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combinations of the above processes with a degree of homogeneity greater
than zero. An ARIMA process of order (p,d,q), where p, d, and q are the
order of the AR, difference, and MA components respectively, is given as
p d q
I *.(1-B) x R + EU. ,
i=0 1 t_1 j=0 1
%
For d0, this can be expressed as
Xt *lVl Vt-2 Vt-p E + 5t \\-l \ Vl '
In backshift notation, this is written as
(l jB ^B2 ... xBP)xt R + (l ^B B2B2 ... -
Finally, the above expressions, in differenced form, appear as
4>(B) xt = R + 8(BKt
where <(>(B) and B(B) are converging invertible polynomials in the lag
operator B. Since xfc has been differenced (is now homogeneous station
ary), the process can be modeled using an AR of order p and an MA of
order q. Thus, is an integrated (I) ARMA, or an ARIMA (p,d,q) pro
cess.
Identification and Estimation of an ARIMA Model
The discussion above has shown that a homogenous nonstationary time
series can be described as an ARIMA process of order p, d, and q.
However, the correct specification of an ARIMA process necessitates
selecting the proper values of p, d, and q to accurately describe the
underlying stochastic process that generated the original time series.
This task is accomplished by examining the autocorrelation function and
partial autocorrelation function of the time series.