SHAZAM Index Numbers

## Index Numbers

Statistical agencies often report time series data in the form of index numbers. For example, the consumer price index is an important economic indicator. Therefore, it is useful to understand how index numbers are constructed and how to interpret them. An introduction to index numbers is in Paul Newbold [Statistics for Business & Economics, Fourth Edition, Prentice-Hall, 1995, Chapter 17, pp. 678 - 688].

First, consider computing a price index for a single item. Suppose that price data is P1, P2, ..., PT and P0 is the price in some arbitrarily chosen base year. A price index is calculated as:

100 (Pt / P0)     for   t = 1, 2, ..., T

The price index expresses the price in every period as a percentage of the base period price.

Example: The table below shows the average Canadian farm price per pound for wool (in cents). The data was retrieved from the CANSIM Statistics Canada data base (series code: D226903). The final column shows a price index where all prices are expressed as a percentage of the price in 1986. That is, the base year is 1986.

 ``` YEAR PRICE PRICE INDEX 1980 71.7 127.81 1981 64.2 114.44 1982 61.4 109.45 1983 54.6 97.33 1984 64.5 114.97 1985 58.2 103.74 1986 56.1 100.00 1987 65.6 116.93 1988 87.9 156.68 1989 75.2 134.05 1990 43.0 76.65 1991 29.3 52.23 1992 40.1 71.48 1993 36.7 65.42 ```

The price index variable shows that the wool price in 1993 was 65% of the price in 1986.

An aggregate price index for a group of commodities can be constructed by using some weighted average of the prices where the weights are quantities. Statistical agencies use this method to obtain the consumer price index. Various formula have been proposed by researchers and an illustration with SHAZAM is given in the next section below.

When price index variables are used as explanatory variables in a regression equation the estimated coefficients must be interpreted appropriately. Consider time series Y and X where X is a price index that is 100 in the base year. The linear regression equation is:

Yt = 0 + 1 Xt + et

where et is a random error. The coefficient 1 measures the change in Y for a 1 percent change in the base period price. A revision to the base period will result in an adjustment to 1.

Now consider the log-linear regression equation:

ln(Yt) = 0 + 1 ln(Xt) + ut

where ut is a random error. With this specification the coefficient 1 has an interpretation as an elasticity. This measure has the appeal that it does not depend on units of measurement.

### Calculating price indexes

The `INDEX` command in SHAZAM can be used to calculate a price index from a set of price and quantity data on a number of commodities. A number of alternative index formula are available. Further description is in the SHAZAM User's Reference Manual. In general, the format of the `INDEX` command is:

 `INDEX p1 q1 p2 q2 p3 q3 . . . / options `

where `p1, p2, ...` are the prices, `q1, q2, ...` are the quantities, and `options` is a list of desired options.

#### Example

This example uses a data set provided by Newbold. The SHAZAM commands (filename: `PRINDEX.SHA`) that follow are used to compute some price indexes. For illustration purposes, observation 8 is selected for the base period. First, a price index for the stock price for a single car manufacturer is computed. Then an aggregate price index is computed using the prices and quantities of all 4 car manufacturers in the data set.

