Traditional Problem Definition - Computational Intelligence in Time Series Forecasting - page 47

For checking the mutual correlation of residuals the correlogram of residuals is
evaluated. The presence of spikes in the correlogram indicates that the residuals
might be correlated and that the model developed is not adequate.
The residual values 12 ZZ Z of an ARMA

process are obtained by substituting in the likelihood function all the estimated
values of Į and ȕ into each of the related time series y ( t ), y ( t+ 1), y ( t+ 2), ..., y ( t+n-
1) and by solving the resulting system of equations. This is generally a difficult
issue. It is much easier to extract the residual sequence for the AR( p ) and the
MA( q ) part of the ARMA( p , q ) process separately. For instance, in the case of an
AR( p ) model

z of the noise sequence zz , ,..., n , ,...,

01 n

p

Yt () PD [ ( Yt i ) Zt ()] ,

¦

i

i 1

for t = p +1, …, n , the residuals are

p

ˆ ˆ ˆ

zt () [() yt P ] ¦ D [( yt i ) P ] ,

i

i 1

whereby for t d p the residuals are not defined. In the case of an MA( q ) model

q

Yt () PE Zt i ( ) Zt () ,

¦

i

i 1

that can be rewritten as

q

Zt () [ () Yt PE ] ¦ Zt i ( ) ,

i

i 1

and the residuals are defined as

zy
z

P
PE

ˆ
ˆ

1 1

y Z

2 2 1 1

... ... ....

q

zy ¦ PE Z

t t i t i

i 1

where the last equality holds for t > q .

2.9 Forecasting Methods

Once the time series model has been built, it can be employed in forecasting the
future values using an adequate forecasting method. Viewed historically, the term