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 where Y t = value at time t Y 0 = 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 – Use 1990 as the base year (prior to the Gulf War). Calculate the simple index number for 1990, 1998, and Year $

Simple Index Number Solution 1990 Index Number (base period) 1998 Index Number Indicates price had dropped by 20.7% (100 – 79.3) between 1990 and 1998.

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

Simple Index Numbers 1990–2006

Class Exercise CopperSteel PeriodPrice ($/T) Tons (T) Price ($/T) Tons (T)Simple Composite Base …………… Current 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 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:

Simple Composite Index Solution

Class Exercise CopperSteel PeriodPrice ($/T) Tons (T) Price ($/T) Tons (T)Simple Composite Base …………… Current 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 P it = price for each commodity at time t Q it = quantity of each commodity at time t t 0 = base period where Note: t 0 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–ChryslerGMFord Shares Purchased /31/2005 Price /29/2006 Price

Base Value Weighted total for base period (1/31/2005): Weighted total for current period 12/29/2006: Daimler–ChryslerGMFord Shares Purchased (1/31/2005) /31/2005 Price /29/2006 Price

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

Class Exercise CopperSteel PeriodPrice ($/T) Tons (T) Price ($/T) Tons (T) Laspeyres Base …………… Current 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 P it = price for each commodity at time t Q it = quantity of each commodity at time t t 0 = 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–ChryslerFordGM PriceVolumePriceVolumePriceVolume 1/31/ /29/

Paasche Index Solution

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 CopperSteel PeriodPrice ($/T) Tons (T) Price ($/T) Tons (T) Laspeyres Base …………… Current 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%