Transcription of 6.2 ACF and PACF of ARMA(p,q)
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6.2 ACF and PACF of ARMA(p,q)
110 CHAPTER 6. ARMA MODELS. ACF and PACF of ARMA(p,q). ACF of ARMA(p,q). In Section we have derived the ACF for ARMA(1,1) process. We have used the linear process representation and the fact that . X. 2. ( ) = j j+ . j=0. We have calculated the coefficients j from the relation (B). (B) = , (B). which (as in the above example) gives the values j = j 1. 1 ( 1 + 1 ). This allows us to calculate the ACF of the process ( ). ( ) = . (0). Another way of finding the coefficients is using the homogeneous difference equations. However, we may obtain such equation directly in terms of ( ) or ( ). For ARMA(1,1). Xt Xt 1 = Zt + Zt 1. we can write ( ) = cov(Xt+ , Xt ). = E(Xt+ Xt ). = E[( Xt+ 1 + Zt+ + Zt+ 1 )Xt ]. = E[ Xt+ 1 Xt + Zt+ Xt + Zt+ 1 Xt ]. = E[Xt+ 1 Xt ] + E[Zt+ Xt ] + E[Zt+ 1 Xt ].
6.2.2 PACF of ARMA(p,q) We have seen earlier that the autocorrelation function of MA(q) models is zero for all lags greater than qas these are q-correlated processes. Hence, the ACF is a good indication of the order of the process. However AR(p) and ARMA(p,q) pro-
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