Unformatted text preview: average model— MA(q))
The moving average model of order q is defined to be xt = µ + wt + θ1wt −1 + θ 2 wt − 2 + ... + θ q wt − q
where θ1, θ2, θq are parameters in ℜ. μ is the expectation
of xt (often assumed to equal 0) The above model can be compactly written as xt = µ + θ ( B) wt
where θ(B) is the moving average operator.
Definition (Moving Average Operator)
The moving average operator is θ ( B) = 1 + θ1 B + θ 2 B 2 + ... + θ p B p
should be q, not p
CIE/IE 500 Transportation Analytics  Fall 2012 11 MA(1)
Consider the mean zero MA(1) process
Then the autocovariance function is computed as
3min quiz: calculate
autocovariance for MA(1) CIE/IE 500 Transportation Analytics  Fall 2012 12 MA(1)— ACF Computation
Therefore, the autocorrelation function is Note ρ(1) ≤ 1/2 for all values of θ (why?). Also, xt is correlated
with xt−1, but not with xt−2, xt−3, . . . . Contrast this with the case
of the AR(1) model in which the correlation between xt and xt−k
is never zero. CIE/IE 500 Transportation Analytics  Fall 2012 13 MA(1) Examples >par(mfrow=c(2,1))
>plot(arima.sim(list(order=c(0,0,1), ma=.5),
n=100),ylab="x",main=(expression("MA(1)
"*theta*" = +.5")))
>plot(arima.sim(list(order=c(0,0,1), ma=.5),
n=100),ylab="x",main=(expression("MA(1)
"*theta*" = .5"))) θ = .5 Quick question: Why series with θ =
.5 is smoother than the series with θ
= −.5? θ = .5 When θ = .5, for example, xt and xt−1 are positively
correlated, and ρ(1) = .4. When θ = −.5, xt and xt−1
are negatively correlated,
CIE/IE 500 Transportation Analytics  Fall 2012 ρ(1) = −.4. 14 Nonuniqueness of MA Models
We note that for an MA(1) model, ρ(h) is the same for θ and 1/ θ;
try 5 and 1/5, for example. In addition, the pair σ2w = 1 and θ = 5
yield the same autocovariance function as the pair σ2w = 25 and θ
= 1/5, namely, Thus, the MA(1) processes and
are the same because of normality (i.e., all finite distributions are
the same). We can only observe the time series xt and not the
noise, wt or vt, so we cannot distinguish between the models. CIE/IE 500 Transportation Analytics  Fall 2012 15 Invertibility of MA Models
For convenience, by mimicking the criterion of causality for AR
models, we will choose the model with an infinite AR
representation. Such a process is called an invertible process.
Write the MA(1) model as
if θ < 1, then led to
which is the desired infinite AR representation of the model.
Hence, given a choice, we will choose the model with σ2w = 25
and θ = 1/5 because it is invertible. CIE/IE 500 Transportation Analytics  Fall 2012 16 ARMA(p,q) Model
Definition (ARMA(p, q) Model)
A time series is ARMA(p,q) if it is stationary and satisfies xt = α + φ1 xt −1 + ... + φ p xt − p + wt + θ1wt −1 + ... + θ q wt − q
The ARMA model above with α = 0 can be expressed more
simply as
φ ( B) xt = θ ( B) wt
where φ ( B) = 1 − φ1 B − φ2 B 2 − ... − φ p B p
θ ( B) = 1 + θ1 B + θ 2 B 2 + ... + θ p...
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 Fall '09

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