Statistics for Managers Using Microsoft Excel, 2/e © 1999 Prentice-Hall, Inc.
Chapter 16 Student Lecture Notes 16-1
© 2004 Prentice-Hall, Inc. Chap 16-1
Basic Business Statistics(9th Edition)
Chapter 16Time-Series Forecasting and
Index Numbers
© 2004 Prentice-Hall, Inc. Chap 16-2
Chapter Topics
The Importance of Forecasting
Component Factors of the Time-Series Model
Smoothing of Annual Time SeriesMoving averages
Exponential smoothing
Least-Squares Trend Fitting and ForecastingLinear, quadratic and exponential models
© 2004 Prentice-Hall, Inc. Chap 16-3
Chapter Topics
Holt-Winters Method for Trend Fitting and ForecastingAutoregressive ModelsChoosing Appropriate Forecasting ModelsTime-Series Forecasting of Monthly or Quarterly DataPitfalls Concerning Time-Series ForecastingIndex Numbers
(continued)
Statistics for Managers Using Microsoft Excel, 2/e © 1999 Prentice-Hall, Inc.
Chapter 16 Student Lecture Notes 16-2
© 2004 Prentice-Hall, Inc. Chap 16-4
The Importance of Forecasting
Government Needs to Forecast Unemployment, Interest Rates, Expected Revenues from Income Taxes to Formulate PoliciesMarketing Executives Need to Forecast Demand, Sales, Consumer Preferences in Strategic Planning
© 2004 Prentice-Hall, Inc. Chap 16-5
The Importance of Forecasting
College Administrators Need to Forecast Enrollments to Plan for Facilities, for Student and Faculty RecruitmentRetail Stores Need to Forecast Demand to Control Inventory Levels, Hire Employees and Provide Training
(continued)
© 2004 Prentice-Hall, Inc. Chap 16-6
What is a Time Series?
Numerical Data Obtained at Regular Time IntervalsThe Time Intervals Can Be Annually, Quarterly, Monthly, Daily, Hourly, Etc.Example:
Year: 1994 1995 1996 1997 1998Sales: 75.3 74.2 78.5 79.7 80.2
Statistics for Managers Using Microsoft Excel, 2/e © 1999 Prentice-Hall, Inc.
Chapter 16 Student Lecture Notes 16-3
© 2004 Prentice-Hall, Inc. Chap 16-7
Time-Series Components
Time-Series
Cyclical
Irregular
Trend
Seasonal
© 2004 Prentice-Hall, Inc. Chap 16-8
Upward trend
Trend Component
Overall Upward or Downward MovementData Taken Over a Period of Years
Sales
Time
© 2004 Prentice-Hall, Inc. Chap 16-9
Cyclical Component
Upward or Downward SwingsMay Vary in LengthUsually Lasts 2 - 10 Years
Sales 1 Cycle
Statistics for Managers Using Microsoft Excel, 2/e © 1999 Prentice-Hall, Inc.
Chapter 16 Student Lecture Notes 16-4
© 2004 Prentice-Hall, Inc. Chap 16-10
Seasonal Component
Upward or Downward SwingsRegular PatternsObserved Within 1 Year
Sales
Time (Monthly or Quarterly)
WinterSpring
Summer
Fall
© 2004 Prentice-Hall, Inc. Chap 16-11
Irregular or Random Component
Erratic, Nonsystematic, Random, “Residual” FluctuationsDue to Random Variations of
NatureAccidents
Short Duration and Non-Repeating
© 2004 Prentice-Hall, Inc. Chap 16-12
Example: 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
Statistics for Managers Using Microsoft Excel, 2/e © 1999 Prentice-Hall, Inc.
