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Statistics for Managers
Using Microsoft® Excel
4th Edition
Chapter 15
Time-Series Forecasting and
Index Numbers
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.

Chap 15-1
Chapter Goals
After completing this chapter, you should be
able to:


Develop and implement basic forecasting models



Identify the components present in a time series



Use smoothing-based forecasting models, including
moving average and exponential smoothing



Apply trend-based forecasting models, including linear
trend and nonlinear trend

 complete time-series forecasting of seasonal data
Statistics for Managers Using
Microsoft Excel, 4einterpret basic index numbers
 compute and © 2004
Chap 15-2
Prentice-Hall, Inc.
The Importance of Forecasting


Governments forecast unemployment, interest
rates, and expected revenues from income taxes
for policy purposes



Marketing executives forecast demand, sales, and
consumer preferences for strategic planning



College administrators forecast enrollments to plan
for facilities and for faculty recruitment



Retail stores forecast demand to control inventory
levels, hire employees and provide training

Statistics for Managers Using
Microsoft Excel, 4e © 2004
Prentice-Hall, Inc.

Chap 15-3
Time-Series Data






Numerical data obtained at regular time
intervals
The time intervals can be annually, quarterly,
daily, hourly, etc.
Example:
Year:

1999 2000 2001 2002 2003

Sales:

75.3

Statistics for Managers Using
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74.2

78.5

79.7

80.2

Chap 15-4
Time-Series Plot
A time-series plot is a two-dimensional
plot of time series data
U.S. Inflation Rate

2001

1999

1997

1995

1993

1991

1989

1987

1985

1983

1981

1979

the horizontal axis
corresponds to the
time periods
Statistics for Managers Using
Microsoft Excel, 4e © 2004
Prentice-Hall, Inc.


1977

16.00
14.00
12.00
10.00
8.00
6.00
4.00
2.00
0.00
1975

the vertical axis
measures the variable
of interest

Inflation Rate (%)



Year

Chap 15-5
Time-Series Components
Time Series
Trend
Component

Seasonal
Component

Statistics for Managers Using
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Cyclical
Component

Irregular
Component

Chap 15-6
Trend Component


Long-run increase or decrease over time
(overall upward or downward movement)



Data taken over a long period of time

Sales

Statistics for Managers Using
Microsoft Excel, 4e © 2004
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d
rd tren
Up w a

Time 15-7
Chap
Trend Component
(continued)



Trend can be upward or downward
Trend can be linear or non-linear

Sales

Sales

Time

Statistics for Managers Using
Downward linear trend
Microsoft Excel, 4e © 2004
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Time
Upward nonlinear trend

Chap 15-8
Seasonal Component




Short-term regular wave-like patterns
Observed within 1 year
Often monthly or quarterly

Sales
Summer

Winter
Summer
Spring

Winter
Spring

Fall

Fall

Statistics for Managers Using
Microsoft Excel, 4e © 2004
Time (Quarterly)
Prentice-Hall, Inc.

Chap 15-9
Cyclical Component




Long-term wave-like patterns
Regularly occur but may vary in length
Often measured peak to peak or trough to
trough
1 Cycle

Sales

Statistics for Managers Using
Microsoft Excel, 4e © 2004
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Year
Chap 15-10
Irregular Component



Unpredictable, random, “residual” fluctuations
Due to random variations of





Nature
Accidents or unusual events

“Noise” in the time series

Statistics for Managers Using
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Chap 15-11
Multiplicative Time-Series Model
for Annual Data



Used primarily for forecasting
Observed value in time series is the product of
components

Yi = Ti × Ci × Ii
where

Ti = Trend value at year i
Ci = Cyclical value at year i

Ii = Irregular (random) value at year i
Statistics for Managers Using
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Chap 15-12
Prentice-Hall, Inc.
Multiplicative Time-Series Model
with a Seasonal Component



Used primarily for forecasting
Allows consideration of seasonal variation

Yi = Ti × Si × Ci × Ii
where

Ti = Trend value at time i
Si = Seasonal value at time i

Ci = Cyclical value at time i
Statistics for Managers Using
Ii =
Microsoft Excel, 4e © 2004 Irregular (random) value at time i
Chap 15-13
Prentice-Hall, Inc.
Smoothing the
Annual Time Series



Calculate moving averages to get an overall
impression of the pattern of movement over
time

Moving Average: averages of consecutive
time series values for a
chosen period of length L
Statistics for Managers Using
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Chap 15-14
Moving Averages






Used for smoothing
A series of arithmetic means over time
Result dependent upon choice of L (length of
period for computing means)
Examples:




For a 5 year moving average, L = 5
For a 7 year moving average, L = 7
Etc.

Statistics for Managers Using
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Chap 15-15
Moving Averages
(continued)


Example: Five-year moving average


First average:
MA(5) =



Y1 + Y2 + Y3 + Y4 + Y5
5

Second average:
MA(5) =

Y2 + Y3 + Y4 + Y5 + Y6
5

 etc.
Statistics for Managers Using
Microsoft Excel, 4e © 2004
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Chap 15-16
Example: Annual Data

1
2
3
4
5
6
7
8
9
10
11
Statistics
etc…

Sales
23
40
25
27
32
48
33
37
37
50
40
for Managers
etc…

Annual Sales
60
50

…

40
Sales

Year

30
20
10
0

Using
Microsoft Excel, 4e © 2004
Prentice-Hall, Inc.

