© 2003 Prentice-Hall, Inc.Chap 12-1 Business Statistics: A First Course (3 rd Edition) Chapter 12...
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Transcript of © 2003 Prentice-Hall, Inc.Chap 12-1 Business Statistics: A First Course (3 rd Edition) Chapter 12...
© 2003 Prentice-Hall, Inc. Chap 12-1
Business Statistics: A First Course
(3rd Edition)
Chapter 12Time-Series Forecasting
© 2003 Prentice-Hall, Inc. Chap 12-2
Chapter Topics
The Importance of Forecasting Component Factors of the Time-Series
Model Smoothing of Annual Time Series
Moving averages Exponential smoothing
Least-Squares Trend Fitting and Forecasting Linear, quadratic and exponential models
© 2003 Prentice-Hall, Inc. Chap 12-3
Chapter Topics
Autoregressive Models Choosing Appropriate Forecasting
Models Time-Series Forecasting of Monthly or
Quarterly Data Pitfalls Concerning Time-Series
Forecasting
(continued)
© 2003 Prentice-Hall, Inc. Chap 12-4
The Importance of Forecasting
Government Needs to Forecast Unemployment, Interest Rates, Expected Revenues from Income Taxes to Formulate Policies
Marketing Executives Need to Forecast Demand, Sales, Consumer Preferences in Strategic Planning
© 2003 Prentice-Hall, Inc. Chap 12-5
The Importance of Forecasting
College Administrators Need to Forecast Enrollments to Plan for Facilities, for Student and Faculty Recruitment
Retail Stores Need to Forecast Demand to Control Inventory Levels, Hire Employees and Provide Training
(continued)
© 2003 Prentice-Hall, Inc. Chap 12-6
What is a Time Series?
Numerical Data Obtained at Regular Time Intervals
The Time Intervals can be Annually, Quarterly, Daily, Hourly, Etc.
Example:Year: 1994 1995 1996 1997
1998Sales: 75.3 74.2 78.5
79.7 80.2
© 2003 Prentice-Hall, Inc. Chap 12-7
Time-Series Components
Time-Series
Cyclical
Random
Trend
Seasonal
© 2003 Prentice-Hall, Inc. Chap 12-8
Upward trend
Trend Component
Overall Upward or Downward Movement Data Taken Over a Period of Years
Sales
Time
© 2003 Prentice-Hall, Inc. Chap 12-9
Cyclical Component
Upward or Downward Swings May Vary in Length Usually Lasts 2 - 10 Years
Sales 1 Cycle
Time
© 2003 Prentice-Hall, Inc. Chap 12-10
Seasonal Component
Upward or Downward Swings Regular Patterns Observed Within 1 Year
Sales
Time (Monthly or Quarterly)
WinterSpring
Summer
Fall
© 2003 Prentice-Hall, Inc. Chap 12-11
Random or Irregular Component
Erratic, Nonsystematic, Random, “Residual” Fluctuations
Due to Random Variations of Nature Accidents
Short Duration and Non-repeating
© 2003 Prentice-Hall, Inc. Chap 12-12
E.g. Quarterly Retail Sales with Seasonal Components
Quarterly with Seasonal Components
0
5
10
15
20
25
0 5 10 15 20 25 30 35
Time
Sale
s
© 2003 Prentice-Hall, Inc. Chap 12-13
E.g. Quarterly Retail Sales with Seasonal Components
Removed
Quarterly without Seasonal Components
0
5
10
15
20
25
0 5 10 15 20 25 30 35
Time
Sa
les
Y(t)
© 2003 Prentice-Hall, Inc. Chap 12-14
Multiplicative Time-Series Model
Used Primarily for Forecasting Observed Value in Time Series is the
product of Components For Annual Data:
For Quarterly or Monthly Data:i i i iY TC I
i i i i iY T S C I
Ti = Trend
Ci = Cyclical
Ii = Irregular
Si = Seasonal
© 2003 Prentice-Hall, Inc. Chap 12-15
Moving Averages
Used for Smoothing Series of Arithmetic Means Over Time Result Dependent Upon Choice of L
(Length of Period for Computing Means) To Smooth Out Cyclical Component, L
Should be Multiple of the Estimated Average Length of the Cycle
For Annual Time Series, L Should be Odd
© 2003 Prentice-Hall, Inc. Chap 12-16
Moving Averages
Example: 3-year Moving Average
First average:
Second average:
1 2 3(3)3
Y Y YMA
2 3 4(3)3
Y Y YMA
(continued)
© 2003 Prentice-Hall, Inc. Chap 12-17
Moving Average Example
Year Units Moving Ave
1994 2 NA
1995 5 3
1996 2 3
1997 2 3.67
1998 7 5
1999 6 NA
John is a building contractor with a record of a total of 24 single family homes constructed over a 6 year period. Provide John with a 3-year Moving Average Graph.
