Basic Business Statistics
(9th Edition)
Chapter 16
Time-Series Forecasting and
Index Numbers
© 2004 Prentice-Hall, Inc.
Chap 16-1
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
© 2004 Prentice-Hall, Inc.
Chap 16-2
Chapter Topics






(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
© 2004 Prentice-Hall, Inc.
Chap 16-3
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
© 2004 Prentice-Hall, Inc.
Chap 16-4
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
© 2004 Prentice-Hall, Inc.
Chap 16-5
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:1994 1995 1996 1997 1998
Sales:
75.3 74.2 78.5 79.7 80.2
© 2004 Prentice-Hall, Inc.
Chap 16-6
Time-Series Components
Trend
Cyclical
Time-Series
Seasonal
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Irregular
Chap 16-7
Trend Component


Overall Upward or Downward Movement
Data Taken Over a Period of Years
Sales
© 2004 Prentice-Hall, Inc.
Time
Chap 16-8
Cyclical Component



Upward or Downward Swings
May Vary in Length
Usually Lasts 2 - 10 Years
Sales
© 2004 Prentice-Hall, Inc.
Chap 16-9
Seasonal Component



Upward or Downward Swings
Regular Patterns
Observed Within 1 Year
Sales
Summer
Winter
Spring
Fall
Time (Monthly or Quarterly)
© 2004 Prentice-Hall, Inc.
Chap 16-10
Irregular or Random Component


Erratic, Nonsystematic, Random, “Residual”
Fluctuations
Due to Random Variations of



Nature
Accidents
Short Duration and Non-Repeating
© 2004 Prentice-Hall, Inc.
Chap 16-11
Example: Quarterly Retail Sales
with Seasonal Components
Quarterly with Seasonal Components
25
20
Sales
15
10
5
0
0
© 2004 Prentice-Hall, Inc.
5
10
15
20
Time
25
30
35
Chap 16-12
Example: Quarterly Retail Sales with
Seasonal Components Removed
Quarterly without Seasonal Com ponents
25
20
Sales
15
Y(t)
10
5
0
0
© 2004 Prentice-Hall, Inc.
5
10
15
20
Tim e
25
30
35
Chap 16-13
Multiplicative Time-Series Model




Used Primarily for Forecasting
Observed Value in Time Series is
of Components
For Annual Data:
Yi  Ti Ci I i
For Quarterly or Monthly Data:
Yi  Ti Si Ci I i
the Product
Ti = Trend
Ci = Cyclical
Ii = Irregular
Si = Seasonal
© 2004 Prentice-Hall, Inc.
Chap 16-14
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
© 2004 Prentice-Hall, Inc.
Chap 16-15
Moving Averages

(continued)
Example: 3-Year Moving Average

Y1  Y2  Y3
First average: MA(3) 
3

Y2  Y3  Y4
Second average: MA(3) 
3
© 2004 Prentice-Hall, Inc.
Chap 16-16
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.
© 2004 Prentice-Hall, Inc.
Year
Units
Moving
Ave
1994
2
NA
1995
5
3
1996
2
3
1997
2
3.67
1998
7
5
1999
6
NA
Chap 16-17
Moving Average Example
Solution
Year
Response
Moving
Ave
Sales
L=3
1994
2
NA
8
1995
5
3
6
1996
2
3
4
1997
2
3.67
2
1998
7
5
1999
6
NA
0
94 95 96 97 98 99
No MA for the first and last (L-1)/2 years
© 2004 Prentice-Hall, Inc.
Chap 16-18
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
© 2004 Prentice-Hall, Inc.
Chap 16-19
Example: 5-Period Moving
Averages of Quarterly Retail Sales
Quarterly 5-Period Moving Averages
25
Sales
20
15
MA(5)
Y(t)
10
5
0
0
© 2004 Prentice-Hall, Inc.
5
10
15
20
Time
25
30
35
Chap 16-20
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


© 2004 Prentice-Hall, Inc.
Close to 0 for smoothing out unwanted cyclical and
irregular components
Close to 1 for forecasting
Chap 16-21
Exponential Weight: Example
Ei  WYi  (1  W ) Ei 1
Year
Response Smoothing Value
Forecast
(W = .2, (1-W)=.8)
1994
2
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
© 2004 Prentice-Hall, Inc.
2
NA
Chap 16-22
Exponential Weight:
Example Graph
Sales
8
Data
6
4
Smoothed
2
0
94
© 2004 Prentice-Hall, Inc.
95
96
97
98
99
Year
Chap 16-23
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
© 2004 Prentice-Hall, Inc.
Chap 16-24
Example: Exponential Smoothing
of Real GNP

