naginterfaces.library.tsa.uni_​arima_​resid

naginterfaces.library.tsa.uni_arima_resid(v, mr, m, par, ishow, io_manager=None)

uni_arima_resid is a diagnostic checking function suitable for use after fitting a Box–Jenkins ARMA model to a univariate time series using uni_arima_estim() or uni_arima_estim_easy(). The residual autocorrelation function is returned along with an estimate of its asymptotic standard errors and correlations. Also, uni_arima_resid calculates the Box–Ljung portmanteau statistic and its significance level for testing model adequacy.

For full information please refer to the NAG Library document for g13as

https://www.nag.com/numeric/nl/nagdoc_29.3/flhtml/g13/g13asf.html

Parameters

vfloat, array-like, shape $\left(n\right)$

$v[\text{t}-1]$ must contain an estimate of ${ϵ}_{\text{t}}$, for $\text{t}=1,2,\dots ,n$.

If uni_arima_resid is used following a call to uni_arima_estim() then the actual argument $v$ must be $\text{exr}\left[\text{icount}\right[0\left]\right]$ as returned by uni_arima_estim().

If uni_arima_resid is used following a call to uni_arima_estim_easy() then the actual argument $v$ must be $\text{res}$ as returned by uni_arima_estim_easy().

mrint, array-like, shape $\left(7\right)$

The orders vector ($p$, $d$, $q$, $P$, $D$, $Q$, $s$) as supplied to uni_arima_estim() or uni_arima_estim_easy().

mint

The value of $m$, the number of residual autocorrelations to be computed. See Further Comments for advice on the value of $m$.

parfloat, array-like, shape $\left(\text{npar}\right)$

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.

ishowint

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 $siglev$.

io_managerFileObjManager, optional

Manager for I/O in this routine.

Returns

rfloat, ndarray, shape $\left(m\right)$

An estimate of the residual autocorrelation coefficient at lag $\text{l}$, for $\text{l}=1,2,\dots ,m$. If $errno$ = 3 on exit then all elements of $r$ are set to zero.

rcmfloat, ndarray, shape $(m,m)$

The estimated standard errors and correlations of the elements in the array $r$. The correlation between $r[i-1]$ and $r[j-1]$ is returned as $rcm[i-1,j-1]$ except that if $i=j$ then $rcm[i-1,j-1]$ contains the standard error of $r[i-1]$. If on exit, $errno$ >= 5, then all off-diagonal elements of $rcm$ are set to zero and all diagonal elements are set to $1/\sqrt{n}$.

chifloat

The value of the portmanteau statistic, $Q_{\left(m\right)}$. If $errno$ = 3 on exit then $chi$ is returned as zero.

idfint

The number of degrees of freedom of $chi$.

siglevfloat

The significance level of $chi$ based on $idf$ degrees of freedom. If $errno$ = 3 on exit, $siglev$ is returned as one.

Raises

NagValueError

(errno $1$)

On entry, $m=⟨value⟩$ and $n=⟨value⟩$.

Constraint: $m<n$.

(errno $1$)

On entry, $m=⟨value⟩$ and $\text{npar}=⟨value⟩$.

Constraint: $m>\text{npar}$.

(errno $1$)

On entry, $\text{npar}=⟨value⟩$.

Constraint: $\text{npar}\ne 0$ and $\text{npar}=p+q+P+Q$.

(errno $1$)

On entry, $mr\left[3\right]=⟨value⟩$ and $mr\left[5\right]=⟨value⟩$.

Constraint: $mr\left[3\right]=0$ and $mr\left[5\right]=0$.

(errno $1$)

On entry, $mr\left[6\right]=⟨value⟩$.

Constraint: $mr\left[6\right]\ge 0$.

(errno $1$)

On entry, $mr\left[5\right]=⟨value⟩$.

Constraint: $mr\left[5\right]\ge 0$.

(errno $1$)

On entry, $mr\left[3\right]=⟨value⟩$.

Constraint: $mr\left[3\right]\ge 0$.

(errno $1$)

On entry, $mr\left[2\right]=⟨value⟩$.

Constraint: $mr\left[2\right]\ge 0$.

(errno $1$)

On entry, $mr\left[0\right]=⟨value⟩$.