© 2003 Prentice-Hall, Inc. Chap 12-18
Moving Average Example Solution
Year Response Moving Ave
1994 2 NA
1995 5 3
1996 2 3
1997 2 3.67
1998 7 5
1999 6 NA94 95 96 97 98 99
8
6
4
2
0
SalesL = 3
No MA for the first and last (L-1)/2 years
© 2003 Prentice-Hall, Inc. Chap 12-19
Moving Average Example Solution in Excel
Use Excel formula “=average(cell range containing the data for the years to average)”
Excel Spreadsheet for the Single Family Home Sales Example
Microsoft Excel Worksheet
© 2003 Prentice-Hall, Inc. Chap 12-20
E.g. 5-point Moving Averages of Quarterly Retail
Sales
Quarterly 5-point Moving Averages
0
5
10
15
20
25
0 5 10 15 20 25 30 35
Time
Sale
s
MA(5)
Y(t)
© 2003 Prentice-Hall, Inc. Chap 12-21
Exponential Smoothing Weighted Moving Average
Weights decline exponentially Most recent observation weighted most
Used for Smoothing and Short Term Forecasting
Weights are: Subjectively chosen Ranges from 0 to 1
Close to 0 for smoothing out unwanted cyclical and irregular components
Close to 1 for forecasting
© 2003 Prentice-Hall, Inc. Chap 12-22
Exponential Weight: Example
Year Response Smoothing Value Forecast(W = .2, (1-W)=.8)
1994 2 2 NA
1995 5 (.2)(5) + (.8)(2) = 2.6 2
1996 2 (.2)(2) + (.8)(2.6) = 2.48 2.6
1997 2 (.2)(2) + (.8)(2.48) = 2.384 2.48
1998 7 (.2)(7) + (.8)(2.384) = 3.307 2.384
1999 6 (.2)(6) + (.8)(3.307) = 3.846 3.307
1(1 )i i iE WY W E
© 2003 Prentice-Hall, Inc. Chap 12-23
Exponential Weight: Example Graph
94 95 96 97 98 99
8
6
4
2
0
Sales
Year
Data
Smoothed
© 2003 Prentice-Hall, Inc. Chap 12-24
Exponential Smoothing in Excel
Use Tools | Data Analysis | Exponential Smoothing The damping factor is (1-W )
Excel Spreadsheet for the Single Family Home Sales Example
Microsoft Excel Worksheet
© 2003 Prentice-Hall, Inc. Chap 12-25
Example: Exponential Smoothing of Real GNP
The EXCEL Spreadsheet with the Real GDP Data and the Exponentially Smoothed Series
Microsoft Excel Worksheet
© 2003 Prentice-Hall, Inc. Chap 12-26
Linear Trend Model
Year Coded X Sales (Y)
95 0 2
96 1 5
97 2 2
98 3 2
99 4 7
00 5 6
0 1i iY b b X
Use the method of least squares to obtain the linear trend forecasting equation
© 2003 Prentice-Hall, Inc. Chap 12-27
Linear Trend Model(continued)
0 1ˆ 2.143 .743i i iY b b X X
Excel Output
CoefficientsIntercept 2.14285714X Variable 1 0.74285714
0
1
2
3
4
5
6
7
8
0 1 2 3 4 5 6X
Sale
s
Projected to year 2001
Linear trend forecasting equation:
© 2003 Prentice-Hall, Inc. Chap 12-28
The Quadratic Trend Model
Year Coded X Sales (Y)
95 0 2
96 1 5
97 2 2
98 3 2
99 4 7
00 5 6
20 1 2i i iY b b X b X
Use the method of least squares to obtain the quadratic trend forecasting equation
© 2003 Prentice-Hall, Inc. Chap 12-29
The Quadratic Trend Model(continued)
2 20 1 2
ˆ 2.857 .33 .214i i i i iY b b X b X X X
CoefficientsIntercept 2.85714286X Variable 1 -0.3285714X Variable 2 0.21428571
Excel Output
0
1
2
3
4
5
6
7
8
0 1 2 3 4 5 6 X
Sale
s Projected to year 2001
© 2003 Prentice-Hall, Inc. Chap 12-30
The Exponential Trend Model
CoefficientsIntercept 0.33583795X Variable 10.08068544
0 1ˆ iXiY b b or
Excel Output of Values in logs
ˆ (2.17)(1.2) iXiY
