ARMA process estimationΒΆ
From the order or a range of orders
,
where
and
, the methods aims to find the best model
that fits the data and estimate the
corresponding coefficients. The best model is considered with
respect to the
criteria (corrected Akaike Information
Criterion), defined by:
where is half the number of points of the time grid of the
process sample (if the data are a process sample) or in a block of the
time series (if the data are a time series).
Two other criteria are computed for each order :
the AIC criterion:
and the BIC criterion:
The BIC criterion leads to a model that gives a better prediction;
the AIC criterion selects the best model that fits the given data;
the criterion improves the previous one by penalizing a
too high order that would artificially fit to the data.
For each order
, the estimation of the coefficients
,
and the
variance
is done using the Whittle estimator which is
based on the maximization of the likelihood function in the frequency
domain.
The principle is detailed hereafter for the case of a time series: in
the case of a process sample, the estimator is similar except for the
periodogram which is computed differently.
We consider a time series associated to the time grid
and a particular order
.
The spectral density function of the
process writes:
(1)ΒΆ
where and
is the frequency value.
The Whittle log-likelihood writes:
(2)ΒΆ
where:
is the non parametric estimate of the spectral density, expressed in the Fourier space (frequencies in
instead of
). By default the Welch estimator is used.
is the
Fourier frequency,
,
with
the largest integer
.
We estimate the scalar coefficients by maximizing the
log-likelihood function. The corresponding equations lead to the
following relation:
(3)ΒΆ
where maximizes:
(4)ΒΆ
The Whitle estimation requires that:
the determinant of the eigenvalues of the companion matrix associated to the polynomial
are outside the unit disc,, which guarantees the stationarity of the process;
the determinant of the eigenvalues of the companion matrix associated to the polynomial
are outside the unit disc, which guarantees the invertibility of the process.
Multivariate estimationΒΆ
Let be a multivariate
time series of dimension
generated by an ARMA process
where
are supposed to
be known. We assume that the white noise
is
distributed according to the normal distribution with zero mean and
with covariance matrix
where
.
The normality of the white noise implies the normality of the process.
If we note
,
then
is normal with zero mean. Its covariance matrix
writes
which depends on the coefficients
for
and
and on the matrix
.
The likelihood of writes:
(5)ΒΆ
where
,
and where
denotes the determinant.
The difficulty arises from the great size () of
which is a dense matrix in the general case.
[mauricio1995] proposes an efficient algorithm to evaluate the likelihood
function. The main point is to use a change of variable that leads to a
block-diagonal sparse covariance matrix.
The multivariate Whittle estimation requires that:
the determinant of the eigenvalues of the companion matrix associated to the polynomial
are outside the unit disc, which guarantees the stationarity of the process;
the determinant of the eigenvalues of the companion matrix associated to the polynomial
are outside the unit disc, which guarantees the invertibility of the process.