13ASF is used following a call to
G13AEF then the actual argument
V must be
${\mathbf{EXR}}\left({\mathbf{ICOUNT}}\left(1\right)+1\right)$ as returned by
G13AEF.
If G13ASF is used following a call to
G13AFF then the actual argument
V must be
RES as returned by
G13AFF.
Constraint:
${\mathbf{V}}$ must contain at least two distinct elements.
 3: MR($7$) – INTEGER arrayInput
On entry: the orders vector (
$p$,
$d$,
$q$,
$P$,
$D$,
$Q$,
$s$) as supplied to
G13AEF or
G13AFF.
Constraints:
 $p,q,P,Q,s\ge 0$;
 $p+q+P+Q>0$;
 if $s=0$, then $P=0$ and $Q=0$.
 4: M – INTEGERInput
On entry: the value of
$m$, the number of residual autocorrelations to be computed. See
Section 8.2 for advice on the value of
M.
Constraint:
${\mathbf{NPAR}}<{\mathbf{M}}<{\mathbf{N}}$.
 5: PAR(NPAR) – REAL (KIND=nag_wp) arrayInput
On entry: the parameter estimates in the order ${\varphi}_{1},{\varphi}_{2},\dots ,{\varphi}_{p}$, ${\theta}_{1},{\theta}_{2},\dots ,{\theta}_{q}$, ${\Phi}_{1},{\Phi}_{2},\dots ,{\Phi}_{P}$, ${\Theta}_{1},{\Theta}_{2},\dots ,{\Theta}_{Q}$ only.
Constraint:
the elements in
PAR must satisfy the stationarity and invertibility conditions.
 6: NPAR – INTEGERInput
On entry: the total number of $\varphi $, $\theta $, $\Phi $ and $\Theta $ parameters, i.e., ${\mathbf{NPAR}}=p+q+P+Q$.
Constraint:
${\mathbf{NPAR}}={\mathbf{MR}}\left(1\right)+{\mathbf{MR}}\left(3\right)+{\mathbf{MR}}\left(4\right)+{\mathbf{MR}}\left(6\right)$.
 7: ISHOW – INTEGERInput
On entry: must be nonzero if the residual autocorrelations, their standard errors and the portmanteau statistics are to be printed and zero otherwise.
These quantities are available also as output variables in
R,
RCM,
CHI,
IDF and
SIGLEV.
 8: R(M) – REAL (KIND=nag_wp) arrayOutput
On exit: an estimate of the residual autocorrelation coefficient at lag
$\mathit{l}$, for
$\mathit{l}=1,2,\dots ,m$. If
${\mathbf{IFAIL}}={\mathbf{3}}$ on exit then all elements of
R are set to zero.
 9: RCM(LDRCM,M) – REAL (KIND=nag_wp) arrayOutput
On exit: the estimated standard errors and correlations of the elements in the array
R. The correlation between
${\mathbf{R}}\left(i\right)$ and
${\mathbf{R}}\left(j\right)$ is returned as
${\mathbf{RCM}}\left(i,j\right)$ except that if
$i=j$ then
${\mathbf{RCM}}\left(i,j\right)$ contains the standard error of
${\mathbf{R}}\left(i\right)$. If on exit,
${\mathbf{IFAIL}}\ge {\mathbf{5}}$, then all offdiagonal elements of
RCM are set to zero and all diagonal elements are set to
$1/\sqrt{n}$.
 10: LDRCM – INTEGERInput
On entry: the first dimension of the array
RCM as declared in the (sub)program from which G13ASF is called.
Constraint:
${\mathbf{LDRCM}}\ge {\mathbf{M}}$.
 11: CHI – REAL (KIND=nag_wp)Output
On exit: the value of the portmanteau statistic,
${Q}_{\left(m\right)}$. If
${\mathbf{IFAIL}}={\mathbf{3}}$ on exit then
CHI is returned as zero.
 12: IDF – INTEGEROutput
On exit: the number of degrees of freedom of
CHI.
 13: SIGLEV – REAL (KIND=nag_wp)Output
On exit: the significance level of
CHI based on
IDF degrees of freedom. If
${\mathbf{IFAIL}}={\mathbf{3}}$ on exit,
SIGLEV is returned as one.
 14: IW(LIW) – INTEGER arrayWorkspace
 15: LIW – INTEGERInput
On entry: the dimension of the array
IW as declared in the (sub)program from which G13ASF is called.
Constraint:
${\mathbf{LIW}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{MR}}\left(1\right),{\mathbf{MR}}\left(3\right),{\mathbf{MR}}\left(4\right),{\mathbf{MR}}\left(6\right)\right)$.
 16: WORK(LWORK) – REAL (KIND=nag_wp) arrayWorkspace
 17: LWORK – INTEGERInput
On entry: the dimension of the array
WORK as declared in the (sub)program from which G13ASF is called.