antilog(.33583795) = 2.17antilog(.08068544) = 1.2
0 1 1ˆlog log logiY b X b
Year Coded X Sales (Y)
95 0 2
96 1 5
97 2 2
98 3 2
99 4 7
00 5 6
After taking the logarithms, use the method of least squares to get the forecasting equation
© 2003 Prentice-Hall, Inc. Chap 12-31
The Least-Squares Trend Models in PHStat
Use PHStat | Simple Linear Regression for Linear Trend and Exponential Trend Models and PHStat | Multiple Regression for Quadratic Trend Model
Excel Spreadsheet for the Single Family Home Sales Example
Microsoft Excel Worksheet
© 2003 Prentice-Hall, Inc. Chap 12-32
Model Selection Using Differences
Use a Linear Trend Model if the First Differences are More or Less Constant
Use a Quadratic Trend Model if the Second Differences are More or Less Constant
2 1 3 2 1n nY Y Y Y Y Y
3 2 2 1 1 1 2n n n nY Y Y Y Y Y Y Y
© 2003 Prentice-Hall, Inc. Chap 12-33
Model Selection Using Differences
3 2 12 1
1 2 1
100% 100% 100%n n
n
Y Y Y YY Y
Y Y Y
Use an Exponential Trend Model if the Percentage Differences are More or Less Constant
(continued)
© 2003 Prentice-Hall, Inc. Chap 12-34
Autoregressive Modeling
Used for Forecasting Takes Advantage of Autocorrelation
1st order - correlation between consecutive values
2nd order - correlation between values 2 periods apart
Autoregressive Model for p-th order:0 1 1 2 2i i i p i p iY A AY A Y A Y
Random Error
© 2003 Prentice-Hall, Inc. Chap 12-35
Autoregressive Model: Example
Year Units
93 4 94 3 95 2 96 3 97 2 98 2 99 4 00 6
The Office Concept Corp. has acquired a number of office units (in thousands of square feet) over the last 8 years. Develop the 2nd order Autoregressive model.
© 2003 Prentice-Hall, Inc. Chap 12-36
Autoregressive Model: Example Solution
Year Yi Yi-1 Yi-2
93 4 --- --- 94 3 4 --- 95 2 3 4 96 3 2 3 97 2 3 2 98 2 2 3 99 4 2 2 00 6 4 2
CoefficientsIntercept 3.5X Variable 1 0.8125X Variable 2 -0.9375
Excel Output
1 2ˆ 3.5 .8125 .9375i i iY Y Y
Develop the 2nd order table
Use Excel to estimate a regression model
© 2003 Prentice-Hall, Inc. Chap 12-37
Autoregressive Model Example: Forecasting
Use the 2nd order model to forecast number of units for 2001:
1 2
2001 2000 1999
ˆ 3.5 .8125 .9375
ˆ 3.5 .8125 .9375
3.5 .8125 6 .9375 4
4.625
i i iY Y Y
Y Y Y
© 2003 Prentice-Hall, Inc. Chap 12-38
Autoregressive Model in PHStat
PHStat | Multiple Regression
Excel Spreadsheet for The Office Units Example
Microsoft Excel Worksheet
© 2003 Prentice-Hall, Inc. Chap 12-39
Autoregressive Modeling Steps
1. Choose p : Note that df = n - 2p - 12. Form a Series of “Lag Predictor” Variables
Yi-1 , Yi-2 , … ,Yi-p
3. Use Excel to Run Regression Model Using all p Variables
4. Test Significance of Ap
If null hypothesis rejected, this model is selected
If null hypothesis not rejected, decrease p by 1 and repeat
© 2003 Prentice-Hall, Inc. Chap 12-40
Selecting A Forecasting Model
Perform a Residual Analysis Look for pattern or direction
Measure Sum of Square Error - SSE (residual errors)
Measure Residual Error Using MAD (mean absolute deviation)
Use Simplest Model Principle of parsimony
© 2003 Prentice-Hall, Inc. Chap 12-41
Residual Analysis
Random errors
Trend not accounted for
