for Statistics Managers
Using Microsoft Excel
5th Edition
Chapter 16
Time-Series Forecasting and Index
Numbers
Statistics for Managers Using Microsoft Excel, 5e 2008 Prentice-Hall, Inc.
Chap 16-1
Learning Objectives
In this chapter, you learn:
About seven different time-series forecasting
models: moving averages, exponential smoothing,
the linear trend, the quadratic trend, the exponential
trend, the autoregressive, and the least-squares
models for seasonal data.
To choose the most appropriate time-series
forecasting model
About price indexes and the difference between
aggregated and simple indexes
Statistics for Managers Using Microsoft Excel, 5e
Chap 16-2
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, 5e
Chap 16-3
Time Series Plot
A time-series plot is a twodimensional plot of time-series data
U.S. Inflation Rate
Year
Statistics for Managers Using Microsoft Excel, 5e
Chap 16-4
2001
1999
1997
1995
1993
1991
1989
1987
1985
1983
1981
1979
corresponds to the
time periods
1977
the horizontal axis
16.00
14.00
12.00
10.00
8.00
6.00
4.00
2.00
0.00
1975
measures the variable
of interest
Inflation Rate (%)
the vertical axis
Time-Series Components
Time Series
Trend
Component
Seasonal
Component
Cyclical
Component
Statistics for Managers Using Microsoft Excel, 5e
Irregular
Component
Chap 16-5
Trend Component
Long-run increase or decrease over time (overall
upward or downward movement)
Data taken over a long period of time
Sales
t re
pward
U
Time
Statistics for Managers Using Microsoft Excel, 5e
Chap 16-6
nd
Trend Component
Trend can be upward or downward
Trend can be linear or non-linear
Sales
Sales
Time
Downward linear trend
Time
Upward nonlinear trend
Statistics for Managers Using Microsoft Excel, 5e
Chap 16-7
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
Time (Quarterly)
Statistics for Managers Using Microsoft Excel, 5e
Chap 16-8
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
Year
Statistics for Managers Using Microsoft Excel, 5e
Chap 16-9
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, 5e
Chap 16-10
Multiplicative Time-Series
Model for Annual Data
Used primarily for forecasting
Observed value in time series is the product of
components
Yi = Ti Ci I i
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, 5e
Chap 16-11
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
Ii = Irregular (random) value at time i
Statistics for Managers Using Microsoft Excel, 5e
Chap 16-12
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, 5e
Chap 16-13
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
Statistics for Managers Using Microsoft Excel, 5e
Chap 16-14
Moving Averages
Example: Five-year moving average
First average:
MA(5) =
Y1 + Y2 + Y3 + Y4 + Y5
5
Second average:
Y2 + Y3 + Y4 + Y5 + Y6
MA(5) =
5
Statistics for Managers Using Microsoft Excel, 5e
Chap 16-15
Example: Annual Data
1
2
3
4
5
6
7
8
9
10
11
etc
Sales
23
40
25
27
32
48
33
37
37
50
40
etc
Annual Sales
60
50
40
Sales
Year
30
20
10
0
1
2
3
4
Statistics for Managers Using Microsoft Excel, 5e
5
6
7
8
9
10
Year
Chap 16-16
11
Example: Annual Data
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
10
50
11
