This homework gives you some experience at manipulating models and data in the frequency domain. Please turn in your homework on paper, in class.
Question 4.1.
A. Compute and plot the spectral density function of a stationary AR2 model, \[ X_n = 1.5 X_{n-1} - 0.8X_{n-2} + \epsilon_n,\] where \(\{\epsilon_n\}\) is white noise with \({\mathrm{Var}}(\epsilon_n)=\sigma^2\). You can use software to do this, or carry out some computations analytically. It is up to you how much (or little) algebra you choose to work through, but please explain fully how you carried out your calculation. Also, plot the autocovariance function.
B. Compute and plot the spectral density function of an MA(2) moving mean, \[ X_n = \epsilon_{n-2} + \epsilon_{n-1}+\epsilon_n,\] where \(\{\epsilon_n\}\) is white noise with \({\mathrm{Var}}(\epsilon_n)=\sigma^2\). As in part (A), you can use software to do this or carry out some computations analytically. Also, plot the autocovariance function.
C. Comment briefly on what you find in parts A and B.
Question 4.2. Present an estimated spectral density of the sunspot time series in sunspots.txt. Comment on your choice of estimator. Comment on the resulting estimate. These data, as well as some background on the historical and current interest in sunspot activity, are described at http://solarscience.msfc.nasa.gov/SunspotCycle.shtml.