Cyclical effects not accounted for
Seasonal effects not accounted for
T T
T T
e e
e e
0 0
0 0
© 2003 Prentice-Hall, Inc. Chap 12-42
Measuring Errors
Choose a Model that Gives the Smallest Measuring Errors
Sum of Square Error (SSE)
Sensitive to outliers
2
1
ˆn
i ii
SSE Y Y
© 2003 Prentice-Hall, Inc. Chap 12-43
Measuring Errors
Mean Absolute Deviation (MAD)
Not Sensitive to Extreme Observations
(continued)
1
ˆn
i ii
Y YMAD
n
© 2003 Prentice-Hall, Inc. Chap 12-44
Principal of Parsimony
Suppose 2 or More Models Provide Good Fit to Data
Select the Simplest Model Simplest model types:
Least-squares linear Least-squares quadratic 1st order autoregressive
More complex types: 2nd and 3rd order autoregressive Least-squares exponential
© 2003 Prentice-Hall, Inc. Chap 12-45
Forecasting With Seasonal Data
Use Categorical Predictor Variables with Least-Squares Trending Fitting
Forecasting Equation (Exponential Model with Quarterly Data):
The bi provides the multiplier for the i-th quarter relative to the 4th quarter.
Qi = 1 if i-th quarter and 0 if not Xj = the coded variable denoting the time
period
1 2 3
0 1 2 3 4ˆ iX Q Q QY b b b b b
© 2003 Prentice-Hall, Inc. Chap 12-46
Forecasting With Quarterly Data: Example
445.77444.27462.69459.27
500.71544.75584.41615.93
645.5670.63687.31740.74
757.12885.14947.28970.43
I234
Quarter 1994 1995 1996 1997
Standards and Poor’s Composite Stock Price Index:
Excel Output
Appears to be an excellent fit.
r2 is .98
Regression StatisticsMultiple R 0.989936819R Square 0.979974906Adjusted R Square 0.972693054Standard Error 0.043867069Observations 16
© 2003 Prentice-Hall, Inc. Chap 12-47
Forecasting With Quarterly Data: Example
(continued)
Excel Output
0 1 1 2
1
ˆln ln ln ln
6.011 .0553 0.0104i i
i
Y b X b Q b
X Q
Regression Equation for the first quarter:
Coefficients Standard ErrorIntercept 6.011187697 0.031115484Coded X 0.055372493 0.002452244Q1 0.010421639 0.031879168Q2 0.023885562 0.031404042Q3 0.019342411 0.031115484
© 2003 Prentice-Hall, Inc. Chap 12-48
Forecasting with Quarterly Data in PHStat
Use PHStat | Multiple Regression
Excel Spreadsheet for The Stock Price Index Example
Microsoft Excel Worksheet
© 2003 Prentice-Hall, Inc. Chap 12-49
Pitfalls Concerning Time-Series Forecasting
Taking for Granted the Mechanism that Governs the Time Series Behavior in the Past will Still Hold in the Future
Using Mechanical Extrapolation of the Trend to Forecast the Future Without Considering Personal Judgments, Business Experiences, Changing Technologies, Habits, etc.
© 2003 Prentice-Hall, Inc. Chap 12-50
Chapter Summary
Discussed the Importance of Forecasting Addressed Component Factors of the
Time-Series Model Performed Smoothing of Data Series
Moving averages Exponential smoothing
Described Least-Squares Trend Fitting and Forecasting Linear, quadratic and exponential models
© 2003 Prentice-Hall, Inc. Chap 12-51
Chapter Summary
Addressed Autoregressive Models Described Procedure for Choosing
Appropriate Models Addressed Time-Series Forecasting of
Monthly or Quarterly Data (Use of Dummy Variables)
Discussed Pitfalls Concerning Time-Series Forecasting
(continued)