## TS Module 7 Stationary mixed processes

 Author Message NEAS Supreme Being         Group: Administrators Posts: 4.2K, Visits: 1.2K TS Module 7 Stationary mixed processes (The attached PDF file has better formatting.)         Mixed autoregressive moving average processes        Invertibility  Read Section 4.4, “Mixed autoregressive moving average processes,” on pages 77-79.  Know equations 4.4.3, 4.4.4, and 4.4.5 on page 78 for the ARMA(1,1) process. Read Section 4.5, “Invertibility,” on pages 79-81. Know the statement on page 80:  “If |è| < 1, the MA(1) model can be inverted into an infinite order autoregressive model. We say that the MA(1) model is invertible if and only if |è| < 1.” The authors emphasize parsimony and simplicity. The previous textbook for the time series course modeled some time series with complex processes, with many moving average and autoregressive parameters. Cryer and Chan concentrate on simple models. If you model a time series with more than four or five parameters, you don’t have a good model. Most student projects conclude that an AR(1), AR(2), ARMA(1,1), or MA(1) model works best, or that first or second differences of the series can be modeled by one of these processes.  Attachments TS Module 7 Stationary mixed processes.pdf (1.6K views, 30.00 KB) Luke Grady Junior Member         Group: Forum Members Posts: 12, Visits: 1 At the bottom of page 77 where it's deriving Yule-Walker type equations I don't understand why E(e_t*Y_t) = sigma^2_eIs it because the middle term of the right-hand side would factor to E(e_t * e_t) which is Cov(e_t,e_t) = Var(e_t)?The error term in Period t is independent of the error term in Period t-1 (by definition). The observation in Period t-1 (Yt-1) depends on the mean and the error terms in periods t-1 and previous. The covariance depends on the variance of the error term in period t, which is sigma-squared. See the attached PDF. Attachments Mod7covariance.pdf (918 views, 26.00 KB) dom2114 Forum Newbie         Group: Forum Members Posts: 4, Visits: 1 I having the same issue...Don't understand that + the next equation for E(et-1,Yt)[NEAS: The reasoning is the same as for the previous equation. The error terms are independent of each other. The only part of Y(t-1) which has a non-zero covariance with the error term in period (t-1) is the error term component of Y(t-1), for which the covariance is sigma-squared. The coefficients phi ö and theta è are scalars, so the covariances are multiplied by phi and –theta.] JasonScandopolous Forum Newbie         Group: Forum Members Posts: 4, Visits: 1 I was confused by this initially.... but indeed this is because of the term E(e^2).  Go to first principles:  Var(x) = E(x^2) - E(x)^2.  Since E(e) = 0, E(e^2) = V(e) = sigma^2. CalLadyQED Forum Guru         Group: Forum Members Posts: 62, Visits: 2 I believe that the statement we're supposed to know from section 4.5 is on pg 80 (just after eqn 4.5.2) not pg. 78. [NEAS: Thank you; typo corrected.]
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