Time Series Analysis and Control Examples 
The autocovariance function of the
random variable Y_{t} is defined as

C_{YY}(k) = E(Y_{t+k} Y_{t})
where EY_{t} = 0.
When the real valued process Y_{t} is stationary
and its autocovariance is absolutely summable, the
population spectral density function is obtained using
the Fourier transform of the autocovariance function
where and C_{YY}(k) is the
autocovariance function such that
.Consider the autocovariance generating function
where C_{YY}(k) = C_{YY}(k) and z is a complex scalar.
The spectral density function can be represented as
The stationary ARMA(p,q) 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(p,q) process, .Therefore, the spectral density function of the stationary ARMA(p,q)
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 g=0
and its maximum at if , while the spectral
density function attains its maximum at g=0 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 differencestationary 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 calls compute the power spectrum
using AR coefficients and the white noise variance.
The power spectral density function of Y_{t} is derived using the
Fourier transformation of C_{YY}(k).
where and g denotes frequency.
The autocovariance function can also be written as
Consider the following stationary AR(p) process:
where is a white noise with mean zero and
constant variance .The autocovariance function of white noise
equals
where if k=0;
otherwise, . Therefore, the power spectral
density of the white noise is ,. Note that, with
,
Using the following autocovariance function of Y_{t},
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 Y_{t} 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.
Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.