```Statistics for
7th Edition
Chapter 16
Time-Series Analysis and
Forecasting
Ch. 16-1
Chapter Goals
After completing this chapter, you should be able to:
 Compute and interpret index numbers






Weighted and unweighted price index
Weighted quantity index
Test for randomness in a time series
Identify the trend, seasonality, cyclical, and irregular
components in a time series
Use smoothing-based forecasting models, including
moving average and exponential smoothing
Apply autoregressive models and autoregressive
integrated moving average models
Ch. 16-2
16.1
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
Ch. 16-3
Single Item Price Index
Consider observations over time on the price of a single
item
 To form a price index, one time period is chosen as a
base, and the price for every period is expressed as a
percentage of the base period price
 Let p0 denote the price in the base period
 Let p1 be the price in a second period
 The price index for this second period is
 p1 
100 
 p0 
Ch. 16-4
Index Numbers: Example

Airplane ticket prices from 2000 to 2008:
Index
Year
Price
(base year
= 2005)
2000
272
85.0
2001
288
90.0
2002
295
92.2
2003
311
97.2
2004
322
100.6
2005
320
100.0
2006
348
108.8
2007
366
114.4
2008
384
120.0
I2001
P2001
288
 100
 (100)
 90
P2005
320
Base Year:
P2005
320
I2005  100
 (100)
 100
P2005
320
I2008
P2008
384
 100
 (100)
 120
P2005
320
Ch. 16-5
Index Numbers: Interpretation
I2001
P2001
288

100 
(100)  90
P2005
320
I2005
P2005
320

100 
(100)  100
P2005
320
I2008 
P2008
384
100 
(100)  120
P2005
320

Prices in 2001 were 90%
of base year prices

Prices in 2005 were 100%
of base year prices
(by definition, since 2005 is
the base year)

Prices in 2008 were 120%
of base year prices
Ch. 16-6
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
Laspeyres Index
Ch. 16-7
Unweighted
Aggregate Price Index
 Unweighted aggregate price index for period
t for a group of K items:
 K
  p ti
100 iK1

  p0i
 i1






i = item
t = time period
K = total number of items
K
p
i1
ti
K
p
i1
0i
= sum of the prices for the group of items at time t
= sum of the prices for the group of items in time period 0
Ch. 16-8
Unweighted Aggregate Price
Index: Example
Automobile Expenses:
Monthly Amounts (\$):
Index
Year
Lease payment
Fuel
Repair
Total
(2007=100)
2007
260
45
40
345
100.0
2008
280
60
40
380
110.1
2009
305
55
45
405
117.4
2010
310
50
50
410
118.8
I2010
P

 100
P
2004
2001

410
 (100)
 118.8
345
Unweighted total expenses were 18.8%
higher in 2010 than in 2007
Ch. 16-9
Weighted
Aggregate Price Indexes
 A weighted index weights the individual prices by
some measure of the quantity sold
 If the weights are based on base period quantities the
index is called a Laspeyres price index
 The Laspeyres price index for period t is the total cost of
period t , expressed as a percentage of the total cost of
purchasing these same quantities in the base period
 The Laspeyres quantity index for period t is the total cost of the
quantities traded in period t , based on the base period prices,
expressed as a percentage of the total cost of the base period
quantities
Ch. 16-10
Laspeyres Price Index
 Laspeyres price index for time period t:


  q0ip ti 

100 iK1


  q0ip0i 
 i1

K
q0i = quantity of item i purchased in period 0
