Compute the periodogram of the first three series identified as seasonal in Exercise 2 (Series 4th, 5th, and 9th) with the transformation \(abla \log \left(\mathrm{CPI}_{t}\right)\) and compare the ACF of these series (first 4 lags). Note the peaks in the spectrum at frequencies \(1 / 12\) and \(1 / 6\). Then look at the spectrum and the ACF of the transformation \(abla abla_{12} \log \left(\mathrm{CPI}_{t}\right)\). Which of these three series has stronger seasonality?

Data From Exercise 2:

Simulate the three ARIMA models of Exercise 1 with the command arima. sim and compare the theoretical ACFs with the sample ACFs.

Data From Exercise 1:

Compute the variance and the ACF (lag-1 to lag-4) of the following ARMA models with \(\operatorname{Var}\left(a_{t}\right)=1\) :

(a) \(z_{t}=0.7 z_{t-1}+a_{t}\);

(b) \(z_{t}=0.4 a_{t-1}+a_{t}\);

(c) \(z_{t}=0.7 z_{t-1}+\) \(0.4 a_{t-1}+a_{t}\).