1. TM 745 Forecasting for Business & Technology Dr. Frank Joseph Matejcik
8th Session 4/07/08:
Chapter 7 ARIMA (Box-Jenkins)-Type Forecasting Models
South Dakota School of Mines and Technology, Rapid City

2. Agenda & New Assignment
I will be traveling next week, so no class
Chapter 7 problems 3,4,5(series A)7B
Chapter 7 ARIMA (Box-Jenkins)-Type Forecasting Models

3. Tentative Schedule Chapters Assigned 28-Jan 1 problems 1,4,8
, contact
4-Feb 2 problems 4, 8, 9
11-Feb 3 problems 1,5,8,11
18-Feb President’s Day
25-Feb 4 problems 6,10
3-Mar 5 problems 5,8
10-Mar Exam 1 Ch 1-4 Revised
17-Mar Break
24-Mar Easter
31-Mar 6 problems 4, 7
Chapters Assigned
7-Apr 7 3,4,5(series A) 7B
14-Apr Out of town
No class
21-Apr 8 Problem 6
28-Apr 9
05-May Final

4. Web Resources
Class Web site on the HPCnet system
Streaming video
Answers will be online. Linked from ^
The same class session that is on the DVD is on the stream in lower quality. 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 assumption is 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 resemble the 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 Models Autocorelation
Autocorrelation was in chapter 2.

18. Correlograms: An Alternative Method of Data Exploration

19. AR Models Partial Autocorelation
Degree of association between Yt & Yt-k when 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: A Set of Numerical Examples Example 1
Use Elmo

43. ARIMA: A Set of Numerical Examples Example 2
Use Elmo

44. ARIMA: A Set of Numerical Examples Example 3
Use Elmo

45. ARIMA: A Set of Numerical Examples Example 4
Use Elmo

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

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?