The Excel Spreadsheet with the Real GDP
Data and the Exponentially Smoothed Series
© 2004 Prentice-Hall, Inc.
Chap 16-25
Linear Trend Model
Use the method of least squares to obtain the
linear trend forecasting equation:
Year Coded X Sales (Y)
94
0
2
95
1
5
96
2
2
97
3
2
98
4
7
99
5
6
© 2004 Prentice-Hall, Inc.
Yˆi  b0  b1 X i
Chap 16-26
Linear Trend Model
(continued)
Linear trend forecasting equation:
Yˆi  b0  b1 X i  2.143  .743 X i
8
Excel Output
7
Co efficien ts
6
2.14285714
X V a ria b le 1 0 .7 4 2 8 5 7 1 4
5
Sales
I n te r c e p t
4
Projected to
year 2000
3
2
1
0
0
© 2004 Prentice-Hall, Inc.
1
2
X
3
4
5
6
Chap 16-27
The Quadratic Trend Model
Use the method of least squares to obtain
the quadratic trend forecasting equation:
Year Coded X Sales (Y)
94
0
2
95
1
5
96
2
2
97
3
2
98
4
7
99
5
6
© 2004 Prentice-Hall, Inc.
2
ˆ
Yi  b0  b1 X i  b2 X i
Chap 16-28
The Quadratic Trend Model
(continued)
2
2
ˆ
Yi  b0  b1 X i  b2 X i  2.857  .33X i  .214 X i
Excel Output
8
7
In te rc e p t
2 .8 5 7 1 4 2 8 6
6
X V a ria b le 1
-0 .3 2 8 5 7 1 4
5
X V a ria b le 2
0 .2 1 4 2 8 5 7 1
Sales
Coefficients
Projected to
year 2000
4
3
2
1
0
0
© 2004 Prentice-Hall, Inc.
1
2
X
3
4
5
6
Chap 16-29
The Exponential Trend Model
After taking the logarithms, use the method of least
squares to get the forecasting equation:
Xi
ˆ
Yi  b0b1
or
log Yˆi  log b0  X1 log b1
C o e f f ic ie n t s
Year Coded X Sales (Y)
94
0
2
In t e r c e p t
0 .3 3 5 8 3 7 9 5
X V a ria b le 10 . 0 8 0 6 8 5 4 4
95
1
5
96
2
2
Excel Output of Values in Logs
97
3
2
a n t ilo g (. 3 3 5 8 3 7 9 5 ) =
2.17
98
4
7
a n t ilo g (. 0 8 0 6 8 5 4 4 ) =
1.2
99
5
6
© 2004 Prentice-Hall, Inc.
Xi
ˆ
Yi  (2.17)(1.2)
Chap 16-30
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
© 2004 Prentice-Hall, Inc.
Chap 16-31
Model Selection Using Differences

Use a Linear Trend Model If the First
Differences are More or Less Constant
Y2  Y1  Y3  Y2 

 Yn  Yn1
Use a Quadratic Trend Model If the Second
Differences are More or Less Constant
Y3  Y2   Y2  Y1   
© 2004 Prentice-Hall, Inc.
 Yn  Yn 1   Yn 1  Yn 2 
Chap 16-32
Model Selection Using Differences
(continued)

Use an Exponential Trend Model If the
Percentage Differences are More or Less
Constant
 Y2  Y1 
 Y3  Y2 

