Transcript Chap. 13: Time Series: Descriptive Analyses, Models

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
 Yt
It  
 Y0

Current
Value
fffffffffffffffffffffffffffffffffffffffffffff
100 Current Index  Base Value %

where
Yt = value at time t
Y0 = value at time 0 (base period)
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
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
$
1.299
1.098
1.087
1.067
1.075
1.111
1.224
1.199
1.03
1.136
1.484
1.42
1.345
1.561
1.852
2.27
2.572
Simple Index Number Solution
1990 Index Number (base period)
 1990price 
 1.299 

100  
100  100
 1.299 
 1990price 
1998 Index Number
 1998price 
 1.03 

100  
100  79.3
 1.299 
 1990price 
Indicates price had dropped by 20.7% (100 – 79.3)
between 1990 and 1998.
Simple Index Number Solution
2006 Index Number
 2006price 
 2.572 

100  
100  198
 1.299 
 1990price 
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
2006
2005
2004
2003
2002
2001
2000
1999
1998
1997
1996
1995
1994
1993
1992
1991
1990
250.0
200.0
150.0
100.0
50.0
0.0
Class Exercise
Example US copper and steel prices & production:
Copper
Period
Base
…
Current
Steel
Price
($/T)
1000
Tons
(T)
200
Price
($/T)
130
Tons (T)
…
…
…
…
1010
190
120
9000
Simple
Composite
8700
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 / 06price 

100
 1/ 05price 
 99.64 

100
 95.49 
 104.3
Simple Composite Index
Solution
Simple Composite Index
Solution
Simple Composite Index Numbers 2005 – 2006
120.0
100.0
80.0
60.0
40.0
20.0
-0
6
N
S06
J06
M
-0
6
M
-0
6
J06
-0
5
N
S05
J05
M
-0
5
M
-0
5
J05
0.0
Class Exercise
Example US copper and steel prices & production:
Copper
Period
Base
…
Current
Steel
Price
($/T)
1000
Tons
(T)
200
Price
($/T)
130
Tons (T)
…
…
…
…
1010
190
120
9000
Simple
Composite
8700
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
weighted total for period t
It 
100
weighted total for base period
k

Q
it0
Pit
Q
it0
Pit0
i 1
k
i 1
100
Note: t0 subscript
stands for base period
where
Pit= price for each commodity at time t
Qit= quantity of each commodity at time t
t0 = 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
100
500
200
1/31/2005 Price
45.51
13.17
36.81
12/29/2006 Price
61.41
7.51
30.72
Shares Purchased
Base Value
Daimler–Chrysler
GM
Ford
100
500
200
1/31/2005 Price
45.51
13.17
36.81
12/29/2006 Price
61.41
7.51
30.72
Shares Purchased (1/31/2005)
Weighted total for base period (1/31/2005):
k
Q
i 1
it0
Pit0  100(45.51)  500(13.17)  200(36.81)
 18498
Weighted total for current period 12/29/2006:
k
Q
i 1
it0
Pit  100(61.41)  500(7.51)  200(30.72)
 16040
Laspeyres Index Solution
k
It 
Q
i 1
k
P
i ,1/ 31/ 05 i ,12 / 29 / 06
Q
i 1
100
P
i ,1/ 31/ 05 i ,1/ 31/ 05
16040

100
18498
 86.7
Indicates portfolio value had decreased by 13.3%
(100–86.7) between 1/31/2005 and 12/29/2006.
Class Exercise
Example US copper and steel prices & production:
Copper
Steel
Period
Price
($/T)
Tons
(T)
Price
($/T)
Tons (T)
Base
1000
200
130
8700
…
…
…
…
1010
190
120
9000
…
Current
Laspeyres
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
weighted total for period t
It 
100
weighted total for base period
k

Q P
i 1
k
it it
Q P
i 1
it it0
100
Weights are
quantities for
time period t
where
Pit= price for each commodity at time t
Qit= quantity of each commodity at time t
t0 = base period
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
k
I1/ 31/ 05 
Q
P
Q
P
i 1
k
i 1
i ,1/ 31/ 05 i ,1/ 31/ 05
 100
i ,1/ 31/ 05 i ,1/ 31/ 05
.8(45.51)  7(13.17)  5.6(36.81)

100
.8(45.51)  7(13.17)  5.6(36.81)
 100
Paasche Index Solution
k
I12 / 29 / 06 
Q
i 1
k
P
i12 / 29 / 06 i12 / 29 / 06
Q
i 1
100
P
i12 / 29 / 06 i1/ 31/ 05
.2(61.41)  10(7.51)  6.1(30.72)

100
.2(45.51)  10(13.17)  6.1(36.81)
274.774

100  75.2
365.343
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
Example US copper and steel prices & production:
Copper
Period
Base
…
Current
Steel
Price
($/T)
1000
Tons
(T)
200
Price
($/T)
130
Tons (T)
…
…
…
…
1010
190
120
9000
Laspeyres
8700
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%