Statistics for Business and Economics

Chapter 13

Time Series:Descriptive Analyses, Models, &

Forecasting

Lyn Noble

Revisions by Peter Jurkat

Index Number

• Measures change over time relative to a base period

• Price Index measures changes in price

– e.g. Consumer Price Index (CPI)

• Quantity Index measures changes in quantity

– e.g. Number of cell phones produced annually

Simple Index Number

Based on price/quantity of a single commodity

0

100tt

YI

Y

where

Yt = value at time t

Y0 = value at time 0 (base period)

Current Index Current ValueBase Valuefffffffffffffffffffffffffffffffffffffffffffff%

Simple Index Number Example

The table shows the price per gallon of regular gasoline in the U.S for the years 1990 – 2006. Use 1990 as the base year (prior to the Gulf War). Calculate the simple index number for 1990, 1998, and 2006.

Year $ 1990 1.2991991 1.0981992 1.0871993 1.0671994 1.0751995 1.1111996 1.2241997 1.1991998 1.031999 1.1362000 1.4842001 1.422002 1.3452003 1.5612004 1.8522005 2.272006 2.572

Simple Index Number Solution

1990 Index Number (base period)

1998price 1.03100 100 79.3

1990price 1.299

1998 Index Number

1990price 1.299100 100 100

1990price 1.299

Indicates price had dropped by 20.7% (100 – 79.3) between 1990 and 1998.

Simple Index Number Solution

2006 Index Number2006price 2.572

100 100 1981990price 1.299

Indicates price had risen by 98% (100 – 198) between 1990 and 2006.

Simple Index Numbers 1990–2006

Simple Index Numbers 1990–2006

Gasoline Price Simple Index

0.0

50.0

100.0

150.0

200.0

250.0

1990

1991

1992

1993

1994

1995

1996

1997

1998

1999

2000

2001

2002

2003

2004

2005

2006

Class Exercise

Copper Steel

Period Price

($/T)

Tons

(T)

Price

($/T)

Tons (T) Simple

Composite

Base 1000 200 130 8700

… … … … …

Current 1010 190 120 9000

Example US copper and steel prices & production:

Calculate the simple (un-weighted) copper price index for the current period to closest 10%Enter: A for 90%, B for 100%, C for 110%

Composite Index Number

• Made up of two or more commodities

• A simple index using the total price or total quantity of all the series (commodities)

• Disadvantage: Quantity of each commodity purchased is not considered

Composite Index Number Example

The table on the next slide shows the closing stock prices on the last day of the month for Daimler–Chrysler, Ford, and GM between 2005 and 2006. Construct the simple composite index using January 2005 as the base period. (Source: Nasdaq.com)

Simple Composite Index Solution

First compute the total for the three stocks for each date.

Simple Composite Index Solution

Now compute the simple composite index by dividing each total by the January 2005 total. For example, December 2006:

12 / 06price100

1/ 05price

99.64100

95.49

104.3

Simple Composite Index Solution

Simple Composite Index Solution

Simple Composite Index Numbers 2005 – 2006

0.0

20.0

40.0

60.0

80.0

100.0

120.0

Class Exercise

Copper Steel

Period Price

($/T)

Tons

(T)

Price

($/T)

Tons (T) Simple

Composite

Base 1000 200 130 8700

… … … … …

Current 1010 190 120 9000

Example US copper and steel prices & production:

Calculate the simple (un-weighted) composite price index for copper and steel for the current period to nearest 10%.Enter: A for 90%, B for 100%, C for 110%

Weighted Composite Price Index

• Weights prices by quantities purchased before computing totals

• Weighted totals used to compute composite index

• Laspeyres Index– Uses base period quantities as weights

• Paasche Index– Uses quantities from each period as weights

Laspeyres Index

• Uses base period quantities as weights– Appropriate when quantities remain approximately

constant over time period

• Example: Consumer Price Index (CPI)

Calculating a Laspeyres Index

0

0 0

1

1

weighted total for period100

weighted total for base period

100

t

k

it itik

it iti

tI

Q P

Q P

Pit= price for each commodity at time t

Qit= quantity of each commodity at time t

t0 = base period

where

Note: t0 subscript stands for base period

Laspeyres Index Number Example

The table shows the closing stock prices on 1/31/2005 and 12/29/2006 for Daimler–Chrysler, Ford, and GM. On 1/31/2005 an investor purchased the indicated number of shares of each stock. Construct the Laspeyres Index using 1/31/2005 as the base period.