Chapter 16 Student Lecture Notes 16-5
© 2004 Prentice-Hall, Inc. Chap 16-13
Example: 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
Sale
s
Y(t)
© 2004 Prentice-Hall, Inc. Chap 16-14
Multiplicative Time-Series Model
Used Primarily for ForecastingObserved Value in Time Series is the Product of Components For Annual Data:
For Quarterly or Monthly Data:i i i iY T C I=
i i i i iY T S C I=
Ti = Trend
Ci = Cyclical
Ii = Irregular
Si = Seasonal
© 2004 Prentice-Hall, Inc. Chap 16-15
Moving Averages
Used for SmoothingSeries of Arithmetic Means Over TimeResult 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 CycleFor Annual Time Series, L Should Be Odd
Statistics for Managers Using Microsoft Excel, 2/e © 1999 Prentice-Hall, Inc.
Chapter 16 Student Lecture Notes 16-6
© 2004 Prentice-Hall, Inc. Chap 16-16
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)
© 2004 Prentice-Hall, Inc. Chap 16-17
Moving Average Example
Year Units MovingAve
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.
© 2004 Prentice-Hall, Inc. Chap 16-18
Moving Average Example Solution
Year Response MovingAve
1994 2 NA
1995 5 3
1996 2 3
1997 2 3.67
1998 7 5
1999 6 NA 94 95 96 97 98 99
8
6
4
2
0
Sales L = 3
No MA for the first and last (L-1)/2 years
Statistics for Managers Using Microsoft Excel, 2/e © 1999 Prentice-Hall, Inc.
Chapter 16 Student Lecture Notes 16-7
© 2004 Prentice-Hall, Inc. Chap 16-19
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
© 2004 Prentice-Hall, Inc. Chap 16-20
Example: 5-Period Moving Averages of Quarterly Retail Sales
Quarterly 5-Period Moving Averages
0
5
10
15
20
25
0 5 10 15 20 25 30 35
Time
Sale
s MA(5)Y(t)
© 2004 Prentice-Hall, Inc. Chap 16-21
Exponential SmoothingWeighted Moving Average
Weights decline exponentiallyMost recent observation weighted most
Used for Smoothing and Short-Term ForecastingWeights are:
Subjectively chosenRange from 0 to 1
Close to 0 for smoothing out unwanted cyclical and irregular componentsClose to 1 for forecasting
Statistics for Managers Using Microsoft Excel, 2/e © 1999 Prentice-Hall, Inc.
Chapter 16 Student Lecture Notes 16-8
© 2004 Prentice-Hall, Inc. Chap 16-22
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 −= + −
© 2004 Prentice-Hall, Inc. Chap 16-23
Exponential Weight: Example Graph
94 95 96 97 98 99
8
6
4
2
0
Sales
Year
Data
Smoothed
© 2004 Prentice-Hall, Inc. Chap 16-24
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
Statistics for Managers Using Microsoft Excel, 2/e © 1999 Prentice-Hall, Inc.
Chapter 16 Student Lecture Notes 16-9
© 2004 Prentice-Hall, Inc. Chap 16-25
Example: Exponential Smoothing of Real GNP
The Excel Spreadsheet with the Real GDP Data and the Exponentially Smoothed Series
Microsoft Excel Worksheet
© 2004 Prentice-Hall, Inc. Chap 16-26
Linear Trend Model
Year Coded X Sales (Y)
94 0 2
95 1 5
96 2 2
97 3 2
98 4 7
99 5 6
0 1i iY b b X= +
Use the method of least squares to obtain the linear trend forecasting equation:
© 2004 Prentice-Hall, Inc. Chap 16-27
Linear Trend Model(continued)
0 1ˆ 2.143 .743i i iY b b X X= + = +
Excel OutputCoefficients
Intercept 2.14285714X Variable 0.74285714
0
1
2
3
4
5
6
7
8
0 1 2 3 4 5 6X
Sale
s
Projected to year 2000
Linear trend forecasting equation:
Statistics for Managers Using Microsoft Excel, 2/e © 1999 Prentice-Hall, Inc.
