Could you please explain how index numbers are constructed and indicate any difficulties that might relate to their interpretation.
What is an index number?
Before we look at how index numbers are constructed, let us first examine what they are and what they are attempting to do. Index numbers are often presented in a series. This is the case in, for example, the retail price index (RPI) or the house price index. Other indices which are produced by the Government Statistical Service include the index of producer prices and the Sterling exchange rate index, while on the stock markets there are the Financial Times-Actuaries indices of industrial ordinary shares and the Dow Jones index.
What these indices are attempting to do is to measure the movements in a set of prices (or production figures or other variable) from one time period to another. This is usually done by evaluating some type of “average” and comparing this with the same (or a similar) “average” for another time period or, more commonly, a base time period, ie a reference point. For example, the house price index for April 1986 attempts to compare house prices in April 1986 with house prices at a reference point in time, say January 1981.
In answering the question above, we shall consider the simplest problem possible and, piece by piece, expand it until the problem looks much more like a real problem.
A simple series of index numbers
Suppose that we consider the price of a particular model of car.
(Here, we have only one commodity and no information on how many were sold; we merely have a representative price — not all the cars sold in a year were sold at that price. In many cases, we may only have information on the mean price or the median price. For example, in the UK housing and construction statistics, data on mean rate rebates are quoted and we are not given information on how many rebates were given at each amount.)
Instead of quoting these figures, we can construct an index which merely compares these prices with a reference price. This reference price is the price in the base year. We quote the price in each year as a percentage of the price in the base year. Thus, the prices are now a set of index numbers.
It is necessary to quote (1976 = 100) as an identification of the base year. It is possible that another year had an index of 100. In many situations it is essential to know which year is the base year.
The above index has many interpretative problems. However, the main one, I believe, is a basic problem of economics. Basic microeconomics says that, if the price and consumption (ie sales) of the car are in equilibrium (ie at a point at which the market is clearing or close to clearing), then if the price increases, consumption will fall. Clearly this is dependent on the price actually increasing relative to the general rate of inflation and on other things being equal. However, the point here is that monitoring prices alone reveals very little information. To take the matter to an extreme, if the price rose in 1982 to £15,000, the index for 1982 would be 617.3; but if no cars were sold at that price, then the index has no meaning as regards the market for this model of car.
Another problem is that, for most commodities, price increases will, to some extent, be due to inflation. Therefore if we wish to account for inflation, we could merely remove the effect of inflation. For example, suppose the rate of inflation from 1976 to 1977 was 9.1% then, the price would be “expected” to rise from £2,430 to £2,651.13. Thus, the 1977 price of £2,685 is merely an increase of 1.28% over what would be “expected”. (This is merely making an adjustment for inflation.) Thus, the index number for 1977 would be 101.3.
A move towards realism
To resolve some of the problems and to create a somewhat more realistic situation, we can introduce some figures on sales. Suppose that we have sales of this car for a region in England.
Here, to construct a price index (for it is an index measuring the change in prices) we need to compare prices but using the information on sales. The simplest way is to compare revenue for the six years. For 1977, the index number is (2,685 X 2,717)/(2,430 X 2,831). This equals 106.0. Once again, we can account for inflation and also for a change in population in the area. Similarly, the index number for 1978 is (2,890 X 3,244)/(2,430 X 2,831) which equals 136.3.
So far, all this has been relatively simple; the problems occur when we wish to measure changes in car prices as a whole. For example, we may have prices on three models of car. Suppose again that we only have information on an “average” price.
Taking the prices of these three models as typical of the car market as a whole, we can follow (at least) two different approaches.
The aggregate method involves adding together the three prices for an individual year and comparing the sums. Thus, the index for 1977 is 107.9 (as (2,685 + 2,050 + 3,400)/(2,430 + 1,905 + 3,200)), and we get a series of index numbers of:
This method is a simplistic one as it ignores the sales of the three models in each year (or alternatively, assumes that the yearly sales of each model of car are equal). However, this method is easy to calculate.
The relative method involves constructing an index for each model of car and then taking the mean of the three indices.
Thus, we get another index for the same set of figures. This index is slightly more difficult arithmetically but has the advantage that we have indices for the three models separately as well as an overall index. However, this index still ignores the sales figures and the mean is a simple unweighted mean, ie each model has the same weight in the averaging process. If one model had sales which were four times as large as another model, then its contribution to the overall index ought to be proportionately larger. This leads to what, in statistical terms, is called a weighted average. The idea here is that the model which had sales four times the size of another model has a greater influence. If that model’s price fluctuates, the fluctuation ought to have four times the influence on the overall index.
The main problems here are (1) whether to use a relative method or an aggregate method and (2) how to allocate and incorporate the weights.