100%  
100% 
 Y1 
 Y2 
© 2004 Prentice-Hall, Inc.
 Yn  Yn1 

100%
 Yn1 
Chap 16-33
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
© 2004 Prentice-Hall, Inc.
Chap 16-34
The Holt-Winters Method
Level: Ei  U  Ei 1  Ti 1   1  U  Yi
Trend: Ti  VTi 1  1  V  Ei  Ei 1 
Ei : level of smoothed series in time period i
Ei 1 : level of smoothed series in time period i  1
Ti : value of trend component in time period i
Ti 1 : value of trend component in time period i  1
Yi : observed value of the time series in period i
U : smoothing constant (where 0  U  1)
V : smoothing constant (where 0  V  1)
E2  Y2 and T2  Y2  Y1
© 2004 Prentice-Hall, Inc.
Chap 16-35
The Holt-Winters Method:
Example
Level: Ei  U  Ei 1  Ti 1   1  U  Yi
Trend: Ti  VTi 1  1  V  Ei  Ei 1 
Year Sales
(Yi )
Level (Ei )
U =.2
Trend (Ti )
V = .2
94
2
NA
NA
95
5
5
5-2=3
96
2
.2(5+3)+.8(2)=3.2
.2(3)+.8(3.2-5)=-.84
97
2
.2(3.2-.84)+.8(2)=2.07
.2(-.84)+.8(2.07-3.2)=-1.07
98
7
.2(2.07-1.07)+.8(7)=5.8
.2(-1.07)+.8(5.8-2.07)=2.77
99
6
.2(5.8+2.77)+.8(6)=6.51
.2(2.77)+.8(6.51-5.8)=1.12
© 2004 Prentice-Hall, Inc.
Chap 16-36
The Holt-Winters Method:
Forecasting
Yˆn  j  En  j Tn 
where Yˆn  j : forecasted value j years into the future
En : level of smoothed series in period n
Tn : value of trend component in period n
j : number of years into the future
Year 00: j  1
Yˆ00  E99  1  T99   6.51  1 1.12   7.638
Year 05: j  6
Yˆ  E  6  T
05
© 2004 Prentice-Hall, Inc.
99
99
  6.51  6 1.12   13.26
Chap 16-37
Holt-Winters Method:
Plot of Series and Forecasts

Excel Spreadsheet with the Computation
14
12
10
8
1994
Level (E)
Series (Y)
6
Forecasts for
2000 to 2005
4
2
0
1
© 2004 Prentice-Hall, Inc.
2
3
4
5
6
7
8
9 10 11 12
Chap 16-38
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:
Yi  A0  AY
1 i 1  A2Yi  2 
 ApYi  p   i
Random
Error
© 2004 Prentice-Hall, Inc.
Chap 16-39
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
93
94
95
96
97
98
99
00
© 2004 Prentice-Hall, Inc.
4
3
2
3
2
2
4
6
Chap 16-40
Autoregressive Model:
Example Solution
Develop the 2nd order
table
Use Excel to estimate a
regression model
Excel Output
Coefficients
I n te rc e p t
3.5
X V a ri a b l e 1
0.8125
X V a ri a b l e 2
-0 . 9 3 7 5
Year
93
94
95
96
97
98
99
00
Yi
4
3
2
3
2
2
4
6
Yi-1
--4
3
2
3
2
2
4
Yi-2
----4
3
2
3
2
2
Yˆi  3.5  .8125Yi 1  .9375Yi 2
© 2004 Prentice-Hall, Inc.
Chap 16-41
Autoregressive Model Example:
Forecasting
Use the 2nd order model to forecast number
of units for 2001:
Yˆi  3.5  .8125Yi 1  .9375Yi 2
Yˆ2001  3.5  .8125Y2000  .9375Y1999
 3.5  .8125  6  .9375  4
 4.625
© 2004 Prentice-Hall, Inc.
Chap 16-42
Autoregressive Model in PHStat


PHStat | Multiple Regression
Excel Spreadsheet for the Office Units
Example
© 2004 Prentice-Hall, Inc.
Chap 16-43
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
© 2004 Prentice-Hall, Inc.
Chap 16-44
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
© 2004 Prentice-Hall, Inc.
Chap 16-45
Residual Analysis
e
e
0
0
Time
Random errors
Time
Cyclical effects not accounted for
e
e
0
0
Time
Time
Trend not accounted for
Seasonal effects not accounted for
© 2004 Prentice-Hall, Inc.
Chap 16-46
Measuring Errors


Choose a Model that Gives the Smallest
Measuring Errors
Sum Square
Error
(SSE)
n
2
ˆ
 SSE 
 Yi  Yi
i 1



Sensitive to outliers
© 2004 Prentice-Hall, Inc.
Chap 16-47
Measuring Errors

(continued)
Mean Absolute
Deviation
(MAD)
n
ˆ
Y

Y

i
i

MAD  i 1
n

Not sensitive to extreme observations
© 2004 Prentice-Hall, Inc.
Chap 16-48
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:



© 2004 Prentice-Hall, Inc.
2nd and 3rd order autoregressive
Least-squares exponential
Holt-Winters Model
Chap 16-49
Forecasting with Seasonal Data


Use Categorical Predictor Variables with LeastSquares Trend Fitting
Forecasting Equation (Exponential Model with
Quarterly Data):
Xi
Q1 Q2
Q3
ˆ
Yi  b0b1 b2 b3 b4



