TM 745 Forecasting for Business & Technology Dr. Frank Joseph Matejcik

1. TM 745 Forecasting for Business & TechnologyDr. Frank Joseph Matejcik 8th Session 3/29/10: Chapter 7 ARIMA (Box-Jenkins)-Type Forecasting Models South Dakota School of Mines and Technology, Rapid City

2. Agenda & New Assignment • Chapter 7 problems 3,4,5(series A)7B • Chapter 7 ARIMA (Box-Jenkins)-Type Forecasting Models • Don’t have your exams graded, yet.

3. Tentative Schedule Chapters Assigned 25-Jan 1 problems 1,4,8 e-mail, contact 1-Feb 2 problems 4,8,9 8-Feb 3 problems 1,5,8,11 15-Feb President’s Day 22-Feb 4 problems 6, 10 1-Mar 5 problems 5, 8 8-Mar Break 15-Mar Exam 1 Ch 1-4 Revised 22-Mar 6 problems 4, 7 29-Mar 7 3,4,5(series A) 7B Chapters Assigned 5-Apr Easter The Rest undecided

4. Web Resources • Class Web site on the HPCnet system • http://sdmines.sdsmt.edu/sdsmt/directory/courses/2010sp/tm745M001 • Answers will be online. Linked from ^ • I have gotten D2L and Elluminate! sites, and have gotten started on Elluminate! documentation. • The same class session that is on the DVD is on the stream in lower quality. http://www.flashget.com/ will allow you to capture the stream more readily and review the lecture, anywhere you can get your computer to run.

5. ARIMA (Box-Jenkins)-Type Forecasting Models • Introduction • The Philosophy of Box-Jenkins • Moving-Average Models • Autoregressive Models • Mixed Autoregressive & Moving-Average Models • Stationarity

6. ARIMA (Box-Jenkins)-Type Forecasting Models • The Box-Jenkins Identification Process • Comments from the field INTELSAT • ARIMA: A Set of Numerical Examples • Forecasting Seasonal Time Series • Total Houses Sold • Integrative Case: The Gap • Using ForecastXTM to Make ARIMA (Box-Jenkins) forecasts

7. Introduction • Examples of times series data • Hourly temperatures at your office • Daily closing price of IBM stock • Weekly automobile production of Fords • Data from an individual firm: sales, profits, inventory, back orders • An electrocardiogram • NO causal stuff, just series data

8. Introduction • ARIMA: Autoregressive Integrated Moving Average • Box-Jenkins • Best used for longer range • Used in short, medium & long range • Advantages • Wide variety of models • Much info from a time series

9. The Philosophy of Box-Jenkins • Regression view point • Box-Jenkins view point

10. The Philosophy of Box-Jenkins • What is white noise? • No relationship between previous values • Previous values no help in forecast • Examples are bit lame in text • Dow Jones last digits, Lotto • A good random number generator (for Simulation) is a better • In Stats books the assumptionis iid Normal(0,s 2)

11. The Philosophy of Box-Jenkins • Standard Regression Analysis • 1. Specify the causal variables. • 2. Use a regression model. • 3. Estimate a & b coefficients. • 4. Examine the summary statistics & try other model specs. • 5. Choose the most best model spec. (often based on RMSE).

12. The Philosophy of Box-Jenkins • For Box-Jenkins methodology: • 1. Start with the observed time series. • 2. Pass the series through a black box. • 3. Examine the series that results from passage through the black box. • 4. If the black box is correct, only white noise should remain. • 5. If the remaining series is not white noise, try another black box.

13. The Philosophy of Box-Jenkins • Wait a bit on the distinction of methods • A common regression check is a probability paper plot of the residuals • In Katya’s triangle we look for“white noise” in the residuals • Some regression checks resemblethe Box-Jenkins approach

14. The Philosophy of Box-Jenkins • Three main types on Models • MA: moving average • AR: autoregressive • ARMA: autoregressive moving average • ARIMA what is the I?

15. Moving-Average Models • Weighted moving average, may be a better term than moving average • MA(k) k: number of steps used

16. Moving-Average Models • Example in text table 7.2 of MA(1)

17. MA ModelsAutocorelation • Autocorrelation was in chapter 2.

18. Correlograms: An Alternative Method of Data Exploration

19. AR ModelsPartial Autocorelation • Degree of association between Yt & Yt-kwhen all other lags are held constant solve below for Y ’s

20. Moving-Average Ideal MA(1)

21. Moving-Average Ideal MA(2)

22. Moving-Average Generated ACF

23. Moving-Average Generated PACF

24. Autoregressive Models • How do we check for this model? • Where did we see it before?

25. Autoregressive Models • Let’s check the PACF and ACF plots • AR(k) : k is the number of steps used

26. ACF & PACF Ideal AR(1)

27. ACF & PACF Ideal AR(2)

28. Mixed Autoregressive and Moving-Average Models • We call these are ARMA models • Check out the ACF & PACF plots

29. Mixed Autoregressive and Moving-Average Models Ideal

30. Mixed Autoregressive and Moving-Average Models Ideal

31. Stationarity • There is a fix for some forms of non-stationarity. Where have seen it before?

32. Stationarity • When that doesn’t work. Try it again!

33. Stationarity • When we use the differencing we cal the models ARIMA(p,d,q) .

34. Stationarity Example

35. Stationarity Example

36. Stationarity • When we use the differencing we call the models ARIMA(p,d,q) .

37. Box-Jenkins Identification Process • What do we use for diagnostics?

38. The Box-Jenkins ID Process • 1.If the autocorrelation function abruptly stops at some point-say, after q spikes-then the appropriate model is an MA(q) type. • 2.If the partial autocorrelation function abruptly stops at some point-say, after p spikes-then the appropriate model is a AR(p). • 3.If neither function falls off abruptly, but both decline toward zero in some fashion, the appropriate model is an ARMA(p, q).

41. The Box-Jenkins ID Process • Ljung-Box statistic • Informal measures are also used

42. ARIMA: Set of Numerical Ex 1

43. ARIMA: Set of Numerical Ex 2 • Use Elmo

44. ARIMA: Set of Numerical Ex 2 • Use Elmo

45. ARIMA: A Set of Numerical Examples Example 3 & 4 • Use Elmo was in the old slides, but we have no more. • Let’s skip in discussion

46. Forecasting Seasonal Time Series • It’s complicated call it • treat the season length like it is a times series. • Notation in next example • Use a second (p,d,q) set for seasonals

47. Case: INTELSAT Case: Intelligent Transportation • Communication Satellites 15 years out • Freeway in example in I-75 Atlanta • ARIMA (1,0,1)(0,1,1)672 Best of All

48. Total Houses Sold • Done rather quickly in the text, Why? • Use ELMO?

49. Integrative Case: The Gap • Same Data • ARIMA (2,0,2)(0,2,1) seems to fit, other models do work.

50. Using ForecastXTM to Make ARIMA (Box-Jenkins) forecasts • Can we try it?