Stationarity and/or Invertibility of time series (TS) are usually verified
via the roots of the polynomial derived from the transfer
operators.
In particular, checkTS.VGAMextra computes such roots via
the coefficients estimated by vector generalized TS family functions
available in VGAMextra ( ARXff, and
MAXff).
Specifically, checkTS.VGAMextra verifies whether the TS
analyzed via vglm is stationary or
invertible, accordingly.
Note that an autoregressive process of order-\(p\)
[AR(\(p\))] with coefficients
\(\theta_{1}, \ldots, \theta_{p}\)
can be written in the form
$$ \theta(B) Y_{t} = \varepsilon_{t}, $$
where
$$ \theta(B) = 1 - \sum_{k = 1}^{p} \theta_{k} B^{k} $$
Here, \( \theta(B) \) is referred to as
the transfer operator of the process, and
\(B^{k} Y_{t} = Y_{t - k},\),
for \(k = 0, 1, \ldots,p\), is the lagged single-function.
In general, an autoregressive process of order-\(p\) is
stationary if the roots of
$$ \theta(z) = 1 - \theta_{1} z - \ldots - \theta_{q} z^q $$
lie outside the unit circle, i.e. \(|z| > 1\).
Similarly, a moving-average process of order-q can be formulated
(without loss of generality \(\mu = 0\))
$$ Y_{t} = \psi(B) \varepsilon_{t}, $$
where \( \psi(B) \) is the transfer operator, given by
$$ \psi(B) = 1 + \sum_{k = 1}^{q} \psi_{k} B^{k}, $$
Note that \( \psi_{0} = 1 \), and
\(B^{k} \varepsilon_{t} = \varepsilon{t - k} \).
Hence, a moving-average process of order-\(q\) [MA(\(q\))],
generally given by (note \(\mu = 0\))
$$Y_{t} = \phi_1 \varepsilon_{t - 1} + \ldots +
\phi_q \varepsilon_{t - q} + \varepsilon_{t},$$
is invertible if all the roots of
$$ \phi(B) = 1 + \phi_{1} B + \ldots + \phi_{q} B^q $$
lie outside the unit circle., i.e.m \(|z| > 1\).
Parallel arguments can be stated for autoregressive moving
aberage processes (ARMA). See Box and Jenkins (1970) for
further details.