3. ARMA Modeling

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3. ARMA Modeling
Now:
Important class of stationary processes
•
Definition 3.1: (ARMA(p, q) process)
2
Let (cid:143) WN(0, σ ) be a white noise process. The process
t t Z
{ } ∼
∈
X is called AutoRegressive-Moving-Average process of or-
t t Z
{ }
∈
ders (p, q) [in symbols: ARMA(p, q)], if it satisfies the stochastic
difference equation
X = c + φ X + . . . + φ X + (cid:143) + θ (cid:143) + . . . + θ (cid:143) ,
t 1 t 1 p t p t 1 t 1 q t q
− − − −
where c, φ1, . . . , φp, θ1, . . . , θq R, φp, θq = 0.
∈ (cid:54)
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Significance of ARMA processes:
Every arbitrary stationary process can be approximated by an
•
ARMA process as accurately as desired
For every autocovariance function γ with lim γ(h) = 0
h
•
→∞
and for every integer k N there exists an ARMA process
∈
X satisfying γ (h) = γ(h) for all h = 0, 1, . . . , k
t t Z X
{ }
∈
(approximability of arbitrary autocovariance functions by an
ARMA process)
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3.1 Lag Operator
Now:
Formal instrument for analyzing ARMA(p, q) processes
•
Definition 3.2: (Lag operator)
Let X be an arbitrary stochastic process. The lag operator
t t Z
{ }
∈
L shifts the time index of the process by one period back into
the past:
L X X .
t t 1
{ } ≡ { }
−
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Remarks and computing rules: (I)
Instead of L X = X we often simply write LX = X
t t 1 t t 1
• { } { }
− −
For the constant process Xt = c for all t Z and c R we
• ∈ ∈
have
LX = Lc = c
t
Applying L n times, we have
•
n
L . . . L X = L X = X
t t t n
−
n-mal
(cid:124) (cid:123)(cid:122) (cid:125)
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Remarks and computing rules: (II)
The reverse of the lag operator is the so-called lead operator
•
1
L that shifts the process 1 period ahead:
−
1
L X = X
− t t+1
1
Since L LX = X we define
− t t
•
0
L X 1X = X
t t t
≡
(”1” denotes the identity operator)
For arbitrary integers m, n Z we have
• ∈
m n m+n
L L X = L X = X
t t t m n
− −
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Remarks and computing rules: (III)
For any arbitrary real numbers a, b R, arbitrary integers
• ∈
m, n Z and arbitrary stochastic processes Xt , Yt we have
∈ { } { }
m n
(aL + bL ) (X + Y ) = aX + bX + aY + bY
t t t m t n t m t n
− − − −
polynomials in the lag operator L
−→
Examples of polynomials in L: (I)
2 p
A(L) = a + a L + a L + . . . + a L
0 1 2 p
•
(with constants a0, a1, . . . , ap R)
∈
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Examples of polynomials in L: (II)
2
A(L) = 1 0.5L and B(L) = 1 + 4L
• −
It follows that
2
C(L) = A(L)B(L) = (1 0.5L)(1 + 4L )
−
2 3
= 1 0.5L + 4L 2L
− −
(the usual computing rules for polynomials apply)
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Examples of polynomials in L: (III)
Defining the polynomials
•
p
Φ(L) = 1 φ L . . . φ L
1 p
− − −
q
Θ(L) = 1 + θ L + . . . + θ L ,
1 q
we may represent the ARMA(p, q) process from Definition
3.1 (cf. Slide 42) as
Φ(L)X = c + Θ(L)(cid:143)
t t
In some situations it turns out to be helpful to consider the
•
lag polynomials Φ and Θ as polynomials in the complex-
valued argument z C
∈
(i.e. Φ(z) and Θ(z))
This mathematical device is known as the z-transformation
•
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3.2 Special and Limiting Cases
Now:
Two special ARMA(p, q) processes and one limiting process
•
3.2.1 MA(q) process
Starting point:
We consider the ARMA(0, q) process
•
X = c + (cid:143) + θ (cid:143) + . . . + θ (cid:143)
t t 1 t 1 q t q
− −
2
with (cid:143) WN(0, σ )
t t Z
{ } ∼
∈
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Remarks:
This process is called Moving-Average process of order q [in
•
symbols: MA(q)]
q
Defining the lag polynomial Θ(L) = 1 + θ L + . . . + θ L , we
1 q
•
write the MA(q) process as
X = c + Θ(L)(cid:143)
t t
Often, we choose c = 0:
•
X = (cid:143) + θ (cid:143) + . . . + θ (cid:143) = Θ(L)(cid:143)
t t 1 t 1 q t q t
− −
Question:
Is the MA(q) process (weakly) stationary?
•
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