TS Module 7 Stationary mixed processes


TS Module 7 Stationary mixed processes

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NEAS
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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.

 


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Luke Grady
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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_e

Is 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.


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Mod7covariance.pdf (918 views, 26.00 KB)
dom2114
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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
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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
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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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