ARMA-models - STAT 248: ARMA Models Handout 5 GSI: Gido van...

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Unformatted text preview: STAT 248: ARMA Models Handout 5 GSI: Gido van de Ven October 1st, 2010 1 Autoregressive Moving Average Models [ARMA(p,q)] 1.1 Introduction Classical regression (i.e. regression with deterministic explanatory variables and/or other observed time series) is often insufficient for explaining all of the interesting dynamics of a time series. For example, the ACF of the residuals of a simple linear regression reveals additional structure in the data that the regression did not capture. The introduction of correlation as a phenomenon that may be generated through lagged linear relations leads to proposing the autoregressive (AR) and the autore- gressive moving average (ARMA) models. Adding non-stationary models leads to the autoregressive integrated moving average (ARIMA). ARMA(p,q) class time series { X t ,t = 0 , 1 , 2 ,... } are defined in terms of linear difference equa- tions with constant coefficients . Lets review the concept of a linear difference equation with constant coefficients. 1. The term linear means that each term of the sequence is defined as a linear function of the preceding terms. 2. The order of a linear recurrence relation is the number of preceding terms required by the definition. 3. The general form of a linear recurrence relation of order d , is as follows: a n = c 1 a n- 1 + c 2 a n- 2 + + c d a n- d + c 4. If, for all i , c i , is independent of n , then the recurrence relation is said to have constant coeffi- cients. 5. The linear recurrence, together with seed values (initial conditions) for a ,...,a d- 1 , determines the sequence uniquely. Why are ARMA(p,q) models so extremely important? For any autocovariance function ( . ) such that lim h ( h ) = 0 and for any integer k > 0, it is possible to find an ARMA process { X t ,t = 0 , 1 , 2 ,... } with autocovariance function X ( . ) such that X ( h ) = ( h ), h = 0 , 1 ,...k . The linear structure of ARMA processes leads to a very simple theory of linear prediction. 1 Definition : [ARMA process] The process { X t ,t = 0 , 1 , 2 ,.., } is said to be an ARMA(p,q) process if { X t } is stationary and if for every t , X t- 1 X t- 1- ...- p X t- p = Z t + 1 Z t- 1 + ... + q Z t- q where { Z t } is white noise. These difference equations can be written symbolically in the more compact form ( B ) X t = ( B ) Z t for t = 0 , 1 , 2 ,..., where and are the p th and the q th degree polynomials ( z ) = 1- 1 z- ...- p z p ( z ) = 1 + 1 z + ... + q z q and B is the backward shift operator defined by B j X t = X t- j for j = 0 , 1 , 2 ,..., . The polynomials ( . ) and ( . ) will be referred to as the autoregressive and moving average polynomials of the difference equations....
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ARMA-models - STAT 248: ARMA Models Handout 5 GSI: Gido van...

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