Daimler–Chrysler GM Ford

Shares Purchased 100 500 200

1/31/2005 Price 45.51 13.17 36.81

12/29/2006 Price 61.41 7.51 30.72

Base Value

0 01

100(45.51) 500(13.17) 200(36.81)

18498

k

it iti

Q P

01

100(61.41) 500(7.51) 200(30.72)

16040

k

it iti

Q P

Weighted total for base period (1/31/2005):

Weighted total for current period 12/29/2006:

Daimler–Chrysler GM Ford

Shares Purchased (1/31/2005) 100 500 200

1/31/2005 Price 45.51 13.17 36.81

12/29/2006 Price 61.41 7.51 30.72

Laspeyres Index Solution

,1/31/ 05 ,12 / 29 / 061

,1/31/ 05 ,1/31/ 051

100

16040100

1849886.7

k

i ii

t k

i ii

Q PI

Q P

Indicates portfolio value had decreased by 13.3% (100–86.7) between 1/31/2005 and 12/29/2006.

Class Exercise

Copper Steel

Period Price

($/T)

Tons

(T)

Price

($/T)

Tons (T) Laspeyres

Base 1000 200 130 8700

… … … … …

Current 1010 190 120 9000

Example US copper and steel prices & production:

Calculate the Laspeyres price index for the current period to nearest 1%.Enter: A for 93.6%, B for 95.5%, C for 102.3%

Paasche Index

• Uses quantities for each period as weights– Appropriate when quantities change over time

• Compare current prices to base period prices at current purchase levels

• Disadvantages– Must know purchase quantities for each time

period– Difficult to interpret a change in index when base

period is not used

Calculating a Paasche Index

0

1

1

weighted total for period100

weighted total for base period

100

t

k

it itik

it iti

tI

Q P

Q P

Pit= price for each commodity at time t

Qit= quantity of each commodity at time t

t0 = base period

where

Weights are quantities for time period t

Paasche Index Number Example

The table shows the 1/31/2005 and 12/29/2006 prices and volumes in millions of shares for Daimler–Chrysler, Ford, and GM. Calculate the Paasche Index using 1/31/2005 as the base period. (Source: Nasdaq.com)

Daimler–Chrysler Ford GM

Price Volume Price Volume Price Volume

1/31/2005 45.51 .8 13.17 7.0 36.81 5.6

12/29/2006 61.41 .2 7.51 10.0 30.72 6.1

Paasche Index Solution

,1/31/ 05 ,1/31/ 051

1/31/ 05

,1/31/ 05 ,1/31/ 051

100

.8(45.51) 7(13.17) 5.6(36.81)100

.8(45.51) 7(13.17) 5.6(36.81)

100

k

i iik

i ii

Q PI

Q P

Paasche Index Solution

12/ 29/ 06 12/ 29/ 061

12/ 29/ 06

12/ 29/ 06 1/31/ 051

100

.2(61.41) 10(7.51) 6.1(30.72)100

.2(45.51) 10(13.17) 6.1(36.81)

274.774100 75.2

365.343

k

i iik

i ii

Q PI

Q P

12/29/2006 prices represent a 24.8% (100 – 75.2) decrease from 1/31/2005 (assuming quantities were at 12/29/2006 levels for both periods)

Class Exercise

Copper Steel

Period Price

($/T)

Tons

(T)

Price

($/T)

Tons (T) Laspeyres

Base 1000 200 130 8700

… … … … …

Current 1010 190 120 9000

Example US copper and steel prices & production:

Calculate the Paasche price index for the current period (enter rounded whole number)Enter: 1 for 93.5%, 2 for 95.5%, 3 for 102.3%