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
© 2004 Prentice-Hall, Inc.
Chap 16-50
Forecasting with Quarterly
Data: Example
Standards and Poor’s Composite Stock Price Index:
Quarter
1995
1996
1997
I
4 4 5 .7 7
5 0 0 .7 1
6 4 5 .5
7 5 7 .1 2
2
4 4 4 .2 7
5 4 4 .7 5
6 7 0 .6 3
8 8 5 .1 4
3
4 6 2 .6 9
5 8 4 .4 1
6 8 7 .3 1
9 4 7 .2 8
4
4 5 9 .2 7
6 1 5 .9 3
7 4 0 .7 4
9 7 0 .4 3
Excel Output
© 2004 Prentice-Hall, Inc.
1994
Regression Statistics
Multiple R
0.989936819
R Square
0.979974906
Adjusted R Square 0.972693054
Standard Error
0.019051226
Observations
16
r2 is .98
Appears to be
an excellent fit.
Chap 16-51
Forecasting with Quarterly
Data: Example
(continued)
Excel Output
Intercept
Coded X
Q1
Q2
Q3
Coefficients Standard Error
2.610625646
0.013513283
0.024047968
0.001064996
0.00452606
0.013844947
0.010373368
0.013638602
0.008400302
0.013513283
t Stat
193.1895916
22.58033882
0.326910637
0.760588787
0.621632977
P-value
8.96553E-21
1.44859E-10
0.749871743
0.462894872
0.546850376
Regression equation for the first quarters:
log10 Yˆi  log10 b0  X i log10 b1  Q1 log10 b2
 2.6106  0.02405 X i  0.00453Q1
Yˆi  10
© 2004 Prentice-Hall, Inc.
2.6106
10
 10
0.02405 X1

0.00453 Q1
Chap 16-52
Forecasting with Quarterly
Data: Example
(continued)
1st quarter of 1994:
log10 Yˆ1994,1  2.6106  .02405  0   0.00453 1  2.615152
Yˆ1994,1  102.615152  412.2415
2.6106
0.02405 0
0.00453 1
ˆ
or Y1994,1  10
10  10   412.2415
1st quarter of 1998:
log10 Yˆ1998,1  2.6106  .02405 16   0.00453 1  2.999919
Yˆ1998,1  102.999919  999.814
or
2.6106
0.02405 16
0.00453 1
ˆ
Y1998,1  10
10  10   999.814
© 2004 Prentice-Hall, Inc.
Chap 16-53
Forecasting with Quarterly Data
in PHStat


Use PHStat | Multiple Regression
Excel Spreadsheet for the Stock Price Index
Example
© 2004 Prentice-Hall, Inc.
Chap 16-54
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 16-55
Simple Price Index

 Pi 
Ii  
100
 Pbase 
where I i = simple price index for year i
Pi = price for year i
Pbase = price for the base year

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
© 2004 Prentice-Hall, Inc.
Chap 16-56
Simple Price Index: Example
Given the prices (in dollars per pound) for
apples, construct the simple price index using
1980 as the base year.
Pbase
Base Year
Year
1980
1985
1990
1995
2000
© 2004 Prentice-Hall, Inc.
Price  Pi 
0.692
0.684
0.719
0.835
0.896
Simple Price Index  I i 
(0.692/0.692)100 = 100.00
(0.684/0.692)100 = 98.84
(0.719/0.692)100 = 103.90
(0.835/0.692)100 = 120.66
(0.896/0.692)100 = 129.48
Chap 16-57
Shifting the Base

 I old 
I new  
 100
 I new base 
where I new = new price index
I old = old price index
I new base = value of the old price index
for the new base year
© 2004 Prentice-Hall, Inc.
Chap 16-58
Shifting the Base: Example
Change the base year of the simple price index
of apples from 1980 to 2000:
Year
Price
1980
1985
1990
1995
2000
0.692
0.684
0.719
0.835
0.896
New Base Year
© 2004 Prentice-Hall, Inc.
Simple Price Index
(base = 1980)
100.00
98.84
103.90
120.66
129.48
I new base
Simple Price Index
(base = 2000)
(100.00/129.48)100 =
77.23
(98.84/129.48)100 =
76.34
(103.90/129.48)100 =
80.25
(120.66/129.48)100 =
93.19
(129.48/129.48)100 =
100.00
I old
I new
Chap 16-59
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
© 2004 Prentice-Hall, Inc.
Chap 16-60
Unweighted Aggregate Price
Index

