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H5
The
YuleWalker equations:
AR(1)
:
()
( 1
)
,
0
ρ
φρ
=−
>
hhh
,
with initial condition
(0)
1
=
.
AR(2)
:
12
)
(
2
)
,
1
+−
hh
h
h
>
, with initial conditions
1
2
(1)
1
φ
=
−
,
2
1
2
2
(2)
1
=
+
−
.
Example 1
Suppose for a stationary AR(2) model,
112 2
,
.5,
(2)
.1
tt
t
t
XX
XZ
−−
=
++
=
=
,
Determine
2
.
In general, solutions or the
YuleWalker equations
:
1
( )
(
1)
...
(
)
0
φ ρ
=
−++
− =
p
h
p
, have the form

1
1
p
p
hA
A
ππ
=+
⋅
⋅
⋅
+
.
It turns put that the condition,
1
i
π
<
for all
i
,
is equivalent to that of causality,
h
tends to zero as
increases.
h
In contrast to the correlation function the
Partial Correlation Function
(PACF),
h
α
, starts from lag 1,
=
.
ACF
h
for
MA
processes cuts off at
( )
q
hq
=
, i.e.,
() 0
,
q
=
>
,
PACF
h
for
processes cuts off at
AR( )
q
hp
=
, i.e.,
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This note was uploaded on 09/23/2009 for the course MATH 200 taught by Professor Gur during the Spring '09 term at Accreditation Commission for Acupuncture and Oriental Medicine.
 Spring '09
 gUR
 Equations

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