ARMA Autocorrelation Functions • For a moving average process

21 Pages·2010·0.07 MB·English

ARMA Autocorrelation Functions
• For a moving average process, MA(q):
x = w + θ w + θ w + · · · + θ w .
t t 1 t−1 2 t−2 q t−q
• So (with θ = 1)
0
(cid:16) (cid:17)
γ(h) = cov x , x
t+h t
   
q q
(cid:88) (cid:88)
= E θ w θ w
 j t+h−j  k t−k
j=0 k=0

q−h

 2 (cid:88)
σ θ θ , 0 ≤ h ≤ q

w j j+h
=
j=0


0 h > q.
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• So the ACF is

q−h

 (cid:88)
 θ θ

 j j+h


j=0

 , 0 ≤ h ≤ q
q
ρ(h) =
(cid:88) 2
θ

 j


 j=0



0 h > q.
• Notes:
– In these expressions, θ = 1 for convenience.
0
– γ(q) (cid:54)= 0 but γ(h) = 0 for h > q. This characterizes
MA(q).
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• For an autoregressive process, AR(p):
x = φ x + φ x + · · · + φ x + w .
t 1 t−1 2 t−2 p t−p t
• So
(cid:16) (cid:17)
γ(h) = cov x , x
t+h t
  
p
(cid:88)
= E φ x + w x
 j t+h−j t+h t
j=1
p
(cid:88) (cid:16) (cid:17)
= φ γ(h − j) + cov w , x .
j t+h t
j=1
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• Because x is causal, x is w + a linear combination of w , w , . . . .
t t t t−1 t−2
• So

2
(cid:16) (cid:17) σ h = 0
w
cov w , x =
t+h t
0 h > 0.

• Hence
p
(cid:88)
γ(h) = φ γ(h − j), h > 0
j
j=1
and
p
(cid:88) 2
γ(0) = φ γ(−j) + σ .
j w
j=1
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2
• If we know the parameters φ , φ , . . . , φ and σ , these equa-
1 2 p w
tions for h = 0 and h = 1, 2, . . . , p form p + 1 linear equations
in the p + 1 unknowns γ(0), γ(1), . . . , γ(p).
• The other autocovariances can then be found recursively
from the equation for h > p.
• Alternatively, if we know (or have estimated) γ(0), γ(1), . . . , γ(p),
they form p + 1 linear equations in the p + 1 parameters
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φ , φ , . . . , φ and σ .
1 2 p w
• These are the Yule-Walker equations.
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• For the ARMA(p, q) model with p > 0 and q > 0:
x =φ x + φ x + · · · + φ x
t 1 t−1 2 t−2 p t−p
+ w + θ w + θ w + · · · + θ w ,
t 1 t−1 2 t−2 q t−q
a generalized set of Yule-Walker equations must be used.
• The moving average models ARMA(0, q) = MA(q) are the
only ones with a closed form expression for γ(h).
• For AR(p) and ARMA(p, q) with p > 0, the recursive equation
means that for h > max(p, q + 1), γ(h) is a sum of geomet-
rically decaying terms, possibly damped oscillations.
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• The recursive equation is
p
(cid:88)
γ(h) = φ γ(h − j), h > q.
j
j=1
• What kinds of sequences satisfy an equation like this?
−h
– Try γ(h) = z for some constant z.
– The equation becomes
 
p p
−h (cid:88) −(h−j) −h (cid:88) j −h
0 = z − φ z = z 1 − φ z = z φ(z).
j  j 
j=1 j=1
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−h
• So if φ(z) = 0, then γ(h) = z satisfies the equation.
• Since φ(z) is a polynomial of degree p, there are p solutions,
say z , z , . . . , z .
1 2 p
• So a more general solution is
p
(cid:88) −h
γ(h) = c z ,
l l
l=1
for any constants c , c , . . . , c .
1 2 p
• If z , z , . . . , z are distinct, this is the most general solution;
1 2 p
if some roots are repeated, the general form is a little more
complicated.
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• If all z , z , . . . , z are real, this is a sum of geometrically
1 2 p
decaying terms.
• If any root is complex, its complex conjugate must also be a
root, and these two terms may be combined into geometri-
cally decaying sine-cosine terms.
• The constants c , c , . . . , c are determined by initial condi-
1 2 p
tions; in the ARMA case, these are the Yule-Walker equa-
tions.
• Note that the various rates of decay are the zeros of φ(z),
the autoregressive operator, and do not depend on θ(z), the
moving average operator.
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• Example: ARMA(1, 1)
x = φx + θw + w .
t t−1 t−1 t
• The recursion is
γ(h) = φγ(h − 1), h = 2, 3, . . .
h
• So γ(h) = cφ for h = 1, 2, . . . , but c (cid:54)= 1.
• Graphically, the ACF decays geometrically, but with a differ-
ent value at h = 0.
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