 ```SAMPLE 1 12 * Weekly stock prices for major car manufacturers. READ P1 P2 P3 P4 20.25 4.125 5.25 46.125 19.875 4.125 6.0 45.25 19.0 4.125 5.5 45.25 19.75 4.125 5.625 46.0 20.25 3.875 6.0 48.25 19.875 3.875 5.375 48.625 19.375 4.0 5.375 47.75 19.625 4.0 5.375 50.125 21.125 4.125 5.75 51.5 22.375 4.375 5.375 51.0 25.0 4.75 7.25 54.0 23.0 4.375 6.625 52.75 * Volume of shares, in hundreds of thousands, traded in each week. READ Q1 Q2 Q3 Q4 8.2 4.3 14.4 27.1 6.3 1.5 16.0 12.9 6.7 1.3 6.9 12.1 4.5 1.9 4.4 13.6 4.3 2.7 5.0 21.9 5.4 1.5 3.8 17.3 3.8 1.7 3.1 11.7 4.3 1.5 3.8 23.8 5.4 1.8 4.9 17.0 9.5 3.5 4.1 21.4 13.7 4.4 18.1 25.0 8.3 2.6 11.3 20.5 * Compute a price index for the 1st car manufacturer - base period is week 8. GEN1 PBASE1=P1:8 GENR PINDEX1=100*P1/PBASE1 * Compute an aggregate price index INDEX P1 Q1 P2 Q2 P3 Q3 P4 Q4 / BASE=8 LASPEYRES=PALL GENR PALL=100*PALL GENR OBS=TIME(0) FORMAT(F10.0,5X,2F13.2) PRINT OBS PINDEX1 PALL / FORMAT STOP ```

The `BASE=8` option on the `INDEX` command is used to specify that the base period is observation number 8. The Laspeyres price index is saved in the variable `PALL`. SHAZAM sets the base period price as `1.0`. The price index variable can be multiplied by `100` to express the index in the more familiar form with `100` in the base period.

The SHAZAM output can be viewed. [SHAZAM Guide home]

#### SHAZAM output - Calculating price indexes

``` |_SAMPLE 1 12
|_* Weekly stock prices for major car manufacturers.
4 VARIABLES AND       12 OBSERVATIONS STARTING AT OBS       1

|_* Volume of shares, in hundreds of thousands, traded in each week.
4 VARIABLES AND       12 OBSERVATIONS STARTING AT OBS       1

|_* Compute a price index for the 1st car manufacturer - base period is week 8.
|_GEN1 PBASE1=P1:8
|_GENR PINDEX1=100*P1/PBASE1

|_* Compute an aggregate price index
|_INDEX P1 Q1 P2 Q2 P3 Q3 P4 Q4 / BASE=8 LASPEYRES=PALL

BASE PERIOD IS OBSERVATION     8
LASPEYRE WILL BE STORED AS VARIABLE: PALL
PRICE INDEX                          QUANTITY
DIVISIA PAASCHE LASPEYRES FISHER  DIVISIA    PAASCHE  LASPEYRES    FISHER
1   .930   .935   .929   .932   1623.       1614.       1625.       1619.
2   .925   .941   .914   .927   877.2       862.3       887.6       874.9
3   .911   .920   .909   .915   788.3       780.3       789.6       784.9
4   .929   .932   .926   .929   804.0       801.3       806.8       804.0
5   .972   .971   .970   .970   1218.       1220.       1221.       1221.
6   .974   .975   .973   .974   1000.       999.6       1002.       1001.
7   .957   .958   .956   .957   685.3       684.5       686.1       685.3
8  1.000  1.000  1.000  1.000   1304.       1304.       1304.       1304.
9  1.033  1.034  1.031  1.033   992.8       991.6       994.1       992.9
10  1.031  1.036  1.025  1.031   1301.       1295.       1308.       1302.
11  1.114  1.127  1.095  1.111   1656.       1637.       1685.       1661.
12  1.073  1.077  1.063  1.070   1267.       1262.       1278.       1270.

|_GENR PALL=100*PALL
|_GENR OBS=TIME(0)
|_FORMAT(F10.0,5X,2F13.2)
|_PRINT OBS PINDEX1 PALL / FORMAT
OBS            PINDEX1        PALL
1.            103.18        92.88
2.            101.27        91.38
3.             96.82        90.95
4.            100.64        92.60
5.            103.18        96.95
6.            101.27        97.33
7.             98.73        95.58
8.            100.00       100.00
9.            107.64       103.13
10.            114.01       102.55
11.            127.39       109.48
12.            117.20       106.31
|_STOP
``` [SHAZAM Guide home]