p 0i = price of item i in time period 0
p ti = price of item i in period t
Ch. 16-11
Laspeyres Quantity Index
 Laspeyres quantity index for time period t:
 K

  qtip0i 

100 iK1


  q0ip0i 
 i1

p0i = price of item i in period 0
q0i = quantity of item i in time period 0
q ti = quantity of item i in period t
Ch. 16-12
16.2
The Runs Test for Randomness

The runs test is used to determine whether a
pattern in time series data is random

A run is a sequence of one or more
occurrences above or below the median

Denote observations above the median with “+”
signs and observations below the median with
“-” signs
Ch. 16-13
The Runs Test for Randomness
(continued)




Consider n time series observations
Let R denote the number of runs in the
sequence
The null hypothesis is that the series is random
Appendix Table 14 gives the smallest
significance level for which the null hypothesis
can be rejected (against the alternative of
observations) as a function of R and n
Ch. 16-14
The Runs Test for Randomness
(continued)

If the alternative is a two-sided hypothesis on
nonrandomness,


the significance level must be doubled if it is
less than 0.5
if the significance level, , read from the table
is greater than 0.5, the appropriate
significance level for the test against the twosided alternative is 2(1 - )
Ch. 16-15
Counting Runs
Sales
Median
Time
--+--++++-----++++
Runs: 1 2 3
4
5
6
n = 18 and there are R = 6 runs
Ch. 16-16
Runs Test Example
n = 18 and there are R = 6 runs

Use Appendix Table 14

n = 18 and R = 6

the null hypothesis can be rejected (against the
alternative of positive association between adjacent
observations) at the 0.044 level of significance

Therefore we reject that this time series is random
using  = 0.05
Ch. 16-17
Runs Test: Large Samples


Given n > 20 observations
Let R be the number of sequences above or below
the median
Consider the null hypothesis H0: The series is random

If the alternative hypothesis is positive association
between adjacent observations, the decision rule is:
Reject H0 if
n
R  1
2
z
 z α
2
n  2n
4(n  1)
Ch. 16-18
Runs Test: Large Samples
(continued)
Consider the null hypothesis H0: The series is random

If the alternative is a two-sided hypothesis of
nonrandomness, the decision rule is:
Reject H0 if
n
R  1
2
z
 z α/2
2
n  2n
4(n  1)
n
R  1
2
or z 
 z α/2
2
n  2n
4(n  1)
Ch. 16-19
Example: Large Sample
Runs Test

A filling process over- or under-fills packages,
compared to the median
OOO U OO U O UU OO UU OOOO UU O UU
OOO UUU OOOO UU OO UUU O U OO UUUUU
OOO U O UU OOO U OOOO UUU O UU OOO U
OO UU O U OO UUU O UU OOOO UUU OOO
n = 100 (53 overfilled, 47 underfilled)
R = 45 runs
Ch. 16-20
Example: Large Sample
Runs Test
(continued)


A filling process over- or under-fills packages,
compared to the median
n = 100 , R = 45
n
100
R  1
45 
1
6
2
2
Z


 1.206
2
2
n  2n
100  2(100) 4.975
4(n  1)
4(100  1)
Ch. 16-21
Example: Large Sample
Runs Test
(continued)
H0: Fill amounts are random
H1: Fill amounts are not random
Test using  = 0.05
Rejection Region
/2 = 0.025
Rejection Region
/2 = 0.025
 1.96
0
1.96
Since z = -1.206 is not less than -z.025 = -1.96,
we do not reject H0
Ch. 16-22
16.3




Time-Series Data
Numerical data ordered over time
The time intervals can be annually, quarterly,
daily, hourly, etc.