Chapter 16 Student Lecture Notes 16-10
© 2004 Prentice-Hall, Inc. Chap 16-28
The Quadratic Trend Model
Year Coded X Sales (Y)
94 0 2
95 1 5
96 2 2
97 3 2
98 4 7
99 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:
© 2004 Prentice-Hall, Inc. Chap 16-29
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 2000
© 2004 Prentice-Hall, Inc. Chap 16-30
The Exponential Trend Model
Coeff ic ientsInterc ept 0.33583795X V ariable 0.08068544
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)
94 0 2
95 1 5
96 2 2
97 3 2
98 4 7
99 5 6
After taking the logarithms, use the method of least squares to get the forecasting equation:
Statistics for Managers Using Microsoft Excel, 2/e © 1999 Prentice-Hall, Inc.
Chapter 16 Student Lecture Notes 16-11
© 2004 Prentice-Hall, Inc. Chap 16-31
The Least-Squares TrendModels in PHStat
Use PHStat | Simple Linear Regression for Linear Trend and Exponential Trend Models and PHStat | Multiple Regression for Quadratic Trend ModelExcel Spreadsheet for the Single Family Home Sales Example
Microsoft Excel Worksheet
© 2004 Prentice-Hall, Inc. Chap 16-32
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 −− = − = = −L
( ) ( ) ( ) ( )3 2 2 1 1 1 2n n n nY Y Y Y Y Y Y Y− − −− − − = = − − −⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦L
© 2004 Prentice-Hall, Inc. Chap 16-33
Model Selection Using Differences
3 2 12 1
1 2 1
100% 100% 100%n n
n
Y Y Y YY YY Y Y
−
−
⎛ ⎞⎛ ⎞ ⎛ ⎞− −−= = = ⎜ ⎟⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠ ⎝ ⎠L
Use an Exponential Trend Model If the Percentage Differences are More or Less Constant
(continued)
Statistics for Managers Using Microsoft Excel, 2/e © 1999 Prentice-Hall, Inc.
Chapter 16 Student Lecture Notes 16-12
© 2004 Prentice-Hall, Inc. Chap 16-34
The Holt-Winters MethodSimilar to Exponential SmoothingAdvantages Over Exponential Smoothing
Can detect future trend and overall movementCan 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 levelsSmaller values of V give more weight to the current trends and less weight to past trends
© 2004 Prentice-Hall, Inc. Chap 16-35
The Holt-Winters Method( ) ( )
( )( )1 1
1 1
1
1
Level: 1
Trend: 1: level of smoothed series in time period
: level of smoothed series in time period 1: value of trend component in time period
: val
i i i i
i i i i
i
i
i
i
E U E T U Y
T VT V E EE iE iT iT
− −
− −
−
−
= + + −
= + − −
−
2 2 2 2 1
ue of trend component in time period 1: observed value of the time series in period : smoothing constant (where 0 1): smoothing constant (where 0 1)
and
i
iY iU UV VE Y T Y Y
−
< << <
= = −
© 2004 Prentice-Hall, Inc. Chap 16-36
The Holt-Winters Method: Example
.2(2.77)+.8(6.51-5.8)=1.12.2(5.8+2.77)+.8(6)=6.51699
.2(-1.07)+.8(5.8-2.07)=2.77.2(2.07-1.07)+.8(7)=5.8798
.2(-.84)+.8(2.07-3.2)=-1.07.2(3.2-.84)+.8(2)=2.07297
.2(3)+.8(3.2-5)=-.84.2(5+3)+.8(2)=3.2296
5-2=35595
NANA294
Trend (Ti )V = .2
Level (Ei )U =.2
Sales (Yi )
Year
( ) ( )( )( )
1 1
1 1
Level: 1
Trend: 1i i i i
i i i i
E U E T U Y
T VT V E E− −
− −
= + + −
= + − −
Statistics for Managers Using Microsoft Excel, 2/e © 1999 Prentice-Hall, Inc.
