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Lec12 – Time Series Analysis - ARIMA

# Lec12 – Time Series Analysis - ARIMA - Lecture 12 Time...

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Lecture 12 – Time Series Analysis: Part III – ARIMA Model Dr. Qing He [email protected] Fall 2012 University at Buffalo CIE/IE 500 Transportation Analytics - Fall 2012 1

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Static vs. Dynamic, and Stationary vs. Nonstationary ° The trend regression model in Lec 11 is a static model since a value x t only depends on t and not x t-1 , x t-2 , ... ° The random walk (with drift) model is dynamic since computation of x t uses x t-1 , however this model was not stationary (recall its covariance function is given by γ(s, t) = min(s, t)σ 2 ). ° This leads us to a large class of dynamic models which are stationary —the ARMA models. ° Properties of the random walk model and the ARMA model are combined in the ( nonstationary ) AR I MA model CIE/IE 500 Transportation Analytics - Fall 2012 2
Autoregressive Model Example ° Consider the following AR(1) process ° where w t ~ iid N(0,1). We simulate the process in R. CIE/IE 500 Transportation Analytics - Fall 2012 3 > plot(arima.sim(list(order=c(1,0,0), ar=-.9), n=100)) t t t w x x - = - 1 9 . 0

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Definition of AR(p) Model ° An autoregressive model of order p is of the form ° where x t is stationary, φ 1 , φ 2 , . . . , φ p are constants (φ p ≠0), w t is Gaussion white noise with mean zero, variance σ 2 w . It has mean µ given by ° Notice that x t is auto-regressed on x t-1 ,…, x t-p however these regressors have random components. CIE/IE 500 Transportation Analytics - Fall 2012 4 t p t p t t t w x x x x = - - - φ φ φ α ... 2 2 1 1 p φ φ φ α μ - - - - = ... 1 2 1 (assuming the denominator is nonzero)
The Autoregressive Operator ° The general mean zero AR model ° which is equivalent to ° can be reformulated as ° and even more concisely as CIE/IE 500 Transportation Analytics - Fall 2012 5 where Definition (Autoregressive Operator) The autoregressive operator is defined to be p p B B B B φ φ φ φ - - - - = ... 1 ) ( 2 2 1

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AR(1) ° Let’s investigate a general AR(1) process given by ° Iterating backwards gives CIE/IE 500 Transportation Analytics - Fall 2012 6
AR(1)— MA( ) Representation ° Provided |φ| < 1, we can represent the AR(1) model as ° In this representation, we easily see ° and the autocovariance function is computed as ... CIE/IE 500 Transportation Analytics - Fall 2012 7 ± MA( )

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AR(1)— ACF Computation ° Autocovariance CIE/IE 500 Transportation Analytics - Fall 2012 8
AR(1)— ACF Computation cont. ° ACF ° Note that ρ(h) satisfies the recursion CIE/IE 500 Transportation Analytics - Fall 2012 9 (with the usual initial constraint ρ(0) = 1).

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Causality ° From the AR(1) model ° we can solve for x t-1 as ° If |φ| > 1, then the above expression gives a stationary representation of x t-1 in terms of future values of x t . This is useless since we don’t know the future! CIE/IE 500 Transportation Analytics - Fall 2012 10 Definition (Causality) When a process does not depend on the future, such as the AR(1) with |φ| < 1, the process is called causal . If |φ|>1, it is the explosive case of this example, so the process is stationary, but it is also future dependent, and not causal .
Moving Average Model— MA(q) ° Definition (Moving average model— MA(q)) ° The moving average model of order q is defined to be °

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