naginterfaces.library.tsa.multi_​regmat_​partial

naginterfaces.library.tsa.multi_regmat_partial(z, m)

multi_regmat_partial calculates the sample partial autoregression matrices of a multivariate time series. A set of likelihood ratio statistics and their significance levels are also returned. These quantities are useful for determining whether the series follows an autoregressive model and, if so, of what order.

For full information please refer to the NAG Library document for g13dp

https://support.nag.com/numeric/nl/nagdoc_30/flhtml/g13/g13dpf.html

Parameters

zfloat, array-like, shape $(k,n)$

$z[\text{i}-1,\text{t}-1]$ must contain the observation $w_{\text{i}\text{t}}$, for $\text{t}=1,2,\dots ,n$, for $\text{i}=1,2,\dots ,k$.

mint

$m$, the number of partial autoregression matrices to be computed. If in doubt set $m=10$.

Returns

maxlagint

The maximum lag up to which partial autoregression matrices (along with their likelihood ratio statistics and their significance levels) have been successfully computed. On a successful exit $maxlag$ will equal $m$. If $errno$ = 2 on exit then $maxlag$ will be less than $m$.

parlagfloat, ndarray, shape $(k,k,m)$

$parlag[i-1,j-1,l-1]$ contains an estimate of the $(i,j)$th element of the partial autoregression matrix at lag $l$, ${\hat{P}}_l\left(ij\right)$, for $l=1,2,\dots ,maxlag$, $i=1,2,\dots ,k$ and $j=1,2,\dots ,k$.

sefloat, ndarray, shape $(k,k,m)$

$se[i-1,j-1,l-1]$ contains an estimate of the standard error of the corresponding element in the array $parlag$.

qqfloat, ndarray, shape $(k,k,m)$

$qq[\text{i}-1,\text{j}-1,\text{l}-1]$ contains an estimate of the $(\text{i},\text{j})$th element of the corresponding variance-covariance matrix ${\hat{\Sigma }}_{\text{l}}$, for $\text{j}=1,2,\dots ,k$, for $\text{i}=1,2,\dots ,k$, for $\text{l}=1,2,\dots ,maxlag$.

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

$x[\text{l}-1]$ contains $X_{\text{l}}$, the likelihood ratio statistic at lag $\text{l}$, for $\text{l}=1,2,\dots ,maxlag$.

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

$pvalue[l-1]$ contains the significance level of the statistic in the corresponding element of $x$.

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

$loglhd[\text{l}-1]$ contains an estimate of the maximum of the log-likelihood function when an $\text{AR}(\text{l}-1)$ model has been fitted to the series, for $\text{l}=1,2,\dots ,maxlag$.

Raises

NagValueError

(errno $1$)

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

Constraint: $n-m-(k\times m+1)\ge k$.

(errno $1$)

On entry, $m=⟨value⟩$.

Constraint: $m\ge 1$.

(errno $1$)

On entry, $n=⟨value⟩$.

Constraint: $n\ge 4$.

(errno $1$)

On entry, $k=⟨value⟩$.

Constraint: $k\ge 1$.

Warns

NagAlgorithmicWarning

(errno $2$)

The recursive equations used to compute the partial autoregression matrices are ill-conditioned. They have been computed up to lag $⟨value⟩$.

Notes

Let $W_{\text{t}}={(w_{1\text{t}},w_{2\text{t}},\dots ,w_{\text{k}\text{t}})}^T$, for $\text{t}=1,2,\dots ,n$, denote a vector of $k$ time series. The partial autoregression matrix at lag $l$, $P_l$, is defined to be the last matrix coefficient when a vector autoregressive model of order $l$ is fitted to the series. $P_l$ has the property that if $W_t$ follows a vector autoregressive model of order $p$ then $P_l=0$ for $l>p$.

Sample estimates of the partial autoregression matrices may be obtained by fitting autoregressive models of successively higher orders by multivariate least squares; see Tiao and Box (1981) and Wei (1990). These models are fitted using a $QR$ algorithm based on the functions correg.linregm_obs_edit and correg.linregm_var_del. They are calculated up to lag $m$, which is usually taken to be at most $n/4$.

The function also returns the asymptotic standard errors of the elements of ${\hat{P}}_l$ and an estimate of the residual variance-covariance matrix ${\hat{\Sigma }}_l$, for $l=1,2,\dots ,m$. If $S_l$ denotes the residual sum of squares and cross-products matrix after fitting an $\text{AR}\left(l\right)$ model to the series then under the null hypothesis $H_0:P_l=0$ the test statistic

$$X_l=-((n-m-1)-\frac{1}{2}-lk)\log \left(\frac{\left|S_l\right|}{\left|S_{l-1}\right|}\right)$$

is asymptotically distributed as ${\chi }^2$ with $k^2$ degrees of freedom. $X_l$ provides a useful diagnostic aid in determining the order of an autoregressive model. (Note that ${\hat{\Sigma }}_l=S_l/(n-l)$.) The function also returns an estimate of the maximum of the log-likelihood function for each AR model that has been fitted.

References

Tiao, G C and Box, G E P, 1981, Modelling multiple time series with applications, J. Am. Stat. Assoc. (76), 802–816

Wei, W W S, 1990, Time Series Analysis: Univariate and Multivariate Methods, Addison–Wesley