40
etc
3=
29.4 =
1+ 2 + 3 + 4 + 5
5
23 + 40 + 25 + 27 + 32
5
Each moving average is for a
consecutive block of 5 years
Statistics for Managers Using Microsoft Excel, 5e
Chap 16-17
Annual vs. Moving Average
Annual vs. 5-Year Moving Average
The 5-year
50
Sales
moving average
smoothes the
data and shows
the underlying
trend
60
40
30
20
10
0
1
2
3
4
5
6
7
8
9
10
Year
Annual
Statistics for Managers Using Microsoft Excel, 5e
5-Year Moving Average
Chap 16-18
11
Exponential Smoothing
Used for smoothing and short term
forecasting (often one period into the future)
A weighted moving average
Weights decline exponentially
Most recent observation weighted most
Statistics for Managers Using Microsoft Excel, 5e
Chap 16-19
Exponential Smoothing
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, 5e
Chap 16-20
Exponential Smoothing Model
Exponential smoothing model
E1 = Y1
Ei = WYi + (1 W )Ei1
For i = 2, 3, 4,
where:
Ei = exponentially smoothed value for period i
Ei-1 = exponentially smoothed value already
computed for period i - 1
Yi = observed value in period i
W = weight (smoothing coefficient), 0 < W < 1
Statistics for Managers Using Microsoft Excel, 5e
Chap 16-21
Exponential Smoothing
Example
Suppose we use weight W = .2
Time
Period (i)
1
2
3
4
5
6
7
8
9
10
Sales
(Yi)
23
40
25
27
32
48
33
37
37
50
Forecast from
prior period
(Ei-1)
Exponentially Smoothed Value for
this period (Ei)
-23
26.4
26.12
26.296
27.437
31.549
31.840
32.872
33.697
23
(.2)(40)+(.8)(23)=26.4
(.2)(25)+(.8)(26.4)=26.12
(.2)(27)+(.8)(26.12)=26.296
(.2)(32)+(.8)(26.296)=27.437
(.2)(48)+(.8)(27.437)=31.549
(.2)(48)+(.8)(31.549)=31.840
(.2)(33)+(.8)(31.840)=32.872
(.2)(37)+(.8)(32.872)=33.697
(.2)(50)+(.8)(33.697)=36.958
Statistics for Managers Using Microsoft Excel, 5e
E1 = Y1 since
no prior
information
exists
Ei =
WYi + (1 W)E i 1
Chap 16-22
Exponential Smoothing
Example
Fluctuations have
been smoothed
60
50
NOTE: the
40
Sales
smoothed value in
this case is
generally a little
low, since the
trend is upward
sloping and the
weighting factor is
only .2
30
20
10
0
1
2
3
4
Statistics for Managers Using Microsoft Excel, 5e
5
6
7
Time Period
Sales
8
9
Smoothed
Chap 16-23
10
Forecasting Time Period i + 1
The smoothed value in the current period
(i) is used as the forecast value for next
period (i + 1) :
Yi+1 = Ei
Statistics for Managers Using Microsoft Excel, 5e
Chap 16-24
Least Squares Linear TrendBased Forecasting
Estimate a trend line using regression analysis
Year
Time
Period
(X)
1999
2000
2001
2002
2003
2004
0
1
2
3
4
5
Use time (X) as the
Sales
(Y)
20
40
30
50
70
65
independent variable:
Y = b0 + b1X
Statistics for Managers Using Microsoft Excel, 5e
Chap 16-25
Least Squares Linear TrendBased Forecasting
The linear trend forecasting equation is:
1999
2000
2001
2002
2003
2004
Sales trend
0
1
2
3
4
5
20
40
30
50
70
65
sales
Year
Yi = 21.905 + 9.5714 Xi
Time
Period (X) Sales (Y)
80
70
60
50
40
30
20
10
0
0
1
2
3
4
5
Year
Statistics for Managers Using Microsoft Excel, 5e
Chap 16-26
6
Least Squares Linear TrendBased Forecasting
Forecast for time period 6
Sales
(y)
1999
2000
2001
2002
2003
2004
2005
0
1
2
3
4
5
6
20
40
30
50
70
65
??
