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Unformatted text preview: B p
should be q, too CIE/IE 500 Transportation Analytics  Fall 2012 17 Parameter Redundancy
Consider a white noise process xt = wt. Equivalently, we can
write this as .5xt−1 = .5wt−1 by shiZing back one unit of [me and
multiplying by .5. Now, subtract the two representations to
obtain
which looks like an ARMA(1, 1) model. Of course, xt is still white
noise; nothing has changed in this regard. But we have hidden
the fact that xt is white noise because of the parameter
redundancy or overparameterization.
Write the parameter redundant model in operator form as
φ(B)xt = θ(B)wt, or
Apply the operator φ(B)−1 = (1 − .5B)−1 to both sides to obtain If we were unaware of parameter redundancy, we might claim the data
are correlated when in fact they are not
CIE/IE 500 Transportation Analytics  Fall 2012 18 Problems in ARMA(p,q)
(i) Parameter redundant models,
cancel common factors to its simplest operator form (ii) Stationary AR models that depend on the future, and
Causality: if AR models are causal, they don’t depend on the future (iii) MA models that are not unique.
Invertibility: if MA models are invertible, they are unique
Definition: The AR and MA polynomials are defined as and respectively, where zCIE/IE 500 Transportation Analytics  Fall 2012
is a complex number. 19 Causality and Invertibility of ARMA(p,q)
The ARMA(p,q) model given by φ(B)xt = θ(B)wt is causal if and
only if
φ(z)= 0 only when z > 1 i.e. all roots including complex roots of φ(z)=0 lie outside the
unit circle.
The ARMA(p,q) model given by φ(B)xt = θ(B)wt is invertible if
and only if
θ(z)= 0 only when z > 1 i.e. all roots including complex roots of θ(z)=0 lie outside the
unit circle. CIE/IE 500 Transportation Analytics  Fall 2012 20 ExampleParameter Redundancy, Causality, and
Invertibility
Consider the process
or, in operator form,
At first, xt appears to be an ARMA(2, 2) process. Is it true? NO.
The associated polynomials have a common factor that can be canceled. After cancellation,
the polynomials become φ(z) = (1 − .9z) and θ(z) = (1 + .5z), so
the model is an ARMA(1, 1) model, (1 − .9B)xt = (1+.5B)wt, or CIE/IE 500 Transportation Analytics  Fall 2012 21 Example cont’d
The reduced model is φ(B)xt = θ(B)wt where φ ( z ) = 1 − .9 z
θ ( z ) = 1 + .5 z
The model is causal because φ(z) = (1 − .9z) = 0 when z = 10/9,
which is outside the unit circle.
The model is also invertible because the root of θ(z) = (1 + .5z) is
z = −2, which is outside the unit circle. CIE/IE 500 Transportation Analytics  Fall 2012 22 Autocorrelation Function (ACF) and Partial
Autocorrelation Function (PACF)
ψweights for a Causal ARMA(p,q)
ACF
PACF
Examples and Summary of ACF and PACF
Model Selection Rules CIE/IE 500 Transportation Analytics  Fall 2012 23 ψweights for a Causal ARMA(p,q)
Property: For a causal ARMA(p,q) model, φ(B)xt = θ(B)wt, where
the zeros of φ(z) are outside the unit circle, we can write where ψ0 = 1 and ψ(z)φ(z) = θ(z).
For the pure MA(q) model, ψ0 = 1, ψj = θj...
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 Fall '09

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