Chap19 time series-analysis_and_forecasting

Statistics for
Business and Economics
6th Edition

Chapter 19
Time-Series Analysis and Forecasting

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

Chap 19-1

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
Statistics for Business and
integrated moving average models
Economics, 6e © 2007 Pearson
Chap 19-2
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

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 19-3

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 p denote the price in the base period
0


Let p1 be the price in a second period



The price index for this second period is

 p1 

Statistics for Business and 100
p 
Economics, 6e © 2007 Pearson  0 
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Chap 19-4

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

I1996

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

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

I2003
Statistics for384
Business and
2003
120.0
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P2003
384
= 100
= (100)
= 120
P2000
320

Chap 19-5

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



of base year prices (by
definition, since 2000 is the
base year)

P
384
I2003 = 2003 × 100 =
(100 ) = 120
P2000
320



of base year prices

I1996

I2000

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Chap 19-6

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

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Weighted
aggregate
price indexes
Laspeyres Index
Chap 19-7

Unweighted
Aggregate Price Index


Unweighted aggregate price index for period t for a group of K items:

 K

p ti 
∑
=
100 iK1 


 ∑ p 0i 
 i=1


i = item
t = time period
K = total number of items

K

∑p
i=1

ti

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

0i

= sum of the prices for the group of items in time period 0

K

∑p
i=1

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Chap 19-8

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
= 100
∑P

2004
2001

410
= (100)
= 118.8
345


Unweighted total expenses were 18.8%
higher in
Economics, 6e © 2007 Pearson2004 than in 2001
Chap 19-9
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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
purchasing the quantities traded in the base period at prices in
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
Statisticsquantities
for Business and


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Chap 19-10

Laspeyres Price Index


Laspeyres price index for time period t:



 ∑ q0ip ti 
=

100 iK1


 ∑ q0ip0i 
 i=1

K

q0i = quantity of item i purchased in period 0
p0i = price of item i in time period 0

p ti = price of item i in period t
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Chap 19-11

Laspeyres Quantity Index


Laspeyres quantity index for time period t:

 K

 ∑ qtip0i 
=

100 iK1


 ∑ q0ip0i 
 i=1

p 0i = price of item i in period 0
q0i = quantity of item i in time period 0

qti = quantity of item i in period t
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Chap 19-12

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

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Chap 19-13

(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
positive association between adjacent
observations) as
Statistics for Business and a function of R and n


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Chap 19-14

(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, α, read from the table
is greater than 0.5, the appropriate
significance level for the test against the twosided alternative is 2(1 - α)

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Chap 19-15

Counting Runs
Sales
Median

Time

--+--++++-----++++
Runs: 1 2 3
4
5
6
n = 18 and there
Economics, 6e © 2007 Pearson are R = 6 runs
Chap 19-16
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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 α = 0.05
Chap 19-17
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

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:

n
R − −1
2
Reject H0 if
z=
< −z α
2
n − 2n
4(n − 1)

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Chap 19-18

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)

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or

n
R − −1
2
z=
> z α/2
2
n − 2n
4(n − 1)

Chap 19-19

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

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Chap 19-20

Runs Test
(continued)




A filling process over- or under-fills packages,
compared to the median
n = 100 , R = 45

Z=

n
100
−1
45 −
−1
−6
2
2
=
=
= −1.206
2
2
n − 2n
100 − 2(100) 4.975
4(n − 1)
4(100 − 1)

R−

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Chap 19-21

Runs Test
H0: Fill amounts are random

(continued)

H1: Fill amounts are not random
Test using α = 0.05

Rejection Region
α/2 = 0.025

Rejection Region
α/2 = 0.025

− 1.96

0

1.96

Statistics for Business and is not less than -z = -1.96,
Since z = -1.206
.025
Economics, 6e © 2007 Pearson reject H
we do not
0
Chap 19-22
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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:

2001 2002 2003 2004 2005

Sales:

75.3

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74.2

78.5

79.7

80.2

Chap 19-23

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

1975

the horizontal axis
corresponds to the
time periods
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

1979

16.00
14.00
12.00
10.00
8.00
6.00
4.00
2.00
0.00

1977

the vertical axis
measures the variable
of interest

Inflation Rate (%)



Year

Chap 19-24

Time-Series Components
Time Series
Trend
Component

Seasonality
Component

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Cyclical
Component

Irregular
Component

Chap 19-25

Trend Component


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



Data taken over a long period of time

Sales

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d
rd tren
Up w a

Time 19-26
Chap

Trend Component
(continued)



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

Sales

Sales

Time

Downward linear trend
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Time
Upward nonlinear trend

