EC3062 ECONOMETRICS IDENTIFICATION OF ARMA MODELS A stationary stochastic process ...

32 Pages·2010·0.22 MB·English

EC3062 ECONOMETRICS
IDENTIFICATION OF ARMA MODELS
A stationary stochastic process can be characterised, equivalently, by its
autocovariance function or its partial autocovariance function.
It can also be characterised by is spectral density function, which is
{ ± ± }
the Fourier transform of the autocovariances γ ; τ = 0, 1, 2, . . . :
τ
(cid:1)∞ (cid:1)∞
f(ω) = γ cos(ωτ) = γ + 2 γ cos(ωτ).
τ 0 τ
−∞
τ= τ=1
∈
Here, ω [0, π] is an angular velocity, or frequency value, in radians per
period.
The empirical counterpart of the spectral density function is the
periodogram I(ω ), which may be defined as
j
(cid:1)− (cid:1)−
T 1 T 1
1
I(ω ) = c cos(ω τ) = c + 2 c cos(ω τ),
j τ j 0 τ j
2
τ=1−T τ=1
where ω = 2πj/T; j = 0, 1, . . . , [T/2] are the Fourier frequencies and
j (cid:2)
{c ; τ = 0, ±1, . . . , ±(T − 1)}, with c = T−1 T−1(y − y¯)(y − y¯), are
−
τ τ t=τ t t τ
the empirical autocovariances.
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EC3062 ECONOMETRICS
The Periodogram and the Autocovariances
We need to show this definition of the peridogram is equivalent to the
previous definition, which was based on the following frequency decompo-
sition of the sample variance:
T(cid:1)−1 [(cid:1)T/2]
1 1
− 2 2 2
(y y¯) = (α + β ),
t j j
T 2
t=0 j=0
where
(cid:1) (cid:1)
2 2
−
α = y cos(ω t) = (y y¯) cos(ω t),
j t j t j
T T
t t
(cid:1) (cid:1)
2 2
−
β = y sin(ω t) = (y y¯) sin(ω t).
j t j t j
T T
t t
Substituting these into the term T(α2 + β2)/2 gives the periodogram
j j
(cid:3) (cid:6)
(cid:4) (cid:5) (cid:4) (cid:5)
(cid:1)− (cid:1)−
T 1 2 T 1 2
2
− −
I(ω ) = cos(ω t)(y y¯) + sin(ω t)(y y¯) .
j j t j t
T
t=0 t=0
2
EC3062 ECONOMETRICS
The quadratic terms may be expanded to give
(cid:4) (cid:5)
(cid:1) (cid:1)
2
− −
I(ω ) = cos(ω t) cos(ω s)(y y¯)(y y¯)
j j j t s
T
(cid:4)t s (cid:5)
(cid:1) (cid:1)
2
− −
+ sin(ω t) sin(ω s)(y y¯)(y y¯) ,
j j t s
T
t s
−
Since cos(A) cos(B) + sin(A) sin(B) = cos(A B), this can be written as
(cid:4) (cid:5)
(cid:1) (cid:1)
2
− − −
I(ω ) = cos(ω [t s])(y y¯)(y y¯)
j j t s
T
t s
(cid:2)
− − −
On defining τ = t s and writing c = (y y¯)(y y¯)/T, we can
−
τ t t t τ
reduce the latter expression to
(cid:1)−
T 1
I(ω ) = 2 cos(ω τ)c ,
j j τ
−
τ=1 T
which is a Fourier transform of the empirical autocovariances.
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EC3062 ECONOMETRICS
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7.5
5
2.5
0
0 π/4 π/2 3π/4 π
Figure 1. The spectral density function of an MA(2) process
y(t) = (1 + 1.250L + 0.800L2)ε(t) .
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EC3062 ECONOMETRICS
60
40
20
0
0 π/4 π/2 3π/4 π
Figure 2. The graph of a periodogram calculated from 160
observations on a simulated series generated by an MA(2) process
y(t) = (1 + 1.250L + 0.800L2)ε(t).
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EC3062 ECONOMETRICS
30
20
10
0
0 π/4 π/2 3π/4 π
Figure 3. The spectral density function of an AR(2) process
(1 − 0.273L + 0.810L2)y(t) = ε(t).
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EC3062 ECONOMETRICS
125
100
75
50
25
0
0 π/4 π/2 3π/4 π
Figure 4. The graph of a periodogram calculated from 160
observations on a simulated series generated by an AR(2) process
(1 − 0.273L + 0.810L2)y(t) = ε(t).
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EC3062 ECONOMETRICS
60
40
20
0
0 π/4 π/2 3π/4 π
Figure 5. The spectral density function of an ARMA(2, 1)
process (1 − 0.273L + 0.810L2)y(t) = (1 + 0.900L)ε(t).
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EC3062 ECONOMETRICS
100
75
50
25
0
0 π/4 π/2 3π/4 π
Figure 6. The graph of a periodogram calculated from 160
observations on a simulated series generated by an ARMA(2, 1)
process (1 − 0.273L + 0.810L2)y(t) = (1 + 0.900L)ε(t).
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EC3062 ECONOMETRICS
The Methodology of Box and Jenkins
Box and Jenkins proposed to use the autocorrelation and partial autocor-
relation functions for identifying the orders of ARMA models. They paid
little attention to the periodogram.
Autocorrelation function (ACF). Given a sample y , y , . . . , y of
−
0 1 T 1
{ }
T observations, the sample autocorrelation function r is the sequence
τ
r = c /c , τ = 0, 1, . . . ,
τ τ 0
(cid:2)
where c = T−1 (y −y¯)(y −y¯) is the empirical autocovariance at lag
−
τ t t τ
τ and c is the sample variance.
0
As the lag increases, the number of observations comprised in the
empirical autocovariances diminishes.
Partial autocorrelation function (PACF). The sample partial au-
{ }
tocorrelation function p gives the correlation between the two sets of
τ
residuals obtained from regressing the elements y and y on the set
−
t t τ
of intervening values y , y , . . . , y . The partial autocorrelation
− − −
t 1 t 2 t τ+1
measures the dependence between y and y after the effect of the in-
−
t t τ
tervening values has been removed.
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