1

2

3

4

5

6

7

8

9

10

11

Year

Chap 15-17

…
Calculating Moving Averages
Average
Year

5-Year
Moving
Average

Year

Sales

1

23

3

29.4

2

40

4

34.4

3

25

5

33.0

4

27

6

35.4

5

32

7

37.4

6

48

8

41.0

7

33

9

39.4

8

37

…

…

9

37

etc…

3=
29.4 =

1+ 2 + 3 + 4 + 5
5
23 + 40 + 25 + 27 + 32
5


10
50
Statistics for ManagersEach moving average is for a
Using
consecutive block of 5 years
11
40
Microsoft Excel, 4e © 2004
Chap 15-18
Prentice-Hall, Inc.
Annual vs. Moving Average
The 5-year
moving average
smoothes the
data and shows
the underlying
trend

Annual vs. 5-Year Moving Average

60
50
40
Sales



30
20
10
0
1

Statistics for Managers Using
Microsoft Excel, 4e © 2004
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2

3

4

5

6

7

8

9

10

11

Year
Annual

5-Year Moving Average

Chap 15-19
Exponential Smoothing


A weighted moving average





Weights decline exponentially
Most recent observation weighted most

Used for smoothing and short term
forecasting (often one period into the future)

Statistics for Managers Using
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Chap 15-20
Exponential Smoothing
(continued)


The weight (smoothing coefficient) is W






Subjectively chosen
Range from 0 to 1
Smaller W gives more smoothing, larger W gives
less smoothing

The weight is:




Close to 0 for smoothing out unwanted cyclical
and irregular components
Close to 1 for forecasting

Statistics for Managers Using
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Chap 15-21
Exponential Smoothing Model


Exponential smoothing model

E1 = Y1
Ei = WYi + (1 − W )Ei−1
For i = 2, 3, 4, …
where:
Ei = exponentially smoothed value for period i
Ei-1 = exponentially smoothed value already
computed for period i - 1
Statistics for Managers Using value in period i
Yi = observed
Microsoft Excel, 4e ©weight (smoothing coefficient), 0 < W < 1
W = 2004

Prentice-Hall, Inc.

Chap 15-22
Exponential Smoothing Example


Time
Period
(i)

Suppose we use weight W = .2
Sales
(Yi)

Forecast
from prior
period (Ei-1)

Exponentially Smoothed
Value for this period (Ei)

1
23
-23
2
40
23
(.2)(40)+(.8)(23)=26.4
3
25
26.4
(.2)(25)+(.8)(26.4)=26.12
4
27
26.12
(.2)(27)+(.8)(26.12)=26.296
5
32
26.296
(.2)(32)+(.8)(26.296)=27.437
6
48
27.437
(.2)(48)+(.8)(27.437)=31.549
7
33
31.549
(.2)(48)+(.8)(31.549)=31.840
8
37
31.840
(.2)(33)+(.8)(31.840)=32.872
9
37
32.872
(.2)(37)+(.8)(32.872)=33.697
Statistics for Managers Using
10
50
Microsoft Excel, 4e33.697
© 2004 (.2)(50)+(.8)(33.697)=36.958
etc.
etc.
etc.
etc.

Prentice-Hall, Inc.

E1 = Y1
since no
prior
information
exists
Ei =
WYi + (1 − W )Ei−1

Chap 15-23
Sales vs. Smoothed Sales



Fluctuations
have been
smoothed
NOTE: the
smoothed value in
this case is
generally a little low,
since the trend is
upward sloping and
the weighting factor
is only .2

60
50
40

Sales



30
20
10
0
1

Statistics for Managers Using
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2

3

4

5

6

7

Time Period
Sales

8

9

10

Smoothed

Chap 15-24
Forecasting Time Period i + 1
The smoothed value in the current
period (i) is used as the forecast value for
next period (i + 1) :


ˆ
Yi+1 = Ei
Statistics for Managers Using
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Chap 15-25
Exponential Smoothing in Excel


Use tools / data analysis /
exponential smoothing


The “damping factor” is (1 - W)

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Chap 15-26
Trend-Based Forecasting


Estimate a trend line using regression analysis
Time
Period
(X)

Sales
(Y)
1999
0
20
2000
1
40
2001
2
30
2002
3
50
2003
4
70
2004
5
65
Statistics for Managers Using
Microsoft Excel, 4e © 2004
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Year



Use time (X) as the
independent variable:

ˆ
Y = b0 + b1X

Chap 15-27
Trend-Based Forecasting
(continued)

Year

Time
Period
(X)



The linear trend forecasting equation is:

ˆ
Yi = 21.905 + 9.5714 Xi

Sales
(Y)

sales

1999
0
20
80
2000
1
40
70
60
2001
2
30
50
40
2002
3
50
30
20
2003
4
70
10
2004
5
65
0
Statistics for Managers Using0
Microsoft Excel, 4e © 2004
Prentice-Hall, Inc.

Sales trend

1

2

3

4

5

Year

Chap 15-28

6
Trend-Based Forecasting
(continued)


Sales
(y)

1999
2000
2001
2002
2003
2004
2005

0
1
2
3
4
5
6

20
40
30
50
70
65
??

