1. Basic Business Statistics (9th Edition)
Chapter 16
Time-Series Forecasting and Index Numbers
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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
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3. Chapter Topics Holt-Winters Method for Trend Fitting and Forecasting
(continued)
Holt-Winters Method for Trend Fitting and Forecasting
Autoregressive Models
Choosing Appropriate Forecasting Models
Time-Series Forecasting of Monthly or Quarterly Data
Pitfalls Concerning Time-Series Forecasting
Index Numbers
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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
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5. The Importance of Forecasting
(continued)
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
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6. What is a Time Series?
Numerical Data Obtained at Regular Time Intervals
The Time Intervals Can Be Annually, Quarterly, Monthly, Daily, Hourly, Etc.
Example:
Year:
Sales:
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7. Time-Series Components
Trend
Cyclical
Time-Series
Seasonal
Irregular
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8. Trend Component Overall Upward or Downward Movement
Data Taken Over a Period of Years
Upward trend
Sales
Time
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9. Cyclical Component Upward or Downward Swings May Vary in Length
Usually Lasts Years
1 Cycle
Sales
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10. Seasonal Component Upward or Downward Swings Regular Patterns
Observed Within 1 Year
Sales
Summer
Winter
Spring
Fall
Time (Monthly or Quarterly)
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11. Irregular or Random Component
Erratic, Nonsystematic, Random, “Residual” Fluctuations
Due to Random Variations of
Nature
Accidents
Short Duration and Non-Repeating
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12. Example: Quarterly Retail Sales with Seasonal Components
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13. Example: Quarterly Retail Sales with Seasonal Components Removed
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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:
Ti = Trend
Ci = Cyclical
Ii = Irregular
Si = Seasonal
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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
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16. Moving Averages Example: 3-Year Moving Average First average:
(continued)
Example: 3-Year Moving Average
First average:
Second average:
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17. Moving Average Example
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.
Year Units Moving Ave NA NA
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18. Moving Average Example Solution
Year Response Moving Ave NA NA
Sales
L = 3 8 6 4 2
No MA for the first and last (L-1)/2 years
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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
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20. Example: 5-Period Moving Averages of Quarterly Retail Sales
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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
Range from 0 to 1
Close to 0 for smoothing out unwanted cyclical and irregular components
Close to 1 for forecasting
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22. Exponential Weight: Example
Year Response Smoothing Value Forecast (W = .2, (1-W)=.8) NA
(.2)(5) + (.8)(2) =
(.2)(2) + (.8)(2.6) =
(.2)(2) + (.8)(2.48) =
(.2)(7) + (.8)(2.384) =
(.2)(6) + (.8)(3.307) =
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23. Exponential Weight: Example Graph
Sales 8 6 4 2
Data
Smoothed
Year
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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
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25. Example: Exponential Smoothing of Real GNP
The Excel Spreadsheet with the Real GDP Data and the Exponentially Smoothed Series
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26. Linear Trend Model
Use the method of least squares to obtain the linear trend forecasting equation:
Year Coded X Sales (Y)
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27. Linear trend forecasting equation:
Linear Trend Model
(continued)
Linear trend forecasting equation:
Excel Output
Projected to year 2000
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28. The Quadratic Trend Model
Use the method of least squares to obtain the quadratic trend forecasting equation:
Year Coded X Sales (Y)
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29. The Quadratic Trend Model
(continued)
Excel Output
Projected to year 2000
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30. The Exponential Trend Model
After taking the logarithms, use the method of least squares to get the forecasting equation: or
Year Coded X Sales (Y)
Excel Output of Values in Logs
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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
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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
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33. Model Selection Using Differences
(continued)
Use an Exponential Trend Model If the Percentage Differences are More or Less Constant
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34. The Holt-Winters Method
Similar to Exponential Smoothing
Advantages Over Exponential Smoothing
Can detect future trend and overall movement
Can provide intermediate and/or long-term forecasting
Two Weights 0<U<1 and 0<V<1 are to Be Chosen
Smaller values of U give more weight to the more recent levels and less weight to earlier levels
Smaller values of V give more weight to the current trends and less weight to past trends
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35. The Holt-Winters Method
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36. The Holt-Winters Method: Example
Year
Sales (Yi )
Level (Ei ) U =.2
Trend (Ti ) V = .2 94 2 NA 95 5
5-2=3 96
.2(5+3)+.8(2)=3.2
.2(3)+.8(3.2-5)=-.84 97
.2( )+.8(2)=2.07
.2(-.84)+.8( )=-1.07 98 7
.2( )+.8(7)=5.8
.2(-1.07)+.8( )=2.77 99 6
.2( )+.8(6)=6.51
.2(2.77)+.8( )=1.12
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37. The Holt-Winters Method: Forecasting
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38. Holt-Winters Method: Plot of Series and Forecasts
Excel Spreadsheet with the Computation
1994
Forecasts for 2000 to 2005
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39. 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:
Random Error
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40. Autoregressive Model: Example
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.