t  
n


P
t
IU    ni 1 i0  100
 i 1Pi 
where t = time period (0, 1, 2, )
n = total number of items under consideration
in1Pi   = sum of the prices paid for each of the
t
n commodities at time period t
in1Pi   = sum of the prices paid for each of the
0
n commodities at time period 0
IU  = value of the unweighted price index at time t
t
© 2004 Prentice-Hall, Inc.
Chap 16-61
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
© 2004 Prentice-Hall, Inc.
Chap 16-62
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:
in1Pi    0.692  0.342  0.365  1.399
0
Base Year
Year
t
1980
1985
1990
1995
2000
0
1
2
3
4
Apples
0.692
0.684
0.719
0.835
0.896
Price
Unweighted Aggregate
Bananas Oranges
Price Index IUt 
0.342
0.365
100.00
0.367
0.533
113.22
0.463
0.57
125.23
0.49
0.625
139.39
0.491
0.843
159.40

P
i1 i
3
 4
n
i 1 i
 P
 
I
 0.896  0.491  0.843  2.230 U 
© 2004 Prentice-Hall, Inc.
4

P
i1 i
3
4
0
 2.230 

100  159.4
 1.399 
Chap 16-63
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


© 2004 Prentice-Hall, Inc.
The Laspeyres price index
The Paasche price index
Chap 16-64
Laspeyres Price Index

t  0 
n

i 1Pi Qi
t 
I L   n 0 0  100
 i 1Pi Qi 
where t = time period (0, 1, 2,
)
n = total number of items under consideration
Qi  = quantity of item i at time period 0
0
t 
I L = value of the Laspeyres price index at time t

Uses the Consumption Quantities Associated
with the Base Year
© 2004 Prentice-Hall, Inc.
Chap 16-65
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:
   
P
i1 i Qi  0.69219.2  0.342 20.2  0.36514.3  25.4143
3
0
0
Year t
1980
1985
1990
1995
2000
0
1
2
3
4
Apple
P
Q
0.692 19.2
0.684 17.3
0.719 19.6
0.835 18.9
0.896 18.8
Bananas
P
Q
0.342 20.2
0.367 23.5
0.463 24.4
0.49 27.4
0.491 31.4
Oranges Laspeyres
P
Q Price Index
0.365 14.3
100.00
0.533 11.6
110.84
0.57 12.4
123.19
0.625
12
137.20
0.843 8.6
154.15
   
P
i1 i Qi  0.89619.2  0.491 20.2  0.84314.3  39.1763
3
4
0
4
IL
© 2004 Prentice-Hall, Inc.
 39.1763 

 100  154.15
 25.4143 
Chap 16-66
Paasche Price Index

t  t  
n

i 1Pi Qi
t 
I P   n 0 0  100
 i 1Pi Qi 
where t = time period (0, 1, 2,
)
n = total number of items under consideration
Qi  = quantity of item i at time period t
t
I P  = value of the Paasche price index at time t
t

Uses the Consumption Quantities Experienced
in the Year of Interest Instead of Using the
Initial Quantities
© 2004 Prentice-Hall, Inc.
Chap 16-67
Paasche Price Index

Advantage


(continued)
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
© 2004 Prentice-Hall, Inc.
Chap 16-68
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:
   
P
i1 i Qi  0.69219.2  0.342 20.2  0.36514.3  25.4143
3
0
0
Year
t
1980
1985
1990
1995
2000
0
1
2
3
4
Apple
P
Q
0.692 19.2
0.684 17.3
0.719 19.6
0.835 18.9
0.896 18.8
Bananas
P
Q
0.342 20.2
0.367 23.5
0.463 24.4
0.49 27.4
0.491 31.4
Oranges Laspeyres
P
Q Price Index
0.365 14.3
100.00
0.533 11.6
104.82
0.57 12.4
127.71
0.625
12
144.44
0.843 8.6
155.47
   
P
i1 i Qi  0.89618.8  0.49131.4  0.8438.6  39.5120
3
4
4
4
IP
© 2004 Prentice-Hall, Inc.
 39.5120 

 100  155.47
 25.4143 
Chap 16-69
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.
© 2004 Prentice-Hall, Inc.
Chap 16-70
Chapter Summary



Discussed the Importance of Forecasting
Addressed Component Factors of the TimeSeries Model
Performed Smoothing of Data Series



Moving averages
Exponential smoothing
Described Least-Squares Trend Fitting and
Forecasting

Linear, quadratic and exponential models
© 2004 Prentice-Hall, Inc.
Chap 16-71
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
© 2004 Prentice-Hall, Inc.
Chap 16-72