The sequence of the observations is important
Example:
Year:
2005 2006 2007 2008 2009
Sales:
75.3
74.2
78.5
79.7
80.2
Ch. 16-23
Time-Series Plot
A time-series plot is a two-dimensional
plot of time series data

the vertical axis
measures the variable
of interest

the horizontal axis
corresponds to the
time periods
U.S. Inflation Rate
Ch. 16-24
Time-Series Components
Time Series
Trend
Component
Seasonality
Component
Cyclical
Component
Irregular
Component
Ch. 16-25
Trend Component

Long-run increase or decrease over time
(overall upward or downward movement)

Data taken over a long period of time
Sales
Time
Ch. 16-26
Trend Component
(continued)


Trend can be upward or downward
Trend can be linear or non-linear
Sales
Sales
Time
Downward linear trend
Time
Upward nonlinear trend
Ch. 16-27
Seasonal Component



Short-term regular wave-like patterns
Observed within 1 year
Often monthly or quarterly
Year n+1
Year n
Sales
Summer
Winter
Summer
Spring
Winter
Fall
Fall
Spring
Time (Quarterly)
Ch. 16-28
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
Ch. 16-29
Irregular Component


Unpredictable, random, “residual” fluctuations
Due to random variations of



Nature
Accidents or unusual events
“Noise” in the time series
Ch. 16-30
Time-Series Component Analysis



Used primarily for forecasting
Observed value in time series is the sum or product of
components
Xt  Tt  St  Ct It

Multiplicative model (linear in log form)
Xt  TtStCtIt
where
Tt = Trend value at period t
St = Seasonality value for period t
Ct = Cyclical value at time t
It = Irregular (random) value for period t
Ch. 16-31
16.4


Moving Averages:
Smoothing the Time Series
Calculate moving averages to get an overall
impression of the pattern of movement over
time
This smooths out the irregular component
Moving Average: averages of a designated
number of consecutive
time series values
Ch. 16-32
(2m+1)-Point Moving Average



A series of arithmetic means over time
Result depends upon choice of m (the
number of data values in each average)
Examples:




For a 5 year moving average, m = 2
For a 7 year moving average, m = 3
Etc.
Replace each xt with
m
1
X *t 
X t  j (t  m  1, m  2,, n  m)

2m  1 j m
Ch. 16-33
Moving Averages

Example: Five-year moving average

First average:
x 5* 

Second average:
x *6 

x1  x 2  x 3  x 4  x 5
5
x2  x3  x 4  x5  x6
5
etc.
Ch. 16-34
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
5
6
7
8
9
10
11
…
Year
Ch. 16-35
Calculating Moving Averages
 Let m = 2
Year
Sales
Average
Year
5-Year
Moving
Average
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…

29.4 
23  40  25  27  32
5
Each moving average is for a
consecutive block of (2m+1) years
Ch. 16-36
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
2
3
4
5
6
7
8
9
10
11
Year
Annual
5-Year Moving Average
Ch. 16-37
Centered Moving Averages
(continued)

Let the time series have period s, where s is
even number


i.e., s = 4 for quarterly data and s = 12 for monthly data
To obtain a centered s-point moving average
series Xt*:

Form the s-point moving averages
x

*
t .5

s/2

j  (s/2)1
x t j
s s
s
s
(t  ,  1,  2,, n  )
2 2
2
2
Form the centered s-point moving averages
x *t .5  x *t .5
x 
2
*
t
s
s
s
(t   1,  2,, n  )
2
2
2
Ch. 16-38
Centered Moving Averages


Used when an even number of values is used in the moving
average
Average periods of 2.5 or 3.5 don’t match the original
periods, so we average two consecutive moving averages to
get centered moving averages
Average
Period
4-Quarter
Moving
Average
Centered
Period
Centered
Moving
Average
2.5
28.75
3
29.88
3.5
31.00
4
32.00
4.5
33.00
5