ˆ
Y = 21.905 + 9.5714 (6)
= 79.33
Sales trend

sales

Year

Time
Period
(X)

Forecast for time period 6:

80
70
60
50
40
30
20
10
0

Statistics for Managers Using0
Microsoft Excel, 4e © 2004
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1

2

3

4

5

Year

Chap 15-29

6
Nonlinear Trend Forecasting


A nonlinear regression model can be used when
the time series exhibits a nonlinear trend



Quadratic form is one type of a nonlinear model:

Yi = β0 + β1Xi + β2 X + εi
2
i



Compare adj. r2 and standard error to that of
linear model to see if this is an improvement

 Can Managers Using
Statistics fortry other functional forms to get best fit
Microsoft Excel, 4e © 2004
Chap 15-30
Prentice-Hall, Inc.
Exponential Trend Model


Another nonlinear trend model:
Xi
0 1

Yi = β β


εi

Transform to linear form:

log(Yi ) = log(β0 ) + Xi log(β1 ) + log( ε i )
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Chap 15-31
Exponential Trend Model
(continued)


Exponential trend forecasting equation:

ˆ
log(Yi ) = b 0 + b1Xi
where

b0 = estimate of log(β0)
b1 = estimate of log(β1)

Interpretation:
ˆ
(β1 − 1) × 100% is the estimated annual compound

growth rate (in %)
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Chap 15-32
Model Selection Using
Differences


Use a linear trend model if the first differences
are approximately constant

(Y2 − Y1 ) = ( Y3 − Y2 ) =  = ( Yn − Yn-1 )


Use a quadratic trend model if the second
differences are approximately constant
[(Y3 − Y2 ) − ( Y2 − Y1 )] = [(Y4 − Y3 ) − ( Y3 − Y2 )]

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=  = [(Yn − Yn-1 ) − ( Yn-1 − Yn-2 )]
Chap 15-33
Model Selection Using
Differences


(continued)

Use an exponential trend model if the
percentage differences are approximately
constant

(Y3 − Y2 )
(Y2 − Y1 )
(Yn − Yn-1 )
× 100% =
× 100% =  =
× 100%
Y1
Y2
Yn-1

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Chap 15-34
Autoregressive Modeling



Used for forecasting
Takes advantage of autocorrelation





1st order - correlation between consecutive values
2nd order - correlation between values 2 periods
apart

pth order Autoregressive models:

Yi = A 0 + A 1Yi-1 + A 2 Yi-2 +  + + A p Yi-p + δi
Statistics for Managers Using
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Random
Error
Chap 15-35
Autoregressive Model:
Example
The Office Concept Corp. has acquired a number of office
units (in thousands of square feet) over the last eight years.
Develop the second order Autoregressive model.
Year

Units

97
4
98
3
99
2
00
3
01
2
02
2
03
4
04
6
Managers

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Using
Microsoft Excel, 4e © 2004
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Chap 15-36
Autoregressive Model:
Example Solution
 Develop the 2nd order
table
 Use Excel to estimate a
regression model
Excel Output

Coefficients
Intercept
3.5
X Variable 1
0.8125
X Variable 2
-0.9375

Year

Yi

Yi-1

Yi-2

97
98
99
00
01
02
03
04

4
3
2
3
2
2
4
6

-4
3
2
3
2
2
4

--4
3
2
3
2
2

Statistics 3.5 Managers Using
ˆ
Yi = for + 0.8125Yi−1 − 0.9375Yi−2
Microsoft Excel, 4e © 2004
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Chap 15-37
Autoregressive Model
Example: Forecasting
Use the second-order equation to forecast
number of units for 2005:
ˆ
Yi = 3.5 + 0.8125Yi−1 − 0.9375Yi−2
ˆ
Y2005 = 3.5 + 0.8125(Y2004 ) − 0.9375(Y2003 )
= 3.5 + 0.8125(6 ) − 0.9375(4 )
= 4.625

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Chap 15-38
Autoregressive Modeling Steps
1. Choose p (note that df = n – 2p – 1)
2. Form a series of “lagged 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
Statistics for Managers Using
repeat
Microsoft Excel, 4e © 2004
Chap 15-39
Prentice-Hall, Inc.

Selecting A Forecasting Model


Perform a residual analysis







Look for pattern or direction

Measure magnitude of residual error using
squared differences
Measure residual error using MAD
Use simplest model


Principle of parsimony

Statistics for Managers Using
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Chap 15-40
Residual Analysis
e

e

0

0

Random errors

T

T
Cyclical effects not accounted for

e

e

0

0

Statistics for Managers Using
T
Microsoftnot accounted2004
Trend Excel, 4e © for
Prentice-Hall, Inc.

T
Seasonal effects not accounted for

Chap 15-41
Measuring Errors




Choose the model that gives the smallest
measuring errors

Sum of squared errors
(SSE)
n

ˆ
SSE = ∑ (Yi − Yi )



Mean Absolute Deviation
(MAD)
n

2

MAD =

i=1



Sensitive to outliers

Statistics for Managers Using
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

ˆ
∑ Y −Y
i=1

i

i

n

Not sensitive to extreme
observations

Chap 15-42
Principal of Parsimony




Suppose two or more models provide a
good fit for the 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

Statistics for Managers Using
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Chap 15-43
Forecasting With Seasonal Data


Recall the classical time series model with
seasonal variation:

Yi = Ti × Si × Ci × Ii


Suppose the seasonality is quarterly


Define three new dummy variables for quarters:
Q1 = 1 if first quarter, 0 otherwise
Q2 = 1 if second quarter, 0 otherwise

Q3 = 1 if third quarter, 0 otherwise
Statistics for Managers Using
Microsoft Excel, 4e 4 is the default if Q1 = Q2 = Q3 = 0)
(Quarter © 2004
Chap 15-44
Prentice-Hall, Inc.
Exponential Model with
Quarterly Data
Xi
0 1

Q1

Q2

Q3

Yi = β β β 2 β3 β 4 ε i
βi provides the multiplier for the ith quarter relative
to the 4th quarter (i = 2, 3, 4)



Transform to linear form:

log(Yi ) = log(β0 ) + Xilog(β1 ) + Q1log(β2 )
+ Q 2log(β3 ) + Q3log(β 4 ) + log( ε i )

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Chap 15-45
Estimating the Quarterly Model


Exponential forecasting equation:

ˆ
log(Yi ) = b0 + b1Xi + b 2Q1 + b3Q 2 + b 4Q3
where

ˆ
b0 = estimate of log(β0), so 10b0 = β0
b
ˆ
b = estimate of log(β ), so 10 1 = β
1

1

1

etc…

Interpretation:
ˆ
(β1 −1) ×100% = estimated quarterly compound growth rate (in %)
ˆ
β = estimated multiplier for first quarter relative to fourth quarter
2

ˆ
β3 = estimated multiplier for second quarter rel. to fourth quarter
for Managers Using
ˆ
β = estimated multiplier for third quarter relative to fourth

Statistics
4
Microsoft Excel, 4e © 2004
quarter
Prentice-Hall, Inc.

