## Thursday, 17 September 2015

### Index numbers

I am today at the launch of the Global Innovation Index.  It takes a large range of countries, and a  massive range of indicators, from ICT intensity to FDI for each country.  It then weights these various measures within each country by 0.5 or by 1.  The combination of these measures adds up to an index, and it turns out that Switzerland is one and the UK is two.

If one is going to construct an index, one needs to understand a bit of index number theory and mechanics.  Thus turns out to be important, as we know from the ONS RPI and CPI.  We can learn from this and this link is useful, Differences between the RPI and CPI Measures of Inflation

"The basic approach to the measurement of inflation adopted by both the CPI and RPI is the same. Both track the changing cost of a fixed basket of goods and services over time and both are produced by combining together around 180,000 individual prices for over 650 representative items. Differences arise due to coverage, the population base of the indices and the way in which individual price quotes are combined at the first stage of aggregation."

let us focus on the way these are combined.

At the first stage of aggregation, the RPI is constructed using an arithmetic mean (AM).
There are two different methods, applied to different items but, for example, the AM would be calculated as follows;

if one price increased by 25% from the base period (which=100) and another decreased by 20% their new index values would be 125 and 80 respectively. The AM of these is;
125+80 =102.5, 2
indicating an ‘average’ price increase of 2.5%.

Now compare with the CPI

At the same level, the CPI uses a geometric mean (GM) which, taking the values used in the adjacent example is calculated thus,
125*80 = 10000 =100 , indicating that there has been
no change in prices.
An advantageous property of the geometric mean is that it can better reflect changes in consumer spending patterns relative to changes in the price of goods and services.

This formula effect means the CPI is higher relative to the RPI by about 0.8 percentage points for June 2010.

So it very much matters what weights are chosen.  One might experiment with the weights, which is what is done in the Report.  So that goes partway to checking how robust this all is, but one also needs to know how the index is constructed: why use arithmetic, which is what they do, and why chose 0.5 and 1?

Technical note. Why are these two indices in the example above different?  It is not really to do with the spending pattern weights, although that is an advantage of the geometric method.  Rather it is to do with transitivity of indices.  Consider a rise from 100 to 150 and back to 100.  On a percentage basis, inflation has risen by 150-100/100=50% in the first period and fallen by 150-100/150=33% in the second period.  But 50-33 is not equal to zero.  This is the problem with arithmetic means done this way which do not return to their base level, here 100, when adding percentage changes.

A further note on this.
Further note
The way to think of the geometric index is that it is an arithmetic average of the logs of the prices i.e. = ½*ln125+1/2*ln80 where “ln” is a natural log (so the index is calculated as the square root of the product of the two numbers).  One property of logs is that a log relation “flattens” as the number gets larger, so that a log change from 0-10 is much larger than a change from 1,000 to 1,010 even those the absolute change is 10 in both cases.  This “downweighting” of the larger numbers has a desirable economic property for it controls in part for substitution bias.  That is, as goods get more expensive, they are downweighted in the index, but that is correct since more expensive goods are presumably consumed less.  This intuition tells you that a better but more complicated index would weight log changes by their shares in total expenditure which is just what official price indices do: note that the billion dollar price index does not do this.

In case this is still seen as arcane, remember that UK pensions are now updated by the CPI and not the RPI.  That has lowered their updating and in fact counts for the bulk of austerity savings in the UK.

Finally, back to the Innovation Index

This paper, http://arxiv.org/pdf/1104.3009v2.pdf contains a discussion of how adding and multiplying can give different results.  That's only part of the point.  The index adds up a measure of, for example, GDP growth, expenditure on schools and number of YouTube videos uploaded (see  Appendix III, esp page 403).  But without a clear view on the relative weights of these indicators it is hard to interpret the overall index.