Page 89 - Compact Numerical Methods For Computers

78 Compact numerical methods for computers
where
(6.29)

given n classes or types of quantity P, of which the jth has value P tj in period t
and is assigned weight W j in the average. Note