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Unformatted text preview: , for j = 1, . . . , q, and ψj
= 0,
For the general case of ARMA(p, q) models, the task of solving
for the ψweights is much more complicated, To solve for the ψweights in general, we must match the coefficients in ψ(z)φ(z) =
θ(z): CIE/IE 500 Transportation Analytics  Fall 2012 24 ψweights for a Causal ARMA(p,q) cont’d
The first few values are where we would take φj = 0 for j > p, and θj = 0 for j > q. and initial condition CIE/IE 500 Transportation Analytics  Fall 2012 25 ACF of MA(q)
Recall ACF of MA(1) Brushing over the computations, the ACF of MA(q) with
representation xt = µ + wt + θ1wt −1 + θ 2 wt − 2 + ... + θ q wt − q
is given as No statistical significance in the correlogram(ACF plot) after lag q indicates the process
may be an MA(q) process
CIE/IE 500 Transportation Analytics  Fall 2012 26 Autocovariance Function of an ARMA(p,q)
The computation of the autocovariance function of the ARMA
model φ(B)xt = θ(B)wt begins as CIE/IE 500 Transportation Analytics  Fall 2012 27 Autocovariance Function of a Causal ARMA(p,q)
For a causal ARMA(p,q) model, we have
So that Therefore, CIE/IE 500 Transportation Analytics  Fall 2012 28 Autocovariance Recurrence
Note that θj = 0 for j > q, so we have the following recurrence
relationship With initial condition Dividing through by γ(0) gives a similar recursion on ρ(h)= γ(h)/
γ(0) CIE/IE 500 Transportation Analytics  Fall 2012 29 Example: ACF of Causal ARMA(1,1)
Start with the causal ARMA(1,1) given by xt = φxt1 + θwt1 + wt.
From the previous slide, the following recursion holds So the general solution is
We use the initial condition from the previous slide to determine
the constant c.
To continue, we need to know the values of ψ0 and ψ1.
Recall ψ0 =1, ψ1= φ1+θ1 = φ+θ,
This gives two equations and two unknowns (γ(0) and γ(1) ) CIE/IE 500 Transportation Analytics  Fall 2012 30 Example: ACF of Causal ARMA(1,1) cont’d
We can continue to solve for these unknowns to produce To solve for c, note that γ(1) = cφ, in which case c = γ(1)/φ.
Hence, the specific solution is Finally, dividing through by γ(0) yields the ACF CIE/IE 500 Transportation Analytics  Fall 2012 31 Motivation of Partial Autocorrelation Function (PACF)
As we learned, ACF can detect a MA(q) process since NO
statistical significance in the correlogram(ACF plot) after lag q
indicates the process may be an MA(q) process
Thus, the ACF provides a considerable amount of information
about the order of the dependence when the process is a
moving average (MA) process.
Question: How to detect an AR(p) process?
Recall the ACF of an AR(1) process: so a geometric decay of the ACF would indicate AR(1). But this is
hard to observe, Other method to clearly indicate AR(1) or
AR(p)?
The PACF!
CIE/IE 500 Transportation Analytics  Fall 2012 32 PACF Definition
The partial autocorrelation function (PACF) of a stationary
process, xt, denoted φhh, for h = 1, 2, . . . , is
and
where xh1h denotes the regress...
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 Fall '09

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