Y = 21.905 + 9.5714 (6)
= 79.33
Sales trend
sales
Year
Time
Period
(X)
80
70
60
50
40
30
20
10
0
0
1
2
3
4
5
Year
Statistics for Managers Using Microsoft Excel, 5e
Chap 16-27
6
Least Squares Quadratic
Trend-Based Forecasting
A nonlinear regression model can be used
when the time series exhibits a nonlinear
trend
Quadratic Trend Forecasting Equation:
=b +b X +b X 2
Y
0
1i
2i
Test quadratic term for significance
Can try other functional forms to get best fit
Statistics for Managers Using Microsoft Excel, 5e
Chap 16-28
Least Squares Exponential
Trend-Based Forecasting
Exponential trend forecasting equation:
log(Yi ) = b0 + b1Xi
where
b0 = estimate of log(0)
b1 = estimate of log(1)
Interpretation:
(1 1) 100% is the estimated annual compound
growth rate (in %)
Statistics for Managers Using Microsoft Excel, 5e
Chap 16-29
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 )]
= = [(Yn Yn-1 ) ( Yn-1 Yn-2 )]
Statistics for Managers Using Microsoft Excel, 5e
Chap 16-30
Model Selection Using
Differences
Use an exponential trend model if the percentage
differences are approximately constant
(Y3 Y2 )
(Y2 Y1 Yn-1 )
(Yn )
100% =
100% = =
100%
Y1
Y2
Yn-1
Statistics for Managers Using Microsoft Excel, 5e
Chap 16-31
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 + A1Yi-1 + A 2 Yi-2 + + A p Yi-p + i
Random
Error
Statistics for Managers Using Microsoft Excel, 5e
Chap 16-32
Autoregressive Modeling
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
98
99
00
01
02
03
04
4
3
2
3
2
2
4
6
Statistics for Managers Using Microsoft Excel, 5e
Chap 16-33
Autoregressive Modeling
Example
Develop the 2nd order table
Use Excel to estimate a
regression model
Excel Output
Intercept
X Variable 1
X Variable 2
Coefficients
3.5
0.8125
-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
Yi = 3.5 + 0.8125Yi1 0.9375Yi2
Statistics for Managers Using Microsoft Excel, 5e
Chap 16-34
Autoregressive Modeling
Example
Use the second-order equation to
forecast number of units for 2005:
Yi = 3.5 + 0.8125Yi1 0.9375Yi2
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, 5e
Chap 16-35
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 repeat
Statistics for Managers Using Microsoft Excel, 5e
Chap 16-36
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, 5e
Chap 16-37
Residual Analysis
e
e
0
0
T
T
Cyclical effects not accounted for
Random errors
e
e
0
0
T
T
Trend not accounted for
Seasonal effects not accounted for
Statistics for Managers Using Microsoft Excel, 5e
Chap 16-38
Measuring Errors
Choose the model that gives the smallest SSE and/or MAD
Sum of squared errors
Mean Absolute Deviation
(MAD)
(SSE)
n
n
SSE = (Yi Yi )2
i=1
Sensitive to outliers
MAD =
Y Y
i=1
i
i
n
Not sensitive to outliers
Statistics for Managers Using Microsoft Excel, 5e
Chap 16-39
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, 5e
Chap 16-40
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
(Quarter 4 is the default if Q1 = Q2 = Q3 = 0)
Statistics for Managers Using Microsoft Excel, 5e
Chap 16-41
Exponential Model with
Quarterly Data
Xi
1
Q1
Q2
Q3
Yi = 0 2 3 4 i
(1-1)*100% = estimated quarterly compound growth rate (in %)
2 = estimated multiplier for first quarter relative to fourth quarter
3= estimated multiplier for second quarter relative to fourth quarter
4= estimated multiplier for third quarter relative to fourth quarter
Transform to linear form:
log(Yi ) = log( 0 ) + X i log(1 ) + Q1log( 2 )
+ Q 2 log( 3 ) + Q 3log( 4 ) + log( i )
Statistics for Managers Using Microsoft Excel, 5e
Chap 16-42
Estimating the Quarterly Model
Exponential forecasting equation:
log(Yi ) = b 0 + b1X i + b 2 Q1 + b 3Q 2 + b 4 Q 3
where
b0 = estimate of log(0)
b1 = estimate of log(1)
etc
Interpretation:
(1 1) 100% = estimated quarterly compound growth rate (in %)
= estimated multiplier for first quarter relative to fourth quarter
2
3 = estimated multiplier for second quarter relative to fourth quarter
4 = estimated multiplier for third quarter relative to fourth quarter
Statistics for Managers Using Microsoft Excel, 5e
Chap 16-43
Quarterly Model Example