Chapter 16 Student Lecture Notes 16-13
© 2004 Prentice-Hall, Inc. Chap 16-37
The Holt-Winters Method:Forecasting
( )ˆ
ˆwhere : forecasted value years into the future
: level of smoothed series in period : value of trend component in period : number of years int
n j n n
n j
n
n
Y E j T
Y j
E nT nj
+
+
= +
o the future
( ) ( )
( ) ( )
00 99 99
05 99 99
Year 00: 1ˆ 1 6.51 1 1.12 7.638Year 05: 6ˆ 6 6.51 6 1.12 13.26
j
Y E Tj
Y E T
=
= + = + =
=
= + = + =
© 2004 Prentice-Hall, Inc. Chap 16-38
Holt-Winters Method:Plot of Series and Forecasts
Excel Spreadsheet with the Computation
0
2
4
6
8
10
12
14
1 2 3 4 5 6 7 8 9 10 11 12
Level (E)Series (Y)
Forecasts for 2000 to 2005
1994
Microsoft Excel Worksheet
© 2004 Prentice-Hall, Inc. Chap 16-39
Autoregressive Modeling
Used for ForecastingTakes Advantage of Autocorrelation
1st order - correlation between consecutive values2nd 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 A Y A Y A Y δ− − −= + + + + +L
Random Error
Statistics for Managers Using Microsoft Excel, 2/e © 1999 Prentice-Hall, Inc.
Chapter 16 Student Lecture Notes 16-14
© 2004 Prentice-Hall, Inc. Chap 16-40
Autoregressive Model: Example
Year Units 93 494 395 296 3 97 2 98 2 99 400 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.
© 2004 Prentice-Hall, Inc. Chap 16-41
Autoregressive Model: Example Solution
Year Yi Yi-1 Yi-293 4 --- ---94 3 4 ---95 2 3 496 3 2 397 2 3 298 2 2 399 4 2 200 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
© 2004 Prentice-Hall, Inc. Chap 16-42
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 .9375ˆ 3.5 .8125 .9375 3.5 .8125 6 .9375 4 4.625
i i iY Y Y
Y Y Y− −= + −
= + −= + × − ×=
Statistics for Managers Using Microsoft Excel, 2/e © 1999 Prentice-Hall, Inc.
Chapter 16 Student Lecture Notes 16-15
© 2004 Prentice-Hall, Inc. Chap 16-43
Autoregressive Model in PHStat
PHStat | Multiple Regression
Excel Spreadsheet for the Office Units Example
Microsoft Excel Worksheet
© 2004 Prentice-Hall, Inc. Chap 16-44
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 ApIf null hypothesis rejected, this model is selectedIf null hypothesis not rejected, decrease p by 1 and repeat
© 2004 Prentice-Hall, Inc. Chap 16-45
Selecting a Forecasting Model
Perform a Residual AnalysisLook 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
Statistics for Managers Using Microsoft Excel, 2/e © 1999 Prentice-Hall, Inc.
Chapter 16 Student Lecture Notes 16-16
© 2004 Prentice-Hall, Inc. Chap 16-46
Residual Analysis
Random errors
Trend not accounted for
Cyclical effects not accounted for
Seasonal effects not accounted for
Time Time
Time Time
e e
e e
0 0
0 0
© 2004 Prentice-Hall, Inc. Chap 16-47
Measuring Errors
Choose a Model that Gives the Smallest Measuring ErrorsSum Square Error (SSE)
Sensitive to outliers
( )2
1
ˆn
i ii
SSE Y Y=
= −∑
© 2004 Prentice-Hall, Inc. Chap 16-48
Measuring Errors
Mean Absolute Deviation (MAD)
Not sensitive to extreme observations
(continued)
1
ˆn
i ii
Y YMAD
n=
−=∑
Statistics for Managers Using Microsoft Excel, 2/e © 1999 Prentice-Hall, Inc.
