Homework 6 (Due on Wednesday, June 5) Statistics 137 Spring Quarter, 2013 1. Consider the data in Homework 3 (“deaths from homicide and suicide”). a) Plot the periodogram of this data. Also plot three smoothed periodograms (modied Daniell) averaging over 3, 7, 11 neighbors. Which amount of smooth- ing would you choose here? Comment on your ndings. b) Use the auto.arima function in R to select the most appropriate ARMA model using the AICC criterion. Write the estimated parameters and their standard errors for the selected model. Use the ACF plot to investigate if the residuals from this model can be described as white noise. c) Plot the spectral density function of the model selected in part (b) and the smoothed periodogram (with the chosen amount of smoothing in part (a)) on the same graph. Comment on your ndings. 2. Let fXtg be an MA(1) sequence with = 0:5 and 2 = 4: a) Write down the rst di造erence frXtg of the sequence fXtg as an MA(2) model by explicitly wring down its the parameters. b) Obtain an explicit expression for the spectral density function of the rst di造erence of Xt. c) Plot the spectral density functions of Xt and rXt. Comment on the di造erence in the shapes of the plots. 3. Let fXtg be a white noise sequence with 2 = 4: a) Consider the sequence Zt = (Xt+1 + Xt + Xt??1)=p3. Obtain an explicit expression for the spectral density function of Zt. b) Plot the spectral density functions of Xt and Zt. Comment on the di造erence in the shapes of the plots. 1