34.00
5.5
6
36.25
6.5
35.00 etc…
37.50
7
38.13
7.5
38.75
8
39.00
8.5
39.25
9
40.13
9.5
41.00
Ch. 16-39
Calculating the
Ratio-to-Moving Average


Now estimate the seasonal impact
Divide the actual sales value by the centered
moving average for that period
xt
100 *
xt
Ch. 16-40
Calculating a Seasonal Index
Quarter
Sales
Centered
Moving
Average
1
2
3
4
5
6
7
8
9
10
11
…
23
40
25
27
32
48
33
37
37
50
40
…
29.88
32.00
34.00
36.25
38.13
39.00
40.13
etc…
…
…
Ratio-toMoving
Average
83.7
84.4
94.1
132.4
86.5
94.9
92.2
etc…
…
…
x3
25
100 *  (100)
 83.7
x3
29.88
Ch. 16-41
Calculating Seasonal Indexes
(continued)
Fall
Fall
Fall
Quarter
Sales
Centered
Moving
Average
1
2
3
4
5
6
7
8
9
10
11
…
23
40
25
27
32
48
33
37
37
50
40
…
29.88
32.00
34.00
36.25
38.13
39.00
40.13
etc…
…
…
Ratio-toMoving
Average
83.7
84.4
94.1
132.4
86.5
94.9
92.2
etc…
…
…
1. Find the median
of all of the
same-season
values
the average over
all seasons is
100
Ch. 16-42
Interpreting Seasonal Indexes

Suppose we get these
seasonal indexes:
Season
Seasonal
Index
 Interpretation:
Spring sales average 82.5% of the
annual average sales
Spring
0.825
Summer
1.310
Summer sales are 31.0% higher
than the annual average sales
Fall
0.920
etc…
Winter
0.945
 = 4.000 -- four seasons, so must sum to 4
Ch. 16-43
16.5
Exponential Smoothing


A weighted moving average

Weights decline exponentially

Most recent observation weighted most
Used for smoothing and short term
forecasting (often one or two periods into
the future)
Ch. 16-44
Exponential Smoothing
(continued)

The weight (smoothing coefficient) is 




Subjectively chosen
Range from 0 to 1
Smaller  gives more smoothing, larger 
gives less smoothing
The weight is:


Close to 0 for smoothing out unwanted cyclical
and irregular components
Close to 1 for forecasting
Ch. 16-45
Exponential Smoothing Model
 Exponential smoothing model
xˆ 1  x1
xˆ t  α xˆ t 1  (1 α )x t
where:
(0  α  1; t  1,2,, n)
xˆ t = exponentially smoothed value for period t
xˆ t -1 = exponentially smoothed value already
computed for period i - 1
xt = observed value in period t
 = weight (smoothing coefficient), 0 <  < 1
Ch. 16-46
Exponential Smoothing Example

Suppose we use weight  = .2 xˆ t  0.2 xˆ t 1  (1 0.2)x t
Time
Period
(i)
1
2
3
4
5
6
7
8
9
10
etc.
Sales
(Yi)
23
40
25
27
32
48
33
37
37
50
etc.
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
etc.
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
etc.
xˆ 1 = x1
since no
prior
information
exists
Ch. 16-47
Sales vs. Smoothed Sales

Fluctuations
have been
smoothed
60
50

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
Sales
40
30
20
10
0
1
2
3
4
5
6
7
Time Period
Sales
8
9
10
Smoothed
Ch. 16-48
Forecasting Time Period (t + 1)

The smoothed value in the current period (t)
is used as the forecast value for next period
(t + 1)

At time n, we obtain the forecasts of future
values, Xn+h of the series
xˆ nh  xˆ n (h  1,2,3 )
Ch. 16-49
Exponential Smoothing in Excel

Use Data / Data Analysis /
exponential smoothing

The “damping factor” is (1 - )
Ch. 16-50
Forecasting with the Holt-Winters
Method: Nonseasonal Series


To perform the Holt-Winters method of forecasting:
Obtain estimates of level xˆ t and trend Tt as
xˆ 1  x 2
T2  x 2  x1
xˆ t  (1 α)(xˆ t 1  Tt 1 )  αx t
(0  α  1; t  3,4,, n)
Tt  (1 β)Tt 1  β(xˆ t  xˆ t 1 )
(0  β  1; t  3,4,, n)


Where  and  are smoothing constants whose
values are fixed between 0 and 1
Standing at time n , we obtain the forecasts of future
values, Xn+h of the series by
xˆ nh  xˆ n  hTn
Ch. 16-51
Forecasting with the Holt-Winters
Method: Seasonal Series