Chap 19-27

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)
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Chap 19-28

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

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Year
Chap 19-29

Irregular Component



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



Nature
Accidents or unusual events

“Noise” in the time series

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Chap 19-30

Time-Series Component Analysis





Used primarily for forecasting
Observed value in time series is the sum or product of
components
Additive Model

X t = Tt + S t + C t × It


Multiplicative model (linear in log form)

X t = Tt S t C tIt
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
Chap 19-31
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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

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Chap 19-32

(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
Statistics for Business and X
X* =
∑ t+ j (t = m + 1,m + 2,,n − m)
t
2m + 1 j= −m

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Chap 19-33

Moving Averages


Example: Five-year moving average


First average:
*
x5 =



x1 + x 2 + x 3 + x 4 + x 5
5

Second average:
x* =
6

x2 + x3 + x 4 + x5 + x6
5

 etc.
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Chap 19-34

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 Business
etc…

Annual Sales
60
50

…

40
Sales

Year

30
20
10
0
1

and
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2

3

4

5

6

7

8

9

10

11

Year

Chap 19-35

…

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

etc…

29.4 =

23 + 40 + 25 + 27 + 32
5

 Each
10
50
Statistics for Business and moving average is for a
11
40
Economics, 6e © 2007consecutive block of (2m+1) years
Pearson
Chap 19-36
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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

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3

4

5

6

7

8

9

10

11

Year
Annual

5-Year Moving Average

Chap 19-37

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 *−.5 + x *+.5
t
Statistics for Business andt
x =
2
*
t

Education, Inc.

s
s
s
(t = + 1, + 2,, n − )
2
2
2

Chap 19-38

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

9

40.13

8.5
39.25
9.5
41.00
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Chap 19-39

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
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Chap 19-40

Calculating a Seasonal Index
Quarter

Sales

Centered
Moving
Average

Ratio-toMoving
Average

1
23
2
40
3
25
29.88
83.7
4
27
32.00
84.4
5
32
34.00
94.1
6
48
36.25
132.4
7
33
38.13
86.5
8
37
39.00
94.9
9
37
40.13
92.2
10
50
etc…
etc…
Statistics for 40
Business …
and
11
…
Economics, 6e © 2007…
Pearson …
…
…

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x3
25
100 * = (100)
= 83.7
x3
29.88

Chap 19-41

Calculating Seasonal Indexes
(continued)

Quarter

Sales

Centered
Moving
Average

Ratio-toMoving
Average

1
23
2
40
Fall
3
25
29.88
83.7
4
27
32.00
84.4
5
32
34.00
94.1
6
48
36.25
132.4
Fall
7
33
38.13
86.5
8
37
39.00
94.9
9
37
40.13
92.2
10
50
etc…
etc…
Statistics for Business …
and
Fall
11
40
…
Economics, 6e © 2007…
Pearson …
…
…

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1. Find the median
of all of the
same-season
values
2. Adjust so that
the average over
all seasons is
100
Chap 19-42

Interpreting Seasonal Indexes


Suppose we get these
seasonal indexes:
Seasonal
Season
Index



Interpretation:

Spring

0.825

Spring sales average 82.5% of the
annual average sales

Summer

1.310

Summer sales are 31.0% higher
than the annual average sales

Fall

0.920

etc…

Winter
0.945
Σ 2007 -- four seasons, so must sum to 4
Economics, 6e © = 4.000Pearson
Chap 19-43
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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)

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Chap 19-44

Exponential Smoothing
(continued)


The weight (smoothing coefficient) is α






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

The weight is:




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

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Chap 19-45

Exponential Smoothing Model


Exponential smoothing model

ˆ
x1 = x1

ˆ
ˆ
x t = α x t −1 + (1− α )x t
where:

(0 < α < 1; t = 1,2,, n)

ˆ
x t = exponentially smoothed value for period t
ˆ
x t -1 = exponentially smoothed value already

computed for period i - 1
xt = observed
Statistics for Business and value in period t
Economics, 6e © α = weight (smoothing coefficient), 0 < α < 1
2007 Pearson

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Chap 19-46

Exponential Smoothing Example


ˆ
ˆ
Suppose we use weight α = .2 x t = 0.2 x t −1 + (1− 0.2)x t

Time
Period
(i)

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
10
50
33.697
(.2)(50)+(.8)(33.697)=36.958
etc.
etc.
etc.
etc.

