Question

the stationarity condition for a mixed seasonal ARMA model

Answer 1

Answer:

7.2 Seasonal ARMA Let us assume that there is seasonality in the data, but no trend. Then we could model the data as (7.3) where Yi is a stationary process. The seasonality component is such that where h denotes the length of the period and In Section 2.2.3 we have discussed removing the seasonal effect from the data by differencing at lag h. We have introduced the lag-h operator which, for (7.3), gives (7.4) Hence, this operation removes the seasonality effect. This fact leads to introducing the seasonal ARMA model, denoted by ARMA(PQ)h, which is of the form (7.5) where and are, respectively, the seasonal AR operator and the seasonal MA operator, with seasonal period of length h Remark 7.4. Analogously to ARMA(P,q), the ARMA(P, ) model is causal only when the roots of ( lie outside the unit circle, and it is invertible only when the roots of ( lie outside the unit circle. Example 7.1. Seasonal ARMA(1,1)12. Such a model can be written as

4-12 which is a generalization of (7.4). When written as and compared to ARMA(1,1) we see that the seasonal ARMA presents the series in terms of its past values at lag equal to the length of the period (here h=12), while the non-seasonal ARMA does it in terms of its past vales at lag 1. Seasonal ARMA incorporates the seasonality into the model. Similarly as for the non-seasonal ARMA, here too, we require |φ| < 1 for the causality and le1< 1 for invertibility of the Remark 7.5. Note that seasonal ARMA(0, Q)h is a seasonal MA(Q)h, and sea- sonal ARMA(P, 0) is a seasonal AR(P) Example 7.2. ACF of MA(1)12 A seasonal MA model with the period length h 12 can be written as It is a zero mean stationary model and it is easy to calculate its autocovariance, namely for τ Ba2 12. Thus, the only non-zero correlations are ρ(0)-1 and which is of the same form as ρ( 1 ) for a non-seasonal MA(1 ).

Example 7.3. ACF of AR(1)h Using the techniques for calculating ACVF and ACF of the non-seasonal AR(1) we obtain for 0, 0 otherwise. This give the ACF similar to the ACF of a non-seasonal AR(1), namely o(r) = for-thk, k = 1.2. 0 otherwise. The following table summarizes the behaviour of the ACF and PACF of the causal and invertible seasonal ARMA models (see R.H.Shumway and Stoffer (2000)). at uts PACF Cuts off after lag Ph Tails off at lags kh Tails off at lags kh where h is the length of the seasonal period, k 1,2 and PACF are zero at non-seasonal lags τ kh. and the values of ACF 7.2.1 Mixed Seasonal ARMA When we combine seasonal and non-seasonal operators we obtain a model which is called mixed seasonal ARMA and it is denoted by The behavior of the ACF and PACF for such models is a combination of behavior of the seasonal and nonseasonal parts of the model. Example 7.4. ARMA(0,1) x (1,0 Such a model has the following form

Figure 7.1: Simulated ARMA(0, 1) (1, 0)h2 process. Figure 7.2: ACF and PACF of the above ARM A(0, 1)(1,0)12 process. where l < 1 and iol < 1. Here we obtain for τ= 12k, k = 1, 2, or 12k 1, k-1,2 otherwise