Assume a seasonal time series of period s

The Holt-Winters method of forecasting uses
a set of recursive estimates from historical
series

These estimates utilize a level factor, , a
trend factor, , and a multiplicative seasonal
factor, 
Ch. 16-52
Forecasting with the Holt-Winters
Method: Seasonal Series
(continued)

The recursive estimates are based on the following
equations
xt
Ft s
(0  α  1)
Tt  (1 β)Tt 1  β(xˆ t  xˆ t 1 )
(0  β  1)
xˆ t  (1 α)(xˆ t 1  Tt 1 )  α
xt
Ft  (1 γ )Ft s  γ
xˆ t
(0  γ  1)
Where xˆ t is the smoothed level of the series, Tt is the smoothed trend
of the series, and Ft is the smoothed seasonal adjustment for the series
Ch. 16-53
Forecasting with the Holt-Winters
Method: Seasonal Series
(continued)


After the initial procedures generate the level,
trend, and seasonal factors from a historical
series we can use the results to forecast future
values h time periods ahead from the last
observation Xn in the historical series
The forecast equation is
xˆ nh  (xˆ t  hTt )Ft hs
where the seasonal factor, Ft, is the one generated for
the most recent seasonal time period
Ch. 16-54
16.6


Autoregressive Models
Used for forecasting



1st order - correlation between consecutive values
2nd order - correlation between values 2 periods
apart
pth order autoregressive model:
x t  γ  φ1x t 1  φ2 x t 2    φp x t p  εt
Random
Error
Ch. 16-55
Autoregressive Models
(continued)


Let Xt (t = 1, 2, . . ., n) be a time series
A model to represent that series is the autoregressive
model of order p:
x t  γ  φ1x t 1  φ2 x t 2    φp x t p  εt

where

, 1 2, . . .,p are fixed parameters

t are random variables that have

mean 0

constant variance

and are uncorrelated with one another
Ch. 16-56
Autoregressive Models
(continued)

The parameters of the autoregressive model are
estimated through a least squares algorithm, as the
values of , 1 2, . . .,p for which the sum of
squares
SS 
n
2
(x

γ

φ
x

φ
x



φ
x
)
 t
1 t 1
2 t 2
p t p
t p 1
is a minimum
Ch. 16-57
Forecasting from Estimated
Autoregressive Models


Consider time series observations x1, x2, . . . , xt
Suppose that an autoregressive model of order p has been fitted to
these data:
x t  γˆ  φˆ1x t 1  φˆ2 x t 2    φˆp x t p  εt

Standing at time n, we obtain forecasts of future values of the
series from
xˆ t h  γˆ  φˆ1xˆ t h1  φˆ2 xˆ t h2    φˆp xˆ t hp

(h  1,2,3,)
ˆ n j is the forecast of Xt+j standing at time n and
Where for j > 0, x
for j  0 , x
ˆ n j is simply the observed value of Xt+j
Ch. 16-58
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
2002
2003
2004
2005
2006
2007
2008
2009
Units
4
3
2
3
2
2
4
6
Ch. 16-59
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
2002
2003
2004
2005
2006
2007
2008
2009
xt
4
3
2
3
2
2
4
6
xt-1
-4
3
2
3
2
2
4
xt-2
--4
3
2
3
2
2
xˆ t  3.5  0.8125xt 1  0.9375xt 2
Ch. 16-60
Autoregressive Model
Example: Forecasting
Use the second-order equation to forecast
number of units for 2010:
xˆ t  3.5  0.8125x t 1  0.9375x t 2
xˆ 2010  3.5  0.8125(x2009 )  0.9375(x2008 )
 3.5  0.8125(6)  0.9375(4)
 4.625
Ch. 16-61
Autoregressive Modeling Steps


Choose p
Form a series of “lagged predictor”
variables xt-1 , xt-2 , … ,xt-p

Run a regression model using all p
variables

Test model for significance

Use model for forecasting
Ch. 16-62
Chapter Summary





Discussed weighted and unweighted index numbers
Used the runs test to test for randomness in time series
data
Addressed components of the time-series model
Addressed time series forecasting of seasonal data
using a seasonal index
Performed smoothing of data series



Moving averages
Exponential smoothing