STAT 730, Time Series Analysis, Spring 2009
Take-Home Exam

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Problem 1. Given a zero mean stationary time series, X(1),...,X(N), show that the sample periodogram I(w) and the sample autocovariance c(h) constitute a Fourier pair.

Problem 2. Apply the CM algorithm to estimate the frequencies in the following data set made of several sinusoids plus additive WN. To download the data click on the link: SineData.

Problem 3. Let X(t), t=1,2,..., be a stationary time series with mean 0 and autocovariance R(k). Define: Xu(t)=X(t)X(t+u)-R(u), and consider the sum of two linear filters L1 and L2 plus WN:

Y(t)=L1(X(t))+L2(Xu(t))+ WN(0, .25)
The (X(t),Y(t)) data for t=1,...,1000, can be downloaded from Use the (X,Y) data to estimate u.

Problem 4. Using the monthly unemployment data Unemp.Women, make a comparison between Box-Jenkins, Kriging, and BTG prediction, and summarize your findings. The actual data are in the 4th column, where every "13th" observation is the average of the previous 12 observations.

Problem 5. Consider the time series x and y, one of which is WN and the other is WN plus a weak sinusoid. Can you tell which one is the WN series ?

Problem 6. a. Explain the effect on the spectrum of the filter L(B) = (1+aB+B^2)/(1+a*eta*B + eta^2*B^2) where B is the backward shift, a = -2cos(theta), and eta in (0,1).
b. Suggest an application for this filter, and demonstrate it using simulated data.

If you are looking for real time series data, some good sources are: