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. 9. I will do a review in class September 30. Suggest some topics. 3

Name:________________________ October 2, 2008 MIDTERM EXAMINATION Statistics 153 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.

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

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

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

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

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

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

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

Properties of autocovariance function Does not identify model uniquely

Useful models and acf's Purely random Building block

Random walk not stationary

*

Moving average, MA(q) From * stationary

Backward shift operator Linear process. Need convergence condition Stationary

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

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

In general case, Very useful for prediction

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

ARIMA(p,d,q).

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

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

Estimated autocorrelation coefficient asymptotically normal interpretation

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

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

AR(1) Cp.

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

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

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