Constraint:
${\mathbf{LWORK}}\ge {\mathbf{NPAR}}\times \left({\mathbf{M}}+{\mathbf{NPAR}}+1\right)+\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{MR}}\left(1\right),{\mathbf{MR}}\left(3\right),{\mathbf{MR}}\left(4\right),{\mathbf{MR}}\left(6\right)\right)\times \phantom{\rule{0ex}{0ex}}\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{MR}}\left(7\right),1\right)+{\mathbf{M}}$.
 18: IFAIL – INTEGERInput/Output

On entry:
IFAIL must be set to
$0$,
$1\text{ or}1$. 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
$1\text{ or}1$ 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
${\mathbf{IFAIL}}\ne {\mathbf{0}}$ on exit, the recommended value is
$1$.
When the value $\mathbf{1}\text{ or}1$ is used it is essential to test the value of IFAIL on exit.
On exit:
${\mathbf{IFAIL}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see
Section 6).
6 Error Indicators and Warnings
If on entry
${\mathbf{IFAIL}}={\mathbf{0}}$ or
${{\mathbf{1}}}$, explanatory error messages are output on the current error message unit (as defined by
X04AAF).
Note: G13ASF may return useful information for one or more of the following detected errors or warnings.
Errors or warnings detected by the routine:
 ${\mathbf{IFAIL}}=1$
On entry,  ${\mathbf{MR}}\left(1\right)<0$, 
or  ${\mathbf{MR}}\left(3\right)<0$, 
or  ${\mathbf{MR}}\left(4\right)<0$, 
or  ${\mathbf{MR}}\left(6\right)<0$, 
or  ${\mathbf{MR}}\left(7\right)<0$, 
or  ${\mathbf{MR}}\left(7\right)=0$ and either ${\mathbf{MR}}\left(4\right)>0$ or ${\mathbf{MR}}\left(6\right)>0$, 
or  ${\mathbf{MR}}\left(1\right)={\mathbf{MR}}\left(3\right)={\mathbf{MR}}\left(4\right)={\mathbf{MR}}\left(6\right)=0$, 
or  ${\mathbf{M}}\le {\mathbf{NPAR}}$, 
or  ${\mathbf{M}}\ge {\mathbf{N}}$, 
or  ${\mathbf{N}}<3$, 
or  ${\mathbf{NPAR}}\ne {\mathbf{MR}}\left(1\right)+{\mathbf{MR}}\left(3\right)+{\mathbf{MR}}\left(4\right)+{\mathbf{MR}}\left(6\right)$, 
or  ${\mathbf{LDRCM}}<{\mathbf{M}}$, 
or  LIW is too small, 
or  LWORK is too small. 
 ${\mathbf{IFAIL}}=2$
On entry, the autoregressive (or moving average) parameters are extremely close to or outside the stationarity (or invertibility) region. To proceed, you must supply different parameter estimates in the array
PAR.
 ${\mathbf{IFAIL}}=3$
On entry, the residuals are practically identical giving zero (or near zero) variance. In this case
CHI is set to zero and
SIGLEV to one and all the elements of
R are set to zero.
 ${\mathbf{IFAIL}}=4$
This is an unlikely exit brought about by an excessive number of iterations being needed to evaluate the zeros of the AR or MA polynomials. All output parameters are undefined.
 ${\mathbf{IFAIL}}=5$
On entry, one or more of the AR operators has a factor in common with one or more of the MA operators. To proceed, this common factor must be deleted from the model. In this case, the offdiagonal elements of
RCM are returned as zero and the diagonal elements set to
$1/\sqrt{n}$. All other output quantities will be correct.
 ${\mathbf{IFAIL}}=6$
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 offdiagonal 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.
The time taken by G13ASF depends upon the number of residual autocorrelations to be computed, $m$.
The number of residual autocorrelations to be computed,
$m$ should be chosen to ensure that when the ARMA model
(1) is written as either an infinite order autoregressive process:
or as an infinite order moving average process:
then the two sequences
$\left\{{\pi}_{1},{\pi}_{2},\dots \right\}$ and
$\left\{{\psi}_{1},{\psi}_{2},\dots \right\}$ are such that
${\pi}_{j}$ and
${\psi}_{j}$ are approximately zero for
$j>m$. An overestimate of
$m$ is therefore preferable to an underestimate of
$m$. In many instances the choice
$m=10$ will suffice. In practice, to be on the safe side, you should try setting
$m=20$.
When
${\mathbf{IFAIL}}={\mathbf{5}}$ or
${\mathbf{6}}$ all the standard errors in
RCM are set to
$1/\sqrt{n}$. This is the asymptotic standard error of
${\hat{r}}_{l}$ when all the autoregressive and moving average parameters are assumed to be known rather than estimated.
G13ASF may be used for diagnostic checking of suitable univariate ARMA models, as described in
Section 3, fitted by
G13BEF or
G13DDF.
Great care must be taken in obtaining the input values for G13ASF from the output values from
G13BEF or
G13DDF.
9 Example
This example fits an ARIMA$\left(1,1,2\right)$ model to a series of $30$ observations. $10$ residual autocorrelations are computed.
9.1 Program Text
Program Text (g13asfe.f90)
9.2 Program Data
Program Data (g13asfe.d)
9.3 Program Results
Program Results (g13asfe.r)