Estimation of ARMA processes - Department of Economics

55 Pages·2009·0.53 MB·English

Estimation of ARMA processes
Helle Bunzel
ISU
February 17, 2009
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The Box-Jenkins Principle
Historically (in the 1960(cid:146)s) econometric models with many variables
and equations were developed.
These models (cid:133)t the data beautifully.
They are not good at forecasting.
It turns out that low-dimensional ARMA(p,q) processes typically do a
better job with forecasting.
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The Box-Jenkins Principle
The Box-Jenkins Principle for forecasting:
1 Transform the data, if nessesary, so covariance stationarity is a
resonable assumption.
2 Make an initial guess at small values of p and q for an ARMA(p,q)
model that might describe the (transformed) series.
3 Estimate the parameters.
4 Perform diagnostic analysis to con(cid:133)rm that the model is consistent
with the observed features of the data.
For Step 2 we need information on the correlation structure.
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The Autocorrelation Function
Recall that the autocorrelation function is de(cid:133)ned as:
γ
ρ = s
s
V (y )V (y )
t+s t
This is what these look like: p
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The Autocorrelation Function
More examples
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The Partial Autocorrelation Function I
Clearly it is the case that it is hard to tell the processes apart from
the autocorrelation function alone.
Another important tool is the partial autocorrelation function.
De(cid:133)nition:
First de-mean y, such that yt(cid:3) =yt (cid:0)µ
Then the (cid:133)rst partial autocorrelation is φ such that
11
yt(cid:3) =φ11yt(cid:3) 1+ut
(cid:0)
The second partial autocorrelation is φ where
22
yt(cid:3) =φ21yt(cid:3) 1+φ22yt(cid:3) 2+ut
(cid:0) (cid:0)
Clearly for an AR(1), φ = 0.
22
These can be found as functions of the autocorrelations:
Note that the m th autocorrelation simply is the last coe¢ cient in the
0
linear regression of y µ on it(cid:146)s own m most recent values.
t
(cid:0)
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The Partial Autocorrelation Function II
Recall that we can write the coe¢ cients of the projection of y on
t+1
X as
t
1
α0 = E yt+1Xt0 E Xt0Xt (cid:0)
Here (cid:0) (cid:1)(cid:2) (cid:0) (cid:1)(cid:3)
X = [y µ,y µ,...,y µ]
t0 t 1 t 2 t m
(cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) ,
X = [y ,y ,...,y ]
t0 t(cid:3) 1 t(cid:3) 2 t(cid:3) m
(cid:0) (cid:0) (cid:0)
And we(cid:146)re projecting y , not y
t(cid:3) t+1.
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The Partial Autocorrelation Function III
Now note that
E y y E y y ... E y y
t(cid:3) 1 t(cid:3) 1 t(cid:3) 1 t(cid:3) 2 t(cid:3) 1 t(cid:3) m
E y (cid:0) y (cid:0) E y (cid:0) y (cid:0) . E y (cid:0) y (cid:0)
E Xt0Xt = 2 (cid:0) t(cid:3)(cid:0):2 t(cid:3)(cid:0)1(cid:1) (cid:0) t(cid:3)(cid:0):2 t(cid:3)(cid:0)2(cid:1) . (cid:0) t(cid:3)(cid:0):2 t(cid:3)(cid:0)m(cid:1) 3
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
(cid:0) (cid:1) 6 E y y E y y ... E (y y ) 7
6 t(cid:3) m t(cid:3) 1 t(cid:3) m t(cid:3) 2 t(cid:3) m t(cid:3) m 7
4 (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) 5
γ γ ... γ
(cid:0)0 1 (cid:1) (cid:0) m 1 (cid:1)
γ γ . γ (cid:0)
= 2 :1 :0 . m:(cid:0)2 3
6 γ γ ... γ 7
6 m 1 m 2 0 7
4 (cid:0) (cid:0) 5
and
E y X = E y y E y y ... E (y y )
t(cid:3) t0 t(cid:3) t(cid:3) 1 t(cid:3) t(cid:3) 2 t(cid:3) t(cid:3) m
(cid:0) (cid:0) (cid:0)
(cid:0) (cid:1) (cid:2) (cid:0) (cid:1) (cid:0) (cid:1) (cid:3)
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The Partial Autocorrelation Function IV
This means that we could (cid:133)nd the partial autocorrelations by taking
the m th entry of α where
0
1
γ γ ... γ (cid:0)
0 1 m 1
γ γ . γ (cid:0)
α0 = γ1 γ2 ... γm 2 :1 :0 . m:(cid:0)2 3
(cid:2) (cid:3)6 γ γ ... γ 7
6 m 1 m 2 0 7
4 (cid:0) (cid:0) 5
Note that we need to invert an m m matrix to get this one.....
(cid:2)
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The Partial Autocorrelation Function V
The partial autocorrelation of di⁄erent models look like:
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