Chapter 16 Student Lecture Notes 16-17
© 2004 Prentice-Hall, Inc. Chap 16-49
Principle of Parsimony
Suppose 2 or More Models Provide Good Fit to DataSelect the Simplest Model
Simplest model types:Least-squares linearLeast-squares quadratic1st order autoregressive
More complex types:2nd and 3rd order autoregressiveLeast-squares exponentialHolt-Winters Model
© 2004 Prentice-Hall, Inc. Chap 16-50
Forecasting with Seasonal Data
Use Categorical Predictor Variables with Least-Squares Trend FittingForecasting Equation (Exponential Model with Quarterly Data):
The bj provides the multiplier for the j -th quarter relative to the 4th quarterQj = 1 if j -th quarter and 0 if notXi = the coded variable denoting the time period i
1 2 30 1 2 3 4
ˆ iX Q Q QiY b b b b b=
© 2004 Prentice-Hall, Inc. Chap 16-51
Forecasting with QuarterlyData: Example
445 .77444 .27462 .69459 .27
5 0 0 .7 15 4 4 .7 55 8 4 .4 16 1 5 .9 3
6 4 5 .56 7 0 .6 36 8 7 .3 17 4 0 .7 4
7 5 7 .1 28 8 5 .1 49 4 7 .2 89 7 0 .4 3
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 0.019051226Observations 16
Statistics for Managers Using Microsoft Excel, 2/e © 1999 Prentice-Hall, Inc.
Chapter 16 Student Lecture Notes 16-18
© 2004 Prentice-Hall, Inc. Chap 16-52
Forecasting with QuarterlyData: Example
(continued)Excel Output
10 10 0 10 1 1 10 2
1
ˆlog log log log 2.6106 0.02405 0.00453
i i
i
Y b X b Q bX Q
= + += + +
Regression equation for the first quarters:
Coefficients Standard Error t Stat P-valueIntercept 2.610625646 0.013513283 193.1895916 8.96553E-21Coded X 0.024047968 0.001064996 22.58033882 1.44859E-10Q1 0.00452606 0.013844947 0.326910637 0.749871743Q2 0.010373368 0.013638602 0.760588787 0.462894872Q3 0.008400302 0.013513283 0.621632977 0.546850376
( ) ( )1 12.6106 0.02405 0.00453ˆ 10 10 10X Q
iY =
© 2004 Prentice-Hall, Inc. Chap 16-53
Forecasting with QuarterlyData: Example
(continued)
1st quarter of 1998:
( ) ( )
( ) ( )
10 1998,1
2.9999191998,1
16 12.6106 0.02405 0.004531998,1
ˆlog 2.6106 .02405 16 0.00453 1 2.999919ˆ 10 999.814
ˆor 10 10 10 999.814
Y
Y
Y
= + + =
= =
= =
1st quarter of 1994:
( ) ( )
( ) ( )
10 1994,1
2.6151521994,1
0 12.6106 0.02405 0.004531994,1
ˆlog 2.6106 .02405 0 0.00453 1 2.615152ˆ 10 412.2415
ˆor 10 10 10 412.2415
Y
Y
Y
= + + =
= =
= =
© 2004 Prentice-Hall, Inc. Chap 16-54
Forecasting with Quarterly Data in PHStat
Use PHStat | Multiple Regression
Excel Spreadsheet for the Stock Price Index Example
Microsoft Excel Worksheet
Statistics for Managers Using Microsoft Excel, 2/e © 1999 Prentice-Hall, Inc.
Chapter 16 Student Lecture Notes 16-19
© 2004 Prentice-Hall, Inc. Chap 16-55
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
© 2004 Prentice-Hall, Inc. Chap 16-56
Simple Price Index
Selection of the Base PeriodShould be a period of economic stability rather than one at or near the peak of an expanding economy or declining economyShould be recent so that comparisons are not greatly affected by changing technology and consumer attitudes or habits
base
base
100
where = simple price index for year = price for year
= price for the base year
ii
i
i
PIP
I iP iP
⎛ ⎞= ⎜ ⎟⎝ ⎠
© 2004 Prentice-Hall, Inc. Chap 16-57
Simple Price Index: Example
Given the prices (in dollars per pound) for apples, construct the simple price index using 1980 as the base year.
Year Price1980 0.692 (0.692/0.692)100 = 100.001985 0.684 (0.684/0.692)100 = 98.841990 0.719 (0.719/0.692)100 = 103.901995 0.835 (0.835/0.692)100 = 120.662000 0.896 (0.896/0.692)100 = 129.48
Simple Price Index
basePBase Year
( )iP ( )iI
Statistics for Managers Using Microsoft Excel, 2/e © 1999 Prentice-Hall, Inc.