Chap 15-46
Quarterly Model Example
Suppose the forecasting equation is:
ˆ
log(Yi ) = 3.43 + .017X i − .082Q1 − .073Q 2 + .022Q 3



b0 = 3.43, so

ˆ
10b0 = β0 = 2691.53

b1 = .017, so

ˆ
10b1 = β1 = 1.040

b2 = -.082, so
b3 = -.073, so
b4 = .022, so

ˆ
10b2 = β 2 = 0.827
ˆ
10b3 = β3 = 0.845

ˆ
10b 4 = β 4 = 1.052

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Chap 15-47
Quarterly Model Example
(continued)

Value:

Interpretation:

ˆ
β0 = 2691.53

Unadjusted trend value for first quarter of first year

ˆ
β1 = 1.040

4.0% = estimated quarterly compound growth rate

ˆ
β 2 = 0.827

Ave. sales in Q2 are 82.7% of average 4th quarter sales,
after adjusting for the 4% quarterly growth rate

ˆ
β3 = 0.845

Ave. sales in Q3 are 84.5% of average 4th quarter sales,
after adjusting for the 4% quarterly growth rate

ˆ
β 4 = 1.052

Ave. sales in Q4 are 105.2% of average 4th quarter

Statistics for Managers Using
sales, after adjusting for the 4% quarterly growth rate
Microsoft Excel, 4e © 2004
Chap 15-48
Prentice-Hall, Inc.
Index Numbers


Index numbers allow relative comparisons
over time



Index numbers are reported relative to a Base
Period Index



Base period index = 100 by definition



Used for an individual item or measurement

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Chap 15-49
Simple Price Index


Simple Price Index:

Pi
Ii =
× 100
Pbase
where
Ii = index number for year i
Pi = price for year i

Statistics for Managers Using base year
Pbase = price for the
Microsoft Excel, 4e © 2004
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Chap 15-50
Index Numbers: Example


Airplane ticket prices from 1995 to 2003:
Index

Year

Price

(base year
= 2000)

1995

272

85.0

1996

288

90.0

1997

295

92.2

1998

311

97.2

1999

322

100.6

2000

320

100.0

2001

348

108.8

2002

366

114.4

Statistics for384
Managers Using
2003
120.0
Microsoft Excel, 4e © 2004
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I1996

P1996
288
=
× 100 =
(100 ) = 90
P2000
320

Base Year:
P2000
320
I2000 =
× 100 =
(100 ) = 100
P2000
320
I2003

P2003
384
=
× 100 =
(100 ) = 120
P2000
320

Chap 15-51
Index Numbers: Interpretation
P1996
288
=
× 100 =
(100 ) = 90
P2000
320



Prices in 1996 were 90%
of base year prices

P2000
320
=
× 100 =
(100 ) = 100
P2000
320



Prices in 2000 were 100%
of base year prices (by
definition, since 2000 is the
base year)

P
384
I2003 = 2003 × 100 =
(100 ) = 120
P2000
320
Statistics for Managers Using



Prices in 2003 were 120%
of base year prices

I1996

I2000

Microsoft Excel, 4e © 2004
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Chap 15-52
Aggregate Price Indexes


An aggregate index is used to measure the rate
of change from a base period for a group of items
Aggregate
Price Indexes
Unweighted
aggregate
price index

Weighted
aggregate
price indexes

Paasche Index
Statistics for Managers Using
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Laspeyres Index
Chap 15-53
Unweighted
Aggregate Price Index


Unweighted aggregate price index formula:
n

(
IUt ) =

Pi( t )
∑
i=1
n

∑P
i=1

(
IUt )
n

× 100

t = time period
n = total number of items

= unweighted price index at time t

∑P
i=1

(0)
i

i = item

(t)
i

= sum of the prices for the group of items at time t

Statistics nfor( 0Managers Using
Pi ) = sum of the prices for the group of items in time period 0
∑
Microsofti=Excel, 4e © 2004
1
Chap 15-54
Prentice-Hall, Inc.
Unweighted Aggregate Price
Index: Example
Automobile Expenses:
Monthly Amounts ($):

Index

Year

Lease payment

Fuel

Repair

Total

(2001=100)

2001

260

45

40

345

100.0

2002

280

60

40

380

110.1

2003

305

55

45

405

117.4

2004

310

50

50

410

118.8

I2004

∑P
=
∑P

410
× 100 =
(100) = 118.8
345
2001

2004

 Unweighted total expenses were 18.8%
Statistics for Managers Using
higher
Microsoft Excel, 4e © 2004 in 2004 than in 2001
Chap 15-55
Prentice-Hall, Inc.
Weighted
Aggregate Price Indexes


Laspeyres index
n

I =
(t)
L

∑P
i=1
n

(t)
i

∑P
i=1

Q

(0)
i

(0)
i

Q



Paasche index
n

× 100

(0)
i

Q(i 0 ) : weights based on

period 0 quantities

I =
(t)
P

∑P
i=1
n

(t)
i

∑P
i=1

Q

(0)
i

(t)
i

Q

× 100

(t)
i

Q(i t ) : weights based on current
period quantities

Statistics for Managers = price in time period t
Pi( t ) Using
Microsoft Excel, 4e ©i( 02004 in period 0
P ) = price
Prentice-Hall, Inc.

