Time Series Analysis

---

Announcements

---

• Pick a project
  

  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.

  • Bootstrap resampling methods for time series. Bootstrap resampling provides an alternative to the classical approximations to the sampling distributions of estimators.
  • Time series models for categorical data, such as the INAR model, or the spectral method described in S&S.
  • Multivariate time series modeling, such as the use of climate proxy variables for global warming (eg, the glacial varve series in the text) or identifying and evaluating leading economic indictors. Data-driven projects often work well if the objective is clear.
  • Predictive modeling, such as predicting financial or economic time series; properties of predition intervals. Prediction is the true test of a time series model, and anticipating the accuracy of predictions an important challenge.
  • Nonlinear time series models (SETAR or bilinear models).
  • Volatility models (GARCH, ARCH), commonly used in financial models.
  • Bayesian models for time series. A huge area, and related to the next area.
  • Regime switching models, such as hidden Markov models, or methods for testing for the presence of multiple regimes.

---

Tips

---

• The web site for the Shumway and Stoffer textbook is second edition is here and for the third edition is here here .

• If you have trouble accessing the R package that goes with the Shumway and Stoffer text, you can access it via this link to tsa3.rda . (Download the file rather than open it.)

• From UPenn computers, you can view PDF files of the chapters of the Shumway and Stoffer text; go to this link .

---

Lecture notes

---

These lecture notes vary in detail from class to class and are intended to be an outline for what we will cover. The syllabus summarizes what's coming and readings from the course text.

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.

1. Introduction ( R script file for this lecture.)
2. Stationarity ( R script )
3. Descriptive estimators ( R code )
   Read Appendix A (in the text) to fill in details that come with this lecture. The topics continue into Lecture 4.
4. Properties of descriptive estimators ( R code )
5. Regression models ( R code )
6. Harmonic regression ( R code for variable star data )
7. Eigenvectors of stationary processes These notes are a highly condensed version of this on-line manuscript .
8. Autoregressive, moving average models (ARMA) ( R code for simulating ARMA models)
9. Covariances of ARMA models
10. Predicting ARMA processes
11. Central limit theorems
12. Estimating ARMA processes (basic script and more R code for estimating ARMA models)
13. Resampling ARMA processes ( R code )
14. State-space models
15. Kalman filter ( R code )
16. Hilbert spaces
17. Spectral representation
18. Discrete Fourier transform (14 Apr 2011, edits, a figure) ( R code )
19. Spectral estimation (14 Apr 2011, edits, rearrange) ( R code )

---

Assignments

---

In general, you ought to read, and sketch an answer to all of the exercises at the end of chapters. I will pick out a few that seem most relevant, but that does not mean that you should ignore the others.

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.

1. Due Feb 11. (comprises 2 assignments)
   Solution ( R code )
   Chapter 1: Exercises 1.8-10, 1.14, 1.15, 1.18, 1.19, 1.20, 1.24 (we do most of (b) in class, you think about the rest), and as best you can 1.28, 1.30, and 1.31. For the latter problems (and parts of the notes for Lecture 3 as well), look at Appendix A.
   (Exercise 1.24 is 1.25 in the 3rd edition; 1.28 is 1.29; 1.30 is 1.31; 1.31 is 1.32.)

   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.)

2. Due March 4.
   Solution ( R code )
   Chapter 3: Exercises 3.3, 3.7, 3.8, 3.9, 3.10, 3.11, 3.12, 3.17, 3.19, 3.31
   (3rd Edition: 3.4, 3.8, 3.9, 3.10, 3.11, 3.12, 3.13, 3.18, 3.20, 3.33)
   For 3.19 (3.20 in 3rd ed), you don't need to turn in the simulation; just explain the nature of the process that was generated and the estimates from the ARMA(1,1) model.
   For 3.31 (3.33 in 3rd), difference the temperature data and model the differences as an ARMA process. For estimating an ARMA model, use the R function "arima". (Use ?arima to see the on-line help.)

   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!)

3. The final assignment! Due April 12.
   Solution Chapter 6: Exercises 6.1, 6.2, 6.3, 6.5, 6.6, 6.13 (3rd Edition: The same!)
   Plus a one-page (roughly) description of what you are doing for your class project.