Time Series Analysis and Examples |
The autocovariance function of the
random variable is defined as
where
.
When the real valued process
is stationary
and its autocovariance is absolutely summable, the
population spectral density function is obtained by using
the Fourier transform of the autocovariance function
where
and
is the
autocovariance function such that
.
Consider the autocovariance generating function
where
and
is a complex scalar.
The spectral density function can be represented as
The stationary ARMA(
) process is denoted
where
and
do not have common roots.
Note that the autocovariance generating function of the
linear process
is given by
For the ARMA(
) process,
.
Therefore, the spectral density function of the stationary ARMA(
)
process becomes
The spectral density function of a white noise is a constant.
The spectral density function of the AR(1) process
is given by
The spectrum of the AR(1) process has its minimum at
and its maximum at
if
, while the spectral
density function attains its maximum at
and its minimum at
, if
. When the series is positively
autocorrelated, its spectral density function is dominated by low
frequencies. It is interesting to observe that the spectrum approaches
as
.
This relationship shows that the series is difference-stationary if
its spectral density function has a remarkable peak near 0.
The spectrum of AR(2) process equals
Refer to Anderson (1971) for details of the characteristics of
this spectral density function of the AR(2) process.
In practice, the population spectral density function cannot
be computed. There are many ways of computing the sample
spectral density function.
The TSBAYSEA and TSMLOCAR subroutines compute the power spectrum
by using AR coefficients and the white noise variance.
The power spectral density function of is derived by using the
Fourier transformation of .
where
and
denotes frequency.
The autocovariance function can also be written as
Consider the following stationary AR(
) process:
where
is a white noise with mean zero and
constant variance
.
The autocovariance function of white noise
equals
where
if
;
otherwise,
. Therefore, the power spectral
density of the white noise is
,
. Note that, with
,
Using the following autocovariance function of
,
the autocovariance function of the white noise is denoted as
On the other hand, another formula of the
gives
Therefore,
Since
,
the rational spectrum of
is
To compute the power spectrum, estimated values of white noise variance
and AR coefficients
are used. The order of the AR process can be determined by using
the minimum AIC procedure.