As for the former problem, we can try both methods. The relative method is a more accurate procedure but is more complicated. As for the latter problem, the incorporation is quite simple; it is the allocation which causes the problems. There are two basic approaches — to use weights calculated on the figures available for the base year, and to use figures calculated on the figures attained for the current year. (Base-year weighted indices are often referred as Laspeyres indices, after the economist who first introduced them. Similarly, current-year weighted indices are often referred to as Paasche indices.)
As an illustration, we can construct a base-year weighted aggregate index and a current-year weighted index. However, to do this, we need some information with which to allocate weights.
We shall first of all look at a base-year weighted index. Here, we use the base year of 1976 to give us the weights. For example, as approximately three times as many model 3s were sold as model 2s, any fluctuation in price of model 3 ought to have three times the effect of a fluctuation of the same relative size in the price of model 2.
To construct this index, we really consider the 1976 sales figures as a portfolio. In other words, we are going to compare prices on the basis of buying 2,831 model 1s, 614 model 2s and 1,843 model 3s. Thus, the cost of purchasing this portfolio in 1976 was £13,946,600. If we purchased the same portfolio in 1977, the total cost would be £15,123,065. The index number for 1977, therefore, is 108.4. We can repeat this process for the other years to yield:
The above is an example of a base-line weighted index, as we used the figures from the base year of 1976 to provide the weights.
For the current-year weighted index, we use the sales figures for the year under consideration. Thus, to get the price index for 1977, we compare the 1977 sales at 1977 prices with 1977 sales at 1976 prices. Similarly, for 1980, we compare the 1980 sales at 1980 prices with 1980 sales at 1976 prices. Thus, the weights change as the year we are considering changes. The resultant index is:
To construct a base-year weighted relative index, we take the three separate indices and introduce the weights. Thus, we multiply each index number by the weight before totalling them. To explain the mechanics further would serve little purpose here so we can merely examine the resultant index and also the current-year weighted relative index together with the two indices previously constructed.
Some issues in constructing a series of index numbers
We have now constructed four price indices for the same market. Which one should we use? A simple question whose answer would greatly help is “which one is correct?” However, we cannot really say which one is correct as they are all fairly accurate and, as we cannot exactly specify what it is that we are trying to measure, it is difficult to judge the relative accuracy of the four indices above. We can, however, consider various aspects of the construction of these indices and of their interpretation.
A base year weighted index is simpler to calculate than a current-year weighted index but we must be satisfied that the base year was, in some sense, typical and that the weights are not yet out-of-date. The weights used in the RPI, for example, are regularly reviewed and altered. Thus, the retail price index uses 11 major categories of expenditure. The weighting for food was 298 in 1966, 255 in 1970 and 232 in 1975, while transport and vehicles had a weighting of 116, 126, and 149, in the three years respectively. Note that not only are the weights regularly reviewed but, until the late 1960s, there were only 10 categories. An 11th category was added — meals bought and consumed outside the home — to make the index better able to represent accurately how retail consumers spend their money.
Another major problem is what items to include in an index. If we have an index for car prices, should we include all makes of cars? Should we include home-made “kit” cars, invalid cars, etc? In the RPI we cannot include every item for sale in the UK. This can be used to advantage. For example, the French have, for several years, provided subsidies for items which appear in their retail price index and thus have been able to reduce inflation very simply by increasing a subsidy.
In addition, the house price index can be dramatically affected by changes in stamp duty or tax relief on mortgage interest or even by a change in the lending policy of an institution. Thus, is some members of the Building Societies Association place greater emphasis and allocate more funds to first-time buyers (who generally buy at the lower-priced end of the market), the actual stock of housing sold may shift appreciably towards the lower end of the market. Remember that a price index can and should only measure the prices of items actually sold.
Thus we must always be aware that the index may not be as representative as we would like, whether this is due to manipulation or to factors which cannot be controlled. This representativeness has been questioned in recent years and articles in the Scotsman, the Guardian and the Financial Times of May 5 1986 all deal with a study by the Institute of Fiscal Studies which throws doubt on the RPI’s ability to measure inflation accurately. Another article in the Guardian of January 16 1983 actually proposes that there should be a separate index for different groups of the population so that separate indices will measure how the prices have changed on the goods and services which different groups of the population buy and consume.
Quantity indices
The above four indices are price indices where we used whatever other information — here it was information on quantities — to provide the weights. There are many situations where we wish to have a quantity index — an index of turnover of staff or an index of factory productivity, for example. Here, we should use whatever other information we have to provide the weights.
Comparison of indices
Suppose we had an index for car prices in the UK and one for car prices in Spain. Can we compare these? We can make direct comparisons only if the base year is the same and the method of construction is the same. Otherwise, indirect and vague comparisons are all we can make.
Conclusion
This article is not intended as a comprehensive survey of the subject of index numbers and a warning of all the interpretative problems which you might encounter. Students seeking further information on this or other statistical matters as they are applied in the profession are referred to Statistics for Property People, written by David B Edelman and published by The Estates Gazette Ltd.