Chap 15-56
Common Price Indexes


Consumer Price Index (CPI)



Producer Price Index (PPI)



Stock Market Indexes


Dow Jones Industrial Average



S&P 500 Index



NASDAQ Index

Statistics for Managers Using
Microsoft Excel, 4e © 2004
Prentice-Hall, Inc.

Chap 15-57
Pitfalls in
Time-Series Analysis




Assuming 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, and habits, etc.

Statistics for Managers Using
Microsoft Excel, 4e © 2004
Prentice-Hall, Inc.

Chap 15-58
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 square trend fitting and
forecasting

 Linear, quadratic and
Statistics for Managers Using exponential models
Microsoft Excel, 4e © 2004
Chap 15-59
Prentice-Hall, Inc.
Chapter Summary
(continued)







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
analysis

Statistics for Managers Using
Microsoft Excel, 4e © 2004
Prentice-Hall, Inc.

Chap 15-60

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Chap15 time series forecasting & index number

  • 1. Statistics for Managers Using Microsoft® Excel 4th Edition Chapter 15 Time-Series Forecasting and Index Numbers Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 15-1
  • 2. Chapter Goals After completing this chapter, you should be able to:  Develop and implement basic forecasting models  Identify the components present in a time series  Use smoothing-based forecasting models, including moving average and exponential smoothing  Apply trend-based forecasting models, including linear trend and nonlinear trend  complete time-series forecasting of seasonal data Statistics for Managers Using Microsoft Excel, 4einterpret basic index numbers  compute and © 2004 Chap 15-2 Prentice-Hall, Inc.
  • 3. The Importance of Forecasting  Governments forecast unemployment, interest rates, and expected revenues from income taxes for policy purposes  Marketing executives forecast demand, sales, and consumer preferences for strategic planning  College administrators forecast enrollments to plan for facilities and for faculty recruitment  Retail stores forecast demand to control inventory levels, hire employees and provide training Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 15-3
  • 4. Time-Series Data    Numerical data obtained at regular time intervals The time intervals can be annually, quarterly, daily, hourly, etc. Example: Year: 1999 2000 2001 2002 2003 Sales: 75.3 Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. 74.2 78.5 79.7 80.2 Chap 15-4
  • 5. Time-Series Plot A time-series plot is a two-dimensional plot of time series data U.S. Inflation Rate 2001 1999 1997 1995 1993 1991 1989 1987 1985 1983 1981 1979 the horizontal axis corresponds to the time periods Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.  1977 16.00 14.00 12.00 10.00 8.00 6.00 4.00 2.00 0.00 1975 the vertical axis measures the variable of interest Inflation Rate (%)  Year Chap 15-5
  • 6. Time-Series Components Time Series Trend Component Seasonal Component Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Cyclical Component Irregular Component Chap 15-6
  • 7. Trend Component  Long-run increase or decrease over time (overall upward or downward movement)  Data taken over a long period of time Sales Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. d rd tren Up w a Time 15-7 Chap
  • 8. Trend Component (continued)   Trend can be upward or downward Trend can be linear or non-linear Sales Sales Time Statistics for Managers Using Downward linear trend Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Time Upward nonlinear trend Chap 15-8
  • 9. Seasonal Component    Short-term regular wave-like patterns Observed within 1 year Often monthly or quarterly Sales Summer Winter Summer Spring Winter Spring Fall Fall Statistics for Managers Using Microsoft Excel, 4e © 2004 Time (Quarterly) Prentice-Hall, Inc. Chap 15-9
  • 10. Cyclical Component    Long-term wave-like patterns Regularly occur but may vary in length Often measured peak to peak or trough to trough 1 Cycle Sales Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Year Chap 15-10
  • 11. Irregular Component   Unpredictable, random, “residual” fluctuations Due to random variations of    Nature Accidents or unusual events “Noise” in the time series Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 15-11
  • 12. Multiplicative Time-Series Model for Annual Data   Used primarily for forecasting Observed value in time series is the product of components Yi = Ti × Ci × Ii where Ti = Trend value at year i Ci = Cyclical value at year i Ii = Irregular (random) value at year i Statistics for Managers Using Microsoft Excel, 4e © 2004 Chap 15-12 Prentice-Hall, Inc.
  • 13. Multiplicative Time-Series Model with a Seasonal Component   Used primarily for forecasting Allows consideration of seasonal variation Yi = Ti × Si × Ci × Ii where Ti = Trend value at time i Si = Seasonal value at time i Ci = Cyclical value at time i Statistics for Managers Using Ii = Microsoft Excel, 4e © 2004 Irregular (random) value at time i Chap 15-13 Prentice-Hall, Inc.
  • 14. Smoothing the Annual Time Series  Calculate moving averages to get an overall impression of the pattern of movement over time Moving Average: averages of consecutive time series values for a chosen period of length L Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 15-14
