Topics to be covered next week

Statistics fi3F

Fall Øuarter, £0fi8

Monday, Dec 3 Spectral Analysis, linear filters, frequency response function,                    spectrum of ARMA (Sections 4.3-4.4 from the text, Handout fi£).

Wednesday, Dec † Spectral Analysis, spectrum of ARMA (Section 4.3-4.4 from               the text, Handout fi£).

Friday, Dec F Comments on spectral analysis.

Homework F (Due on Friday, Dec F).

You may form a group of three registered in this course and submit one com- pleted homework for the group. The front page should be blank. Please write down the names of the students in the group on the submitted work.

fi. Please use the data set “deaths from homicide and suicide” for this question.

(a) Plot the periodogram of this data. Also plot three smoothed periodograms (modified Daniell) averaging over 3, F, fifi neighbors (ie, with spans F, fi† and £3). Which amount of smoothing would you choose here? Comment on your findings.

(b) Use the auto.arima function in R (‘forecast‘ package) to select the most appropriate ARMA model using the AICC criterion. Plot the spectral density function of the selected model and the smoothed periodogram (with the chosen amount of smoothing in part (a)) on the same graph. Comment on your findings.

£. Let (Et} be an M A(fi) sequence with 8 = 0.† and o2 = 4.

(a)Write  down the first difference Et of the sequence Et as an MA(£) model by explicitly writing down its the parameters.

(b) Obtain an explicit expression for the spectral density function of the first difference of Et.

(c)Plot the spectral density functions of Et and Et. Comment on the differ- ence in the shapes of the plots.

3. Let (Et} be a white noise sequence with o2 = 4.

(a) Consider  the  sequence  Zt  =  (Et‡fi ‡ Et ‡ Et—fi)/’3. Obtain an explicit expression for the spectral density function of Zt.

(b) Plot the spectral density functions of Et and Zt. Comment on the difference in the shapes of the plots.

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