Notes 6
Model Specification
:
Overall strategy  Box Jenkins Approach:
1. To decide a reasonable  but tentative  values for
p, d, q
.
2. Estimate
φ, θ
and
σ
2
a
, for that model.
3. Check the model’s adequacy.
4. If the model appears inadequate, consider the nature of the inadequacy to select
another model.
5. Estimate the new model and check it for accuracy.
To obtain a tentative order
d
(based on the graphical approach)
A basic rule:
The first (and most important) step in fitting an ARIMA model is to determine the
value of
d
(i.e. the order of differencing). Normally, the correct amount of differencing is
the lowest order of differencing that yields a time series which fluctuates around a well
defined mean value and whose autocorrelation function (ACF) plot decays fairly rapidly
to zero. If the series still exhibits a longterm trend, or otherwise lacks a tendency to
return to its mean value, or if its autocorrelations are positive out to a high number of
lags, say 10 or more, then a higher order of differencing is needed.
Properties of the sample autocorrelation function
Theorem
: Suppose that
Z
t
=
μ
+
∞
X
j
=0
ψ
j
a
t

j
where
a
t
iid
∼
(0
, σ
2
a
)
,
0
< σ
2
a
<
∞
.
Assume
∑
∞
j
=0

ψ
j

<
∞
and
∑
∞
j
=0
ψ
2
j
<
∞
.
(This will be satisfied by any stationary
ARMA model).
Then, for any fixed
m
, the joint distribution of
√
n
(
r
1

ρ
1
)
,
√
n
(
r
2

ρ
2
)
, . . . ,
√
n
(
r
m

ρ
m
)
approaches, as
n
→ ∞
, a joint normal distribution with zero means, variances
c
ii
and
covariances
c
ij
where
r
k
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '08
 WU,KaHo
 Autocorrelation, Stationary process, Autoregressive moving average model, Time series analysis, Autoregressive model, Moving average model

Click to edit the document details