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ˆ
x1 = x1

since no
prior
information
exists

Chap 19-47

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

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2

3

4

5

6

7

Time Period
Sales

8

9

10

Smoothed

Chap 19-48

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

ˆ
ˆ
x n +h = x n

(h = 1,2,3 )

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Chap 19-49

Exponential Smoothing in Excel


Use tools / data analysis /
exponential smoothing


The “damping factor” is (1 - α)

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Chap 19-50

Forecasting with the Holt-Winters
Method: Nonseasonal Series



To perform the Holt-Winters method of forecasting:
ˆ
Obtain estimates of level x t and trend T t as

ˆ
x1 = x 2

T2 = x 2 − x1

ˆ
ˆ
x t = α(x t −1 + Tt −1 ) + (1− α)x t

(0 < α < 1; t = 3,4,, n)

ˆ ˆ
Tt = βTt −1 + (1− β)(x t − x t −1 )

(0 < β < 1; t = 3,4,, n)

Where α and β 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
ˆ
x n +h = ˆ
Economics, 6e © 2007 Pearson x n + hTn
Chap 19-51
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

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, α, a
trend factor, β, and a multiplicative seasonal
factor, γ

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Chap 19-52

(continued)


The recursive estimates are based on the following
equations
xt
Ft −s

(0 < α < 1)

ˆ ˆ
Tt = βTt −1 + (1− β)(x t − x t −1 )

(0 < β < 1)

ˆ
ˆ
x t = α(x t −1 + Tt −1 ) + (1− α)

xt
Ft = γFt −s + (1− γ )
ˆ
xt

(0 < γ < 1)

ˆ
Where x t is the smoothed level of the series, Tt is the smoothed trend
Statistics for Businessthe smoothed seasonal adjustment for the series
of the series, and Ft is and

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Chap 19-53

(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

ˆ
ˆ
xn+h = (x t + hTt )Ft +h−s
where the seasonal
Statistics for Business and factor, Ft, is the one generated for
the most recent
Economics, 6e © 2007 Pearsonseasonal time period
Chap 19-54
Education, Inc.

Autoregressive Models



Used for forecasting
Takes advantage of autocorrelation





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
Education, Inc.

Random
Error
Chap 19-55

(continued)


Let Xt (t = 1, 2, . . ., n) be a time series



A model to represent that series is the autoregressive
model of order p:



where


γ, φ1 φ2, . . .,φp are fixed parameters



εt are random variables that have


mean 0

constant variance
Economics, 6e © 2007 Pearson one another
and are uncorrelated with
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


Chap 19-56

(continued)


The parameters of the autoregressive model are
estimated through a least squares algorithm, as the
values of γ, φ1 φ2, . . .,φp for which the sum of squares

SS =

n

(x t − γ − φ1x t −1 − φ2 x t −2 −  − φp x t −p )2
∑

t =p +1

is a minimum

Education, Inc.

Chap 19-57

Forecasting from Estimated



Consider time series observations x1, x2, . . . , xt
Suppose that an autoregressive model of order p has been fitted to
these data:

ˆ ˆ
ˆ
ˆ


Standing at time n, we obtain forecasts of future values of the
series from

ˆ
ˆ ˆˆ
ˆ ˆ
ˆ ˆ
x t +h = γ + φ1x t +h−1 + φ2 x t +h−2 +  + φp x t +h−p

(h = 1,2,3,)

ˆ

n+ j
WhereBusinessxand is the forecast of Xt+j standing at time n and
Statistics for for j > 0,


ˆ

for j 0 x + j is Pearson
Economics,≤6e, © n2007 simply the observed value of Xt+j

Education, Inc.

Chap 19-58

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

Units

1999
2000
2001
2002
2003
2004
2005
2006
Business

4
3
2
3
2
2
4
6
and

Statistics for
Education, Inc.

Chap 19-59

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

xt

xt-1

99
00
01
02
03
04
05
06

4
3
2
3
2
2
4
6

-4
3
2
3
2
2
4

Statistics 3.5 Business and 0.9375x
ˆ
x t = for + 0.8125x t −1 −
t −2
Education, Inc.

xt-2
--4
3
2
3
2
2

Chap 19-60

Autoregressive Model
Example: Forecasting
Use the second-order equation to forecast
number of units for 2007:
ˆ
x t = 3.5 + 0.8125x t −1 − 0.9375x t −2
ˆ
x 2007 = 3.5 + 0.8125(x 2006 ) − 0.9375(x 2005 )
= 3.5 + 0.8125(6) − 0.9375(4)
= 4.625
Education, Inc.

Chap 19-61

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

StatisticsUse model for forecasting
for Business and
Education, Inc.


Chap 19-62

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

Addressed autoregressive models for forecasting
Chap 19-63
Education, Inc.


Chap19 time series-analysis_and_forecasting

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