Chapter 16 Student Lecture Notes 16-20
© 2004 Prentice-Hall, Inc. Chap 16-58
Shifting the Base
oldnew
new base
new
old
new base
100
where = new price index = old price index
= value of the old price index for the new base year
IIIIII
⎛ ⎞= ⎜ ⎟⎝ ⎠
© 2004 Prentice-Hall, Inc. Chap 16-59
Shifting the Base: Example
Year Price
1980 0.692 (100.00/129.48)100 = 77.231985 0.684 (98.84/129.48)100 = 76.341990 0.719 (103.90/129.48)100 = 80.251995 0.835 (120.66/129.48)100 = 93.192000 0.896 (129.48/129.48)100 = 100.00
Simple Price Index Simple Price Index
100.0098.84
(base = 2000)
103.90120.66129.48
(base = 1980)
Change the base year of the simple price index of apples from 1980 to 2000:
new baseI newIoldINew Base Year
© 2004 Prentice-Hall, Inc. Chap 16-60
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 PastAffects the Cost of Living and/or the Quality of Life for a Large Number of ConsumersTwo Basic Types
Unweighted aggregate price indexWeighted aggregate price index
Statistics for Managers Using Microsoft Excel, 2/e © 1999 Prentice-Hall, Inc.
Chapter 16 Student Lecture Notes 16-21
© 2004 Prentice-Hall, Inc. Chap 16-61
Unweighted Aggregate Price Index
( )( )
( )
( )
10
1
1
100
where = time period (0, 1, 2, ) = total number of items under consideration
= sum of the prices paid for each of the comm
tnt i i
U ni i
tni i
PIP
tn
Pn
=
=
=
⎛ ⎞Σ= ⎜ ⎟
Σ⎝ ⎠
Σ
L
( )
( )
01
odities at time period
= sum of the prices paid for each of the commodities at time period 0
= value of the unweighted price index at tim
ni i
tU
t
Pn
I
=Σ
e t
© 2004 Prentice-Hall, Inc. Chap 16-62
Unweighted Aggregate Price Index
Easy to ComputeTwo Distinct Shortcomings
Each commodity in the group is treated as equally important so that the most expensive commodities per unit can overly influence the indexNot all commodities are consumed at the same rate, but they are treated the same by the index
(continued)
© 2004 Prentice-Hall, Inc. Chap 16-63
Year tApples Bananas Oranges
1980 0 0.692 0.342 0.3651985 1 0.684 0.367 0.5331990 2 0.719 0.463 0.571995 3 0.835 0.49 0.6252000 4 0.896 0.491 0.843
113.22125.23139.39159.40
Unweighted Aggregate
100.00
PricePrice Index
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:
( )01 0.692 0.342 0.365 1.399n
i iP=Σ = + + =
( )tUI
Base Year
( )41 0.896 0.491 0.843 2.230n
i iP=Σ = + + =( )
( )
( )
3 44 1
3 01
2.230 100 159.41.399
iiU
ii
PI
P=
=
⎛ ⎞= = =⎜ ⎟⎝ ⎠
∑∑
Statistics for Managers Using Microsoft Excel, 2/e © 1999 Prentice-Hall, Inc.
Chapter 16 Student Lecture Notes 16-22
© 2004 Prentice-Hall, Inc. Chap 16-64
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 ItemAccount 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 indexThe Paasche price index
© 2004 Prentice-Hall, Inc. Chap 16-65
Laspeyres Price Index
Uses the Consumption Quantities Associated with the Base Year
( )( ) ( )
( ) ( )
( )
( )
01
0 01
0
100
where = time period (0, 1, 2, ) = total number of items under consideration
= quantity of item at time period 0
= value of the Laspe
tnt i i i
L ni i i
i
tL
P QIP Q
tn
Q i
I
=
=
⎛ ⎞Σ= ⎜ ⎟
Σ⎝ ⎠L
yres price index at time t
© 2004 Prentice-Hall, Inc. Chap 16-66
Laspeyres Price Index: ExampleGiven 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:
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )3 0 01
0.692 19.2 0.342 20.2 0.365 14.3 25.4143i iiP Q
== + + =∑
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )3 4 01
0.896 19.2 0.491 20.2 0.843 14.3 39.1763i iiP Q
== + + =∑
Year t Laspeyres P Q P Q P Q Price Index
1980 0 0.692 19.2 0.342 20.2 0.365 14.3 100.001985 1 0.684 17.3 0.367 23.5 0.533 11.6 110.841990 2 0.719 19.6 0.463 24.4 0.57 12.4 123.191995 3 0.835 18.9 0.49 27.4 0.625 12 137.202000 4 0.896 18.8 0.491 31.4 0.843 8.6 154.15
Apple Bananas Oranges
( )4 39.1763 100 154.1525.4143LI ⎛ ⎞= =⎜ ⎟
⎝ ⎠
Statistics for Managers Using Microsoft Excel, 2/e © 1999 Prentice-Hall, Inc.