Year Units
94 3
97 2
98 2
99 4
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41. Autoregressive Model: Example Solution
Develop the 2nd order table
Use Excel to estimate a regression model
Year Yi Yi Yi-2
Excel Output
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42. Autoregressive Model Example: Forecasting
Use the 2nd order model to forecast number of units for 2001:
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43. Autoregressive Model in PHStat
PHStat | Multiple Regression
Excel Spreadsheet for the Office Units Example
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44. Autoregressive Modeling Steps
1. Choose p : Note that df = n - 2p - 1
2. 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
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45. Selecting a Forecasting Model
Perform a Residual Analysis
Look for pattern or direction
Measure Residual Error Using SSE (Sum of Square Error)
Measure Residual Error Using MAD (Mean Absolute Deviation)
Use Simplest Model
Principle of parsimony
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46. Cyclical effects not accounted for
Residual Analysis e e
Time
Time
Random errors
Cyclical effects not accounted for e e
Time
Time
Trend not accounted for
Seasonal effects not accounted for
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47. Measuring Errors
Choose a Model that Gives the Smallest Measuring Errors
Sum Square Error (SSE)
Sensitive to outliers
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48. Measuring Errors Mean Absolute Deviation (MAD)
(continued)
Mean Absolute Deviation (MAD)
Not sensitive to extreme observations
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49. Principle 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
Holt-Winters Model
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50. Forecasting with Seasonal Data
Use Categorical Predictor Variables with Least-Squares Trend Fitting
Forecasting Equation (Exponential Model with Quarterly Data):
The bj provides the multiplier for the j -th quarter relative to the 4th quarter
Qj = 1 if j -th quarter and 0 if not
Xi = the coded variable denoting the time period i
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51. Forecasting with Quarterly Data: Example
Standards and Poor’s Composite Stock Price Index:
Quarter
Excel Output
r2 is .98
Appears to be an excellent fit.
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52. Forecasting with Quarterly Data: Example
(continued)
Excel Output
Regression equation for the first quarters:
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53. Forecasting with Quarterly Data: Example
(continued)
1st quarter of 1994:
1st quarter of 1998:
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54. Forecasting with Quarterly Data in PHStat
Use PHStat | Multiple Regression
Excel Spreadsheet for the Stock Price Index Example
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55. Index Numbers
Measure the Value of an Item (Group of Items) at a Particular Point in Time as a Percentage of the Item’s (Group of Items’) Value at Another Point in Time
A price index measures the percentage change in the price of an item (group of items) in a given period of time over the price paid for the item (group of items) at a particular point of time in the past
Commonly Used in Business and Economics as Indicators of Changing Business or Economic Activity
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56. Simple Price Index Selection of the Base Period
Should be a period of economic stability rather than one at or near the peak of an expanding economy or declining economy
Should be recent so that comparisons are not greatly affected by changing technology and consumer attitudes or habits
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57. Simple Price Index: Example
Given the prices (in dollars per pound) for apples, construct the simple price index using 1980 as the base year.
Base Year
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58. Shifting the Base
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59. Shifting the Base: Example
Change the base year of the simple price index of apples from 1980 to 2000:
New Base Year
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60. Aggregate Price Index
Reflects the Percentage Change in Price of a Group of Commodities (Market Basket) in a Given Period of Time Over the Price Paid for that Group of Commodities at a Particular Point of Time in the Past
Affects the Cost of Living and/or the Quality of Life for a Large Number of Consumers
Two Basic Types
Unweighted aggregate price index
Weighted aggregate price index
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61. Unweighted Aggregate Price Index
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62. Unweighted Aggregate Price Index
(continued)
Easy to Compute
Two Distinct Shortcomings
Each commodity in the group is treated as equally important so that the most expensive commodities per unit can overly influence the index
Not all commodities are consumed at the same rate, but they are treated the same by the index
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63. Unweighted Aggregate Price Index: Example
Given the prices (in dollars per pound) for apples, bananas and oranges, compute the unweighted aggregate price index using 1980 as the base year:
Base Year
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64. Weighted Aggregate Price Indexes
Allow for Differences in the Consumption Levels Associated with the Different Items Comprising the Market Basket by Attaching a Weight to Each Item to Reflect the Consumption Quantity of that Item
Account for Differences in the Magnitude of Prices Per Unit and Differences in the Consumption Levels of the Items
Two Types that are Commonly Used
The Laspeyres price index
The Paasche price index
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65. Laspeyres Price Index
Uses the Consumption Quantities Associated with the Base Year
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66. Laspeyres Price Index: Example
Given the prices (in dollars per pound) and per capita consumption (in pounds) for apples, bananas, and oranges, compute the Laspeyres price index using 1980 as the base year:
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67. Paasche Price Index
Uses the Consumption Quantities Experienced in the Year of Interest Instead of Using the Initial Quantities
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68. Paasche Price Index Advantage Disadvantages
(continued)
Advantage
A more accurate reflection of total consumption costs at the point of interest in time
Disadvantages
Accurate consumption values for current purchases are often hard to obtain
If a particular product increases greatly in price compared to other items in the market basket, consumers will avoid the high-priced item out of necessity, not because of changes in preferences
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69. Paasche Price Index: Example
Given the prices (in dollars per pound) and per capita consumption (in pounds) for apples, bananas, and oranges, compute the Paasche price index using 1980 as the base year:
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70. 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.
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71. 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
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72. Chapter Summary
(continued)
Discussed Holt-Winters Method of Trend Fitting and Forecasting
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
Described Index Numbers
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