  • 15. Moving Averages     Used for smoothing A series of arithmetic means over time Result dependent upon choice of L (length of period for computing means) Examples:    For a 5 year moving average, L = 5 For a 7 year moving average, L = 7 Etc. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 15-15
  • 16. Moving Averages (continued)  Example: Five-year moving average  First average: MA(5) =  Y1 + Y2 + Y3 + Y4 + Y5 5 Second average: MA(5) = Y2 + Y3 + Y4 + Y5 + Y6 5  etc. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 15-16
  • 17. Example: Annual Data 1 2 3 4 5 6 7 8 9 10 11 Statistics etc… Sales 23 40 25 27 32 48 33 37 37 50 40 for Managers etc… Annual Sales 60 50 … 40 Sales Year 30 20 10 0 Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. 1 2 3 4 5 6 7 8 9 10 11 Year Chap 15-17 …
  • 18. Calculating Moving Averages Average Year 5-Year Moving Average Year Sales 1 23 3 29.4 2 40 4 34.4 3 25 5 33.0 4 27 6 35.4 5 32 7 37.4 6 48 8 41.0 7 33 9 39.4 8 37 … … 9 37 etc… 3= 29.4 = 1+ 2 + 3 + 4 + 5 5 23 + 40 + 25 + 27 + 32 5  10 50 Statistics for ManagersEach moving average is for a Using consecutive block of 5 years 11 40 Microsoft Excel, 4e © 2004 Chap 15-18 Prentice-Hall, Inc.
  • 19. Annual vs. Moving Average The 5-year moving average smoothes the data and shows the underlying trend Annual vs. 5-Year Moving Average 60 50 40 Sales  30 20 10 0 1 Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. 2 3 4 5 6 7 8 9 10 11 Year Annual 5-Year Moving Average Chap 15-19
  • 20. Exponential Smoothing  A weighted moving average    Weights decline exponentially Most recent observation weighted most Used for smoothing and short term forecasting (often one period into the future) Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 15-20
  • 21. Exponential Smoothing (continued)  The weight (smoothing coefficient) is W     Subjectively chosen Range from 0 to 1 Smaller W gives more smoothing, larger W gives less smoothing The weight is:   Close to 0 for smoothing out unwanted cyclical and irregular components Close to 1 for forecasting Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 15-21
  • 22. Exponential Smoothing Model  Exponential smoothing model E1 = Y1 Ei = WYi + (1 − W )Ei−1 For i = 2, 3, 4, … where: Ei = exponentially smoothed value for period i Ei-1 = exponentially smoothed value already computed for period i - 1 Statistics for Managers Using value in period i Yi = observed Microsoft Excel, 4e ©weight (smoothing coefficient), 0 < W < 1 W = 2004 Prentice-Hall, Inc. Chap 15-22
  • 23. Exponential Smoothing Example  Time Period (i) Suppose we use weight W = .2 Sales (Yi) Forecast from prior period (Ei-1) Exponentially Smoothed Value for this period (Ei) 1 23 -23 2 40 23 (.2)(40)+(.8)(23)=26.4 3 25 26.4 (.2)(25)+(.8)(26.4)=26.12 4 27 26.12 (.2)(27)+(.8)(26.12)=26.296 5 32 26.296 (.2)(32)+(.8)(26.296)=27.437 6 48 27.437 (.2)(48)+(.8)(27.437)=31.549 7 33 31.549 (.2)(48)+(.8)(31.549)=31.840 8 37 31.840 (.2)(33)+(.8)(31.840)=32.872 9 37 32.872 (.2)(37)+(.8)(32.872)=33.697 Statistics for Managers Using 10 50 Microsoft Excel, 4e33.697 © 2004 (.2)(50)+(.8)(33.697)=36.958 etc. etc. etc. etc. Prentice-Hall, Inc. E1 = Y1 since no prior information exists Ei = WYi + (1 − W )Ei−1 Chap 15-23
  • 24. Sales vs. Smoothed Sales  Fluctuations have been smoothed NOTE: the smoothed value in this case is generally a little low, since the trend is upward sloping and the weighting factor is only .2 60 50 40 Sales  30 20 10 0 1 Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. 2 3 4 5 6 7 Time Period Sales 8 9 10 Smoothed Chap 15-24
  • 25. Forecasting Time Period i + 1 The smoothed value in the current period (i) is used as the forecast value for next period (i + 1) :  ˆ Yi+1 = Ei Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 15-25
  • 26. Exponential Smoothing in Excel  Use tools / data analysis / exponential smoothing  The “damping factor” is (1 - W) Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 15-26
  • 27. Trend-Based Forecasting  Estimate a trend line using regression analysis Time Period (X) Sales (Y) 1999 0 20 2000 1 40 2001 2 30 2002 3 50 2003 4 70 2004 5 65 Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Year  Use time (X) as the independent variable: ˆ Y = b0 + b1X Chap 15-27
  • 28. Trend-Based Forecasting (continued) Year Time Period (X)  The linear trend forecasting equation is: ˆ Yi = 21.905 + 9.5714 Xi Sales (Y) sales 1999 0 20 80 2000 1 40 70 60 2001 2 30 50 40 2002 3 50 30 20 2003 4 70 10 2004 5 65 0 Statistics for Managers Using0 Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Sales trend 1 2 3 4 5 Year Chap 15-28 6
  • 29. Trend-Based Forecasting (continued)  Sales (y) 1999 2000 2001 2002 2003 2004 2005 0 1 2 3 4 5 6 20 40 30 50 70 65 ?? ˆ Y = 21.905 + 9.5714 (6) = 79.33 Sales trend sales Year Time Period (X) Forecast for time period 6: 80 70 60 50 40 30 20 10 0 Statistics for Managers Using0 Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. 1 2 3 4 5 Year Chap 15-29 6
