1
ARIMA
The ARIMA procedure computes the parameter estimates for a given seasonal or
non-seasonal univariate ARIMA model. It also computes the fitted values,
forecasting values, and other related variables for the model.
Notation
The following notation is used throughout this chapter unless otherwise stated:
y
t
(t=1, 2, ..., N) Univariate time series under investigation
N Total number of observations
a
t
(t = 1, 2, ... , N) White noise series normally distributed with mean zero and
variance
σ
a
2
p
Order of the non-seasonal autoregressive part of the model
q
Order of the non-seasonal moving average part of the model
d
Order of the non-seasonal differencing
P
Order of the seasonal autoregressive part of the model
Q
Order of the seasonal moving-average part of the model
D
Order of the seasonal differencing
s
Seasonality or period of the model
φ
p
B()
AR polynomial of B of order p,
φϕϕϕ
pp
p
BBBB( ) ...=− − −−1
12
2
θ
q
B()
MA polynomial of B of order q,
θϑϑϑ
qq
q
BBBB( ) ...=− − −−1
12
2
Φ
P
B()
SAR polynomial of B of order P,
ΦΦΦΦ
PP
P
BBBB( ) ...=− − −−1
12
2
Θ
Q
B()
SMA polynomial of B of order Q,
ΘΘΘΘ
QQ
Q
BBB B( ) ...=− − −−1
12
2
∇ Non-seasonal differencing operator ∇= −1 B
∇
s
Seasonal differencing operator with seasonality s,
∇=−
s
s
B1
B
Backward shift operator with
By y
tt
=
−1
and Ba a
tt
=
−1
评论7