Constraint: $mr\left[0\right]\ge 0$.

(errno $2$)

On entry, the seasonal moving average operator is non-invertible.

(errno $2$)

On entry, the seasonal autoregressive operator is nonstationary.

(errno $2$)

On entry, the non-seasonal moving average operator is non-invertible.

(errno $2$)

On entry, the non-seasonal autoregressive operator is nonstationary.

(errno $4$)

Excessive iterations needed to find zeros of determinental polynomials.

(errno $5$)

On entry, one or more of the AR operators has a factor in common with one or more of the MA operators.

(errno $6$)

The matrix $rcm$ could not be computed because one of its diagonal elements was found to be non-positive.

Warns

NagAlgorithmicWarning

(errno $3$)

On entry, the elements of $v$ are nearly identical giving near-zero variance.

Notes

In the NAG Library the traditional C interface for this routine uses a different algorithmic base. Please contact NAG if you have any questions about compatibility.

Consider the univariate multiplicative autoregressive-moving average model

$$\varphi \left(B\right)\Phi \left(B^s\right)(W_t-\mu )=\theta \left(B\right)\Theta \left(B^s\right){ϵ}_t$$

where $W_t$, for $\text{t}=1,2,\dots ,n$, denotes a time series and ${ϵ}_t$, for $\text{t}=1,2,\dots ,n$, is a residual series assumed to be normally distributed with zero mean and variance ${\sigma }^2$ ($>0$). The ${ϵ}_t$’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 $B^r{W}_t=W_{t-r}$. The polynomials in (1) are defined as follows:

$$\varphi \left(B\right)=1-{\varphi }_{1}B-{\varphi }_2{B}^2-\cdots -{\varphi }_p{B}^p$$

is the non-seasonal autoregressive (AR) operator;

$$\theta \left(B\right)=1-{\theta }_{1}B-{\theta }_2{B}^2-\cdots -{\theta }_q{B}^q$$

is the non-seasonal moving average (MA) operator;

$$\Phi \left(B^s\right)=1-{\Phi }_1{B}^s-{\Phi }_2{B}^{2s}-\cdots -{\Phi }_P{B}^{Ps}$$

is the seasonal AR operator; and

$$\Theta \left(B^s\right)=1-{\Theta }_1{B}^s-{\Theta }_2{B}^{2s}-\cdots -{\Theta }_Q{B}^{Qs}$$

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 non-seasonal.

The estimated residual autocorrelation coefficient at lag $l$, ${\hat{r}}_l$, is computed as:

$${\hat{r}}_l=\frac{{\sum }_{t=l+1}^n({\widehat{ϵ}}_{t-l}-\overline{ϵ})({\widehat{ϵ}}_t-\overline{ϵ})}{{\sum }_{t=1}^n{({\widehat{ϵ}}_t-\overline{ϵ})}^2}\text{,}l=1,2,\dots$$

where ${\widehat{ϵ}}_t$ denotes an estimate of the $t$th residual, ${ϵ}_t$, and $\overline{ϵ}={\sum }_{t=1}^n{\widehat{ϵ}}_t/n$. A portmanteau statistic, $Q_{\left(m\right)}$, is calculated from the formula (see Box and Ljung (1978)):

$$Q_{\left(m\right)}=n(n+2)\sum _{l=1}^m{\hat{r}}_l^2/(n-l)$$

where $m$ denotes the number of residual autocorrelations computed. (Advice on the choice of $m$ is given in Further Comments.) Under the hypothesis of model adequacy, $Q_{\left(m\right)}$ has an asymptotic ${\chi }^2$-distribution on $m-p-q-P-Q$ degrees of freedom. Let ${\hat{r}}^T=({\hat{r}}_1,{\hat{r}}_2,\dots ,{\hat{r}}_m)$ then the variance-covariance matrix of $\hat{r}$ is given by:

$$Var\left(\hat{r}\right)=[I_m-X{\left(X^{T}X\right)}^{-1}{X}^T]/n\text{.}$$

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 $Var\left(\hat{r}\right)$ and, therefore, are not required as input to uni_arima_resid.)

Note: for additive models with fixed parameter values (i.e., fitted by multi_varma_estimate()) multi_varma_diag() should be used instead of uni_arima_resid.

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