Statistics for business economics 7th by paul newbold chapter 16

Statistics for
Business and Economics
7th Edition

Chapter 16
Time-Series Analysis and
Forecasting
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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

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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

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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 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 
100 
 p0 
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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

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

P2001
288
I2001 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


Prices in 2001 were 90% of
base year prices

I2005

P2005
320

100 
(100) 100
P2005
320



Prices in 2005 were 100%
of base year prices
(by definition, since 2005 is
the base year)

I2008

P
384
 2008 100 
(100) 120
P2005
320



Prices in 2008 were 120%
of base year prices

P2001
288
I2001 
100 
(100) 90
P2005
320

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

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

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

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

K

 p ti

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

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

i1

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

K

p

0i

i1

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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

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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
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
quantities
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Ch. 16-10


Laspeyres Price Index
 Laspeyres price index for time period t:
K



  q0ip ti 

100 iK1



  q0ip0i 
 i1

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
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

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
qti = quantity of item i in period t
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

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

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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
positive association between adjacent
observations) as a function of R and n

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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 - )

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Ch. 16-15


Counting Runs
Sales
Median

Time

--+--++++-----++++
Runs: 1 2 3

4

5

6

n = 18 and there are R = 6 runs
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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

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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

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

n
R  1
2
z
  zα
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)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

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
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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)

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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
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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

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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

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Ch. 16-24


Time-Series Components
Time Series
Trend
Component

Seasonality
Component

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Cyclical
Component

Irregular
Component

Ch. 16-25