Lec12 – Time Series Analysis - ARIMA

Of course xt is still white noise nothing has changed

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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 over-parameterization. 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 Example-Parameter 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) ψ-weights 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 ψ-weights 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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This document was uploaded on 02/12/2014.

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