Suppose the forecasting equation is:
log(Yi ) = 3.43 + .017X i .082Q1 .073Q 2 + .022Q 3
b0 = 3.43, so
10b0 = 0 = 2691 .53
b1 = .017, so
10b1 = 1 = 1.040
b2 = -.082, so
10 b 2 = 2 = 0.828
b3 = -.073, so
b4 = .022, so
10b3 = 3 = 0.845
10b 4 = 4 = 1.052
Statistics for Managers Using Microsoft Excel, 5e
Chap 16-44
Quarterly Model Example
Value:
Interpretation:
0 = 2691.53
Unadjusted trend value for first quarter of first year
1 = 1.040
4.0% = estimated quarterly compound growth rate
2 = 0.827
Ave. sales in Q2 are 82.7% of average 4th quarter sales,
after adjusting for the 4% quarterly growth rate
3 = 0.845
Ave. sales in Q3 are 84.5% of average 4th quarter sales,
after adjusting for the 4% quarterly growth rate
4 = 1.052
Ave. sales in Q4 are 105.2% of average 4th quarter
sales, after adjusting for the 4% quarterly growth rate
Statistics for Managers Using Microsoft Excel, 5e
Chap 16-45
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
Statistics for Managers Using Microsoft Excel, 5e
Chap 16-46
Simple Price Index
Simple Price Index:
Pi
Ii =
100
Pbase
where
Ii = index number for year i
Pi = price for year i
Pbase = price for the base year
Statistics for Managers Using Microsoft Excel, 5e
Chap 16-47
Index Numbers: Example
Airplane ticket prices from 1998 to 2006:
Year
Price
Index
(base year =
2000)
1998
272
92.2
1999
288
295
100
2001
311
105.4
2002
322
320
348
118.0
2005
366
124.1
2006
384
130.2
I 2006
108.5
2004
P2006
384
=
100 =
(100) = 130.2
P2000
295
109.2
2003
I 2000
P2000
295
=
100 =
(100) = 100
P2000
295
97.6
2000
I1998
P
272
1998
=
100 =
(100) = 92.2
P2000
295
Statistics for Managers Using Microsoft Excel, 5e
Chap 16-48
Index Numbers: Interpretation
I1998
P
272
1998
=
100 =
(100) = 92.2
P2000
295
I 2000
P2000
295
=
100 =
(100) = 100
P2000
295
I 2006
Prices in 2006 were 130.2%
P2006
384
=
100 =
(100) = 130.2
of base year prices
P2000
295
Prices in 1998 were 92.2% of
base year prices
Prices in 2000 were 100% of
base year prices (by
definition, since 2000 is the
base year)
Statistics for Managers Using Microsoft Excel, 5e
Chap 16-49
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, 5e
Laspeyres Index
Chap 16-50
Unweighted
Aggregate Price Index
Unweighted aggregate price index formula:
n
I=
(t)
U
Pi( t )
i=1
n
Pi( 0 )
100
i = item
t = time period
n = total number of items
i=1
(
IUt )
= unweighted price index at time t
n
P
i =1
(t )
i
= sum of the prices for the group of items at time t
n
Pi ( 0 )
i =1
= sum of the prices for the group of items in time period 0
Statistics for Managers Using Microsoft Excel, 5e
Chap 16-51
Unweighted Aggregate Price
Index: Example
Automobile Expenses:
Monthly Amounts ($):
Year
Lease payment
Fuel
Repair
Total
Index (2003=100)
2003
260
45
40
345
100.0
2004
280
60
40
380
110.1
2005
305
55
45
405
117.4
2006
310
50
50
410
118.8
I 2006
P
=
P
410
100 =
(100) = 118.8
345
2003
2006
Unweighted total expenses were 18.8%
higher in 2006 than in 2003
Statistics for Managers Using Microsoft Excel, 5e
Chap 16-52
Weighted
Aggregate Price Indexes
Laspeyres index
Paasche index
n
(
ILt ) =
Pi( t )Q(i 0 )
i=1
n
Pi( 0 )Q(i 0 )
n
100
I
(t )
P
=
i =1
P
i =1
n
P
i =1
Q(i 0 ) = weights based on
period 0 quantities
i
i
(t )
(t )
i
Q
(0)
100
(t )
i
Q
Q(i t ) = weights based on current
period quantities
Pi( t ) = price in time period t
Pi( 0 ) = price in period 0
Statistics for Managers Using Microsoft Excel, 5e
Chap 16-53
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, 5e
Chap 16-54
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, 5e
Chap 16-55
Chapter Summary
In this chapter, we have
Discussed the importance of forecasting
Addressed component factors of the time-series
model
Performed smoothing of data series
Moving averages
Exponential smoothing
Described least squares trend fitting and forecasting
Linear, quadratic and exponential models
Statistics for Managers Using Microsoft Excel, 5e
Chap 16-56
Chapter Summary
In this chapter, we have
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
Discussed index numbers
Statistics for Managers Using Microsoft Excel, 5e
Chap 16-57