Notes 8
Diagnostics Checking
:
Recall
Z
t
=
∞
j
=1
π
j
Z
t

j
+
a
t
Put
ˆ
a
t
=
Z
t

∞
j
=1
ˆ
π
j
Z
t

j
Residual = actual value  estimated value
Residual Analysis
1. Plot residuals ˆ
a
t
against
t
(See whether trend exists)
2. Histogram of ˆ
a
t
(or standardized residuals), normalscore correlation test (Check
normality)
3. Plot ˆ
a
t
against
ˆ
Z
t
(check constant variance)
4. Check autocorrelation of residuals.
We consider the sample autocorrelation of the residuals,
{
ˆ
r
k
}
.
If the residuals
follows a white noise process, then the sample acf are approximately uncorrelated
and each
{
ˆ
r
k
}
is distributed approximately as Normal with mean 0 and variance 1
/n
,
where
n
is the series length. However, residuals, even with a corrected specified model
with efficiently estimated parameters, have different properties.
Properties of the sample acf of residuals for some models
For AR(1) model, for large
n
V ar
(ˆ
r
1
)
≈
φ
2
n
V ar
(ˆ
r
k
)
≈
1

(1

φ
2
)
φ
2(
k

1)
n
, k >
1
.
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 Spring '08
 ?
 Normal Distribution, Chi Square Distribution, general ARMA model

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