Overall, the autoregressive moving average process of order \(c(p, q)\),
shortly denoted as ARMA(\(p\), \(q\)), with intercept
\(\mu\) can be expressed as
$$y_{t} = \mu + \theta_{1} y_{t - 1} + \ldots + \theta_{p} y_{t - p} +
\phi_1 \varepsilon_{t - 1} + \ldots +
\phi_q \varepsilon_{t - q} + \varepsilon_{t}.$$
It is well known that it can be expressed in terms of an autoregressive
process of infinite order, AR(\(\infty\)), by
recursive substitutions. For instance, given a mean-zero ARMA(1, 1),
$$y_{t} = \theta_1 y_{t - 1} + \phi_1 \varepsilon_{t - 1} +
\varepsilon_{t},
\quad \quad (1) $$
one may express
$$\varepsilon_{t - 1} = Y_{t - 1} - ( \theta_{1} y_{t - 2} +
\phi_{1} \varepsilon_{t - 2} $$
Substituting this equation in (1) yields the initial inverted process,
as follows:
$$ y_{t} = \psi_{1} y_{t - 1} + \psi_{2} y_{t - 2} +
f(\varepsilon_{t - 2}, \varepsilon_{t} ). $$
where \(f\) is a function of \(\varepsilon_{t - 2}\) and
\(\varepsilon_{t}\).
Repeated substitutions as above produces the so-called inverted
process,
$$ y_{t} = \sum_{k = 1}^{\infty} \psi_{k} y_{t - k} +
\varepsilon_{t}. \quad \quad (2) $$
\( k = 1, \ldots, \infty\).
Hence, setting an acceptable order (via the moreOrder
argument, \(1\) or \(2\) for instance), an
AR(\(p\) + moreOrd)
inverted model is internally fitted
within WN.InitARMA. Consequently, the unobserved white noise,
\( \{ \varepsilon_{t} \} \), is estimated by computing
the residuals in (2), after regression.
whiteN = TRUE enables this option.
Finally, initial values of the MAXff, and
ARMAXff family functions can be computed by least squares from
the estimated white noise above, \( \{ \varepsilon_{t} \} \)
and the given data, \( \{ t_{t} \} \).
Initial values of ARXff
are also internally computed using \( \{ t_{t} \} \) only.