Chapter 16 Student Lecture Notes 16-23
© 2004 Prentice-Hall, Inc. Chap 16-67
Paasche Price Index
Uses the Consumption Quantities Experienced in the Year of Interest Instead of Using the Initial Quantities
( )( ) ( )
( ) ( )
( )
( )
10 0
1
100
where = time period (0, 1, 2, ) = total number of items under consideration
= quantity of item at time period
= value of the Paasc
t tnt i i i
P ni i i
ti
tP
P QIP Q
tn
Q i t
I
=
=
⎛ ⎞Σ= ⎜ ⎟
Σ⎝ ⎠L
he price index at time t
© 2004 Prentice-Hall, Inc. Chap 16-68
Paasche Price Index
AdvantageA more accurate reflection of total consumption costs at the point of interest in time
DisadvantagesAccurate consumption values for current purchases are often hard to obtainIf 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
(continued)
© 2004 Prentice-Hall, Inc. Chap 16-69
Paasche Price Index: ExampleGiven 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:
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )3 0 01
0.692 19.2 0.342 20.2 0.365 14.3 25.4143i iiP Q
== + + =∑
( )4 39.5120 100 155.4725.4143PI ⎛ ⎞= =⎜ ⎟
⎝ ⎠
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )3 4 41
0.896 18.8 0.491 31.4 0.843 8.6 39.5120i iiP Q
== + + =∑
Year t Laspeyres P Q P Q P Q Price Index
1980 0 0.692 19.2 0.342 20.2 0.365 14.3 100.001985 1 0.684 17.3 0.367 23.5 0.533 11.6 104.821990 2 0.719 19.6 0.463 24.4 0.57 12.4 127.711995 3 0.835 18.9 0.49 27.4 0.625 12 144.442000 4 0.896 18.8 0.491 31.4 0.843 8.6 155.47
Apple Bananas Oranges
Statistics for Managers Using Microsoft Excel, 2/e © 1999 Prentice-Hall, Inc.
Chapter 16 Student Lecture Notes 16-24
© 2004 Prentice-Hall, Inc. Chap 16-70
Pitfalls Concerning Time-Series Forecasting
Taking for Granted the Mechanism that Governs the Time Series Behavior in the Past Will Still Hold in the FutureUsing Mechanical Extrapolation of the Trend to Forecast the Future Without Considering Personal Judgments, Business Experiences, Changing Technologies, Habits, Etc.
© 2004 Prentice-Hall, Inc. Chap 16-71
Chapter Summary
Discussed the Importance of ForecastingAddressed Component Factors of the Time-Series ModelPerformed Smoothing of Data Series
Moving averagesExponential smoothing
Described Least-Squares Trend Fitting and Forecasting
Linear, quadratic and exponential models
© 2004 Prentice-Hall, Inc. Chap 16-72
Chapter Summary
Discussed Holt-Winters Method of Trend Fitting and ForecastingAddressed Autoregressive ModelsDescribed Procedure for Choosing Appropriate ModelsAddressed Time-Series Forecasting of Monthly or Quarterly Data (Use of Dummy-Variables)Discussed Pitfalls Concerning Time-Series ForecastingDescribed Index Numbers
(continued)