1. Statistics 153 Review - Sept 30, 2008
Notes re the 153 Midterm on October 2, 2008
1. The class will be split into two groups by first letter of surname.
Letters A to L will take the exam in 330 Evans.
Letters M to Z will take it in 340 Evans.
2. The exam will cover material through Chapter 4.
3. There will be 2 questions, answer both.
4. The questions will be like the Assignment's, but the exam will be closed book - no books or notes allowed.
5. No questions to the Proctors about the exam content please. If unsure, make an interpretation, state it and answer that.
6. You will have 60 minutes, exactly, to work.
7. The exam itself will be handed out. There will be space on it to answer the questions.
8. The solutions to Assignment 3 will be posted in the glass case, center corridor, third floor Evans, Tuesday after class, but the papers won't have been graded yet.
I will do a review in class September 30. Suggest some topics. 3

2. Name:________________________ October 2, 2008
MIDTERM EXAMINATION Statistics D. R. Brillinger
Answer both questions in the space provided. Show your work.
If you are not sure of the meaning of a question, set down an interpretation, and provide a reasonable answer.
You have exactly 60 minutes for the exam.
Question 1. Let {Zt} be a purely random process.

3. What is a time series?
a sequence of numbers, x, indexed by t

4. Stat 153 - 11 Sept 2008 D. R. Brillinger
Simple descriptive techniques
Trend
Xt =  + t + t

5. Filtering/filters
yt = r=-qs ar xt+r
yt = k hk xt-k p. 189
This form carries stationary into stationary
Filters may be in series
stationarity preserved if filter time invariant

6. Differencing
yt = xt - xt-1 =  xt
"removes" linear trend
Seasonal variation model
Xt = mt + St + t
St  St-s
12 xt = xt - xt-12 , t in months

7. Stationary case, autocorrelation estimate at lag k, rk
t=1N-k (xt- )(xt+k - )
over
t=1N (xt )2
autocovariance estimate at lag k, ck
t=1N-k (xt )(xt+k ) / N

8. Stat 153 - 16 Sept 2008 D. R. Brillinger
Chapter 3
mean function
variance function
autocovariance

9. Strictly stationary
All joint distributions unaffected by simple time shift
Second-order stationary

10. Properties of autocovariance function
Does not identify model uniquely

11. Useful models and acf's
Purely random
Building block

12. Random walk
not stationary

13. *

14. Moving average, MA(q)
From *
stationary

15. Backward shift operator
Linear process.
Need convergence condition Stationary

16. autoregressive process, AR(p)
first-order, AR(1) Markov *
Linear process
For convergence/stationarity root of φ(z)=0 in |z|>1

17. a.c.f. From *
p.a.c.f vanishes for k>p

18. In general case,
Very useful for prediction

19. ARMA(p,q)
Roots of (z)=0 in |z|>1 for stationarity
Roots of θ(z)=0 in |z|>1 for invertibility

20. ARIMA(p,d,q).

21. Yule-Walker equations for AR(p).
Correlate, with Xt-k , each side of
For AR(1)

22. Stat 153 - 23 Sept 2008 D. R. Brillinger
Chapter 4 - Fitting t.s. models in the time domain
sample autocovariance coefficient.
Under stationarity, ...

23. Estimated autocorrelation coefficient
asymptotically normal
interpretation

24. Estimating the mean
Can be bigger or less than 2/N

25. Fitting an autoregressive, AR(p)
Easy. Remember regression and least squares
normal equations

26. AR(1)
Cp.

27. Seasonal ARIMA. seasonal parameter s
SARIMA(p,d,q)(P,D,Q)s
Example

28. Residual analysis.
Paradigm
observation = fitted value plus residual
The parametric models have contained Zt

29. Portmanteau lack-of-fit statistic
ARMA(p,q) appropriate?