Lec12 – Time Series Analysis - ARIMA

# q wt q where 1 2 q are parameters in is the

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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 3-min 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 Non-uniqueness 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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## This document was uploaded on 02/12/2014.

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