  • 30. Nonlinear Trend Forecasting  A nonlinear regression model can be used when the time series exhibits a nonlinear trend  Quadratic form is one type of a nonlinear model: Yi = β0 + β1Xi + β2 X + εi 2 i  Compare adj. r2 and standard error to that of linear model to see if this is an improvement  Can Managers Using Statistics fortry other functional forms to get best fit Microsoft Excel, 4e © 2004 Chap 15-30 Prentice-Hall, Inc.
  • 31. Exponential Trend Model  Another nonlinear trend model: Xi 0 1 Yi = β β  εi Transform to linear form: log(Yi ) = log(β0 ) + Xi log(β1 ) + log( ε i ) Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 15-31
  • 32. Exponential Trend Model (continued)  Exponential trend forecasting equation: ˆ log(Yi ) = b 0 + b1Xi where b0 = estimate of log(β0) b1 = estimate of log(β1) Interpretation: ˆ (β1 − 1) × 100% is the estimated annual compound growth rate (in %) Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 15-32
  • 33. Model Selection Using Differences  Use a linear trend model if the first differences are approximately constant (Y2 − Y1 ) = ( Y3 − Y2 ) =  = ( Yn − Yn-1 )  Use a quadratic trend model if the second differences are approximately constant [(Y3 − Y2 ) − ( Y2 − Y1 )] = [(Y4 − Y3 ) − ( Y3 − Y2 )] Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. =  = [(Yn − Yn-1 ) − ( Yn-1 − Yn-2 )] Chap 15-33
  • 34. Model Selection Using Differences  (continued) Use an exponential trend model if the percentage differences are approximately constant (Y3 − Y2 ) (Y2 − Y1 ) (Yn − Yn-1 ) × 100% = × 100% =  = × 100% Y1 Y2 Yn-1 Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 15-34
  • 35. Autoregressive Modeling   Used for forecasting Takes advantage of autocorrelation    1st order - correlation between consecutive values 2nd order - correlation between values 2 periods apart pth order Autoregressive models: Yi = A 0 + A 1Yi-1 + A 2 Yi-2 +  + + A p Yi-p + δi Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Random Error Chap 15-35
  • 36. Autoregressive Model: Example The Office Concept Corp. has acquired a number of office units (in thousands of square feet) over the last eight years. Develop the second order Autoregressive model. Year Units 97 4 98 3 99 2 00 3 01 2 02 2 03 4 04 6 Managers Statistics for Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 15-36
  • 37. Autoregressive Model: Example Solution  Develop the 2nd order table  Use Excel to estimate a regression model Excel Output Coefficients Intercept 3.5 X Variable 1 0.8125 X Variable 2 -0.9375 Year Yi Yi-1 Yi-2 97 98 99 00 01 02 03 04 4 3 2 3 2 2 4 6 -4 3 2 3 2 2 4 --4 3 2 3 2 2 Statistics 3.5 Managers Using ˆ Yi = for + 0.8125Yi−1 − 0.9375Yi−2 Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 15-37
  • 38. Autoregressive Model Example: Forecasting Use the second-order equation to forecast number of units for 2005: ˆ Yi = 3.5 + 0.8125Yi−1 − 0.9375Yi−2 ˆ Y2005 = 3.5 + 0.8125(Y2004 ) − 0.9375(Y2003 ) = 3.5 + 0.8125(6 ) − 0.9375(4 ) = 4.625 Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 15-38
  • 39. Autoregressive Modeling Steps 1. Choose p (note that df = n – 2p – 1) 2. Form a series of “lagged 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 Statistics for Managers Using repeat Microsoft Excel, 4e © 2004 Chap 15-39 Prentice-Hall, Inc. 
  • 40. Selecting A Forecasting Model  Perform a residual analysis     Look for pattern or direction Measure magnitude of residual error using squared differences Measure residual error using MAD Use simplest model  Principle of parsimony Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 15-40
  • 41. Residual Analysis e e 0 0 Random errors T T Cyclical effects not accounted for e e 0 0 Statistics for Managers Using T Microsoftnot accounted2004 Trend Excel, 4e © for Prentice-Hall, Inc. T Seasonal effects not accounted for Chap 15-41
  • 42. Measuring Errors   Choose the model that gives the smallest measuring errors Sum of squared errors (SSE) n ˆ SSE = ∑ (Yi − Yi )  Mean Absolute Deviation (MAD) n 2 MAD = i=1  Sensitive to outliers Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.  ˆ ∑ Y −Y i=1 i i n Not sensitive to extreme observations Chap 15-42
  • 43. Principal of Parsimony   Suppose two or more models provide a good fit for the 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 Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 15-43
  • 44. Forecasting With Seasonal Data  Recall the classical time series model with seasonal variation: Yi = Ti × Si × Ci × Ii  Suppose the seasonality is quarterly  Define three new dummy variables for quarters: Q1 = 1 if first quarter, 0 otherwise Q2 = 1 if second quarter, 0 otherwise Q3 = 1 if third quarter, 0 otherwise Statistics for Managers Using Microsoft Excel, 4e 4 is the default if Q1 = Q2 = Q3 = 0) (Quarter © 2004 Chap 15-44 Prentice-Hall, Inc.
  • 45. Exponential Model with Quarterly Data Xi 0 1 Q1 Q2 Q3 Yi = β β β 2 β3 β 4 ε i βi provides the multiplier for the ith quarter relative to the 4th quarter (i = 2, 3, 4)  Transform to linear form: log(Yi ) = log(β0 ) + Xilog(β1 ) + Q1log(β2 ) + Q 2log(β3 ) + Q3log(β 4 ) + log( ε i ) Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 15-45
  • 46. Estimating the Quarterly Model  Exponential forecasting equation: ˆ log(Yi ) = b0 + b1Xi + b 2Q1 + b3Q 2 + b 4Q3 where ˆ b0 = estimate of log(β0), so 10b0 = β0 b ˆ b = estimate of log(β ), so 10 1 = β 1 1 1 etc… Interpretation: ˆ (β1 −1) ×100% = estimated quarterly compound growth rate (in %) ˆ β = estimated multiplier for first quarter relative to fourth quarter 2 ˆ β3 = estimated multiplier for second quarter rel. to fourth quarter for Managers Using ˆ β = estimated multiplier for third quarter relative to fourth Statistics 4 Microsoft Excel, 4e © 2004 quarter Prentice-Hall, Inc. Chap 15-46
  • 47. Quarterly Model Example Suppose the forecasting equation is: ˆ log(Yi ) = 3.43 + .017X i − .082Q1 − .073Q 2 + .022Q 3  b0 = 3.43, so ˆ 10b0 = β0 = 2691.53 b1 = .017, so ˆ 10b1 = β1 = 1.040 b2 = -.082, so b3 = -.073, so b4 = .022, so ˆ 10b2 = β 2 = 0.827 ˆ 10b3 = β3 = 0.845 ˆ 10b 4 = β 4 = 1.052 Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 15-47
  • 48. Quarterly Model Example (continued) Value: Interpretation: ˆ β0 = 2691.53 Unadjusted trend value for first quarter of first year ˆ β1 = 1.040 4.0% = estimated quarterly compound growth rate ˆ β 2 = 0.827 Ave. sales in Q2 are 82.7% of average 4th quarter sales, after adjusting for the 4% quarterly growth rate ˆ β3 = 0.845 Ave. sales in Q3 are 84.5% of average 4th quarter sales, after adjusting for the 4% quarterly growth rate ˆ β 4 = 1.052 Ave. sales in Q4 are 105.2% of average 4th quarter Statistics for Managers Using sales, after adjusting for the 4% quarterly growth rate Microsoft Excel, 4e © 2004 Chap 15-48 Prentice-Hall, Inc.
  • 49. Index Numbers  Index numbers allow relative comparisons over time  Index numbers are reported relative to a Base Period Index  Base period index = 100 by definition  Used for an individual item or measurement Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 15-49
  • 50. Simple Price Index  Simple Price Index: Pi Ii = × 100 Pbase where Ii = index number for year i Pi = price for year i Statistics for Managers Using base year Pbase = price for the Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 15-50
  • 51. Index Numbers: Example  Airplane ticket prices from 1995 to 2003: Index Year Price (base year = 2000) 1995 272 85.0 1996 288 90.0 1997 295 92.2 1998 311 97.2 1999 322 100.6 2000 320 100.0 2001 348 108.8 2002 366 114.4 Statistics for384 Managers Using 2003 120.0 Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. I1996 P1996 288 = × 100 = (100 ) = 90 P2000 320 Base Year: P2000 320 I2000 = × 100 = (100 ) = 100 P2000 320 I2003 P2003 384 = × 100 = (100 ) = 120 P2000 320 Chap 15-51
  • 52. Index Numbers: Interpretation P1996 288 = × 100 = (100 ) = 90 P2000 320  Prices in 1996 were 90% of base year prices P2000 320 = × 100 = (100 ) = 100 P2000 320  Prices in 2000 were 100% of base year prices (by definition, since 2000 is the base year) P 384 I2003 = 2003 × 100 = (100 ) = 120 P2000 320 Statistics for Managers Using  Prices in 2003 were 120% of base year prices I1996 I2000 Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 15-52
  • 53. Aggregate Price Indexes  An aggregate index is used to measure the rate of change from a base period for a group of items Aggregate Price Indexes Unweighted aggregate price index Weighted aggregate price indexes Paasche Index Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Laspeyres Index Chap 15-53
  • 54. Unweighted Aggregate Price Index  Unweighted aggregate price index formula: n ( IUt ) = Pi( t ) ∑ i=1 n ∑P i=1 ( IUt ) n × 100 t = time period n = total number of items = unweighted price index at time t ∑P i=1 (0) i i = item (t) i = sum of the prices for the group of items at time t Statistics nfor( 0Managers Using Pi ) = sum of the prices for the group of items in time period 0 ∑ Microsofti=Excel, 4e © 2004 1 Chap 15-54 Prentice-Hall, Inc.
  • 55. Unweighted Aggregate Price Index: Example Automobile Expenses: Monthly Amounts ($): Index Year Lease payment Fuel Repair Total (2001=100) 2001 260 45 40 345 100.0 2002 280 60 40 380 110.1 2003 305 55 45 405 117.4 2004 310 50 50 410 118.8 I2004 ∑P = ∑P 410 × 100 = (100) = 118.8 345 2001 2004  Unweighted total expenses were 18.8% Statistics for Managers Using higher Microsoft Excel, 4e © 2004 in 2004 than in 2001 Chap 15-55 Prentice-Hall, Inc.
  • 56. Weighted Aggregate Price Indexes  Laspeyres index n I = (t) L ∑P i=1 n (t) i ∑P i=1 Q (0) i (0) i Q  Paasche index n × 100 (0) i Q(i 0 ) : weights based on period 0 quantities I = (t) P ∑P i=1 n (t) i ∑P i=1 Q (0) i (t) i Q × 100 (t) i Q(i t ) : weights based on current period quantities Statistics for Managers = price in time period t Pi( t ) Using Microsoft Excel, 4e ©i( 02004 in period 0 P ) = price Prentice-Hall, Inc. Chap 15-56
  • 57. Common Price Indexes  Consumer Price Index (CPI)  Producer Price Index (PPI)  Stock Market Indexes  Dow Jones Industrial Average  S&P 500 Index  NASDAQ Index Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 15-57
  • 58. Pitfalls in Time-Series Analysis   Assuming 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, and habits, etc. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 15-58
  • 59. 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 square trend fitting and forecasting  Linear, quadratic and Statistics for Managers Using exponential models Microsoft Excel, 4e © 2004 Chap 15-59 Prentice-Hall, Inc.
  • 60. Chapter Summary (continued)     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 analysis Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 15-60