Your assignment is to choose a topic for your project. Several of you have spoken to me about this, but for those of you who have not, here are some suggestions of areas that you might find interesting. These are not specific projects, but general areas of current interest that might pose a topic that you'll find interesting. For more information about these topics, consult the index of the text or ask the google.
I try to post a preliminary version of the notes the evening before that class; the version that I will use appears some time the morning of class. In addition, I will generally post an update to the notes after class to fix typos, add comments, and perhaps remove things that we did not get to (most likely, to a later set of notes). I put a time stamp on the latest version.
You will have about week to submit what you've done. You need not use R for any computing, but you will need to have access to some sort of software because some questions call for doing a bit of computing.
Solutions will be posted or handed out when assignments are returned, and then removed from the web page.
I've used the numbering from the second edition of the text for the exercises, and will note if there are changes in the 3rd.
Chapter 2: Exercises 2.1, 2.4, 2.5, 2.9, 2.10
Several
of these consist only of data-analysis, so add graphs
to show what you have done.
(Exercise 2.9 is 2.11 in
the 3rd edition, and 2.10 is 2.9 in the 3rd edition.)
Nonlinear time series can exhibit characteristics that are quite
different from linear processes. Simulate the SETAR model
y[t] = 0.4266 + w[t] if y[t-1] < 0.1
y[t] = 2.0372 - 2.7399 y[t-1] + w[t] otherwise.
w[t] is Gaussian white noise with variance 25. Let the
process run for t=1,...,100, then "cut off" the white noise
(ie, set w[t]=0 for t = 101, 102,..., 150.)
What
happens? How does the behavior of this process differ from
that of a general linear process when the noise "cuts off"?
(This process comes from an application to lemming counts
in Norway.)
Add the following two parts to question 3.9 (3.10 in 3rd):
(c) Are the coverage properties of the 4 prediction
intervals independent? That is, are the four 0/1 random
variables that indicate whether the intervals cover the
future values independent? Explain your answer
briefly.
(d) The question requires 95% prediction intervals
at each lead. How can you get 95% coverage over all
4 weeks?
Hint. Exercise 3.11 (3.12 in 3rd ed) is harder than its length suggests. Here's one approach. Suppose that Gamma_n is singular; then there is an AR(k) process (for some k less than n) that fits perfectly. The exercise does not state it, but the process is assumed stationary. Stationarity means that this recursion applies at every point in time. Show using backsubstitution that this model predicts X_n perfectly from X_1,...,X_k. Find the covariances implied by this relationship, a la the Yule-Walker equations. These lead to a contradiction to the condition that gamma(0)>0. (I'll let you 'assume' that the coefficients of the process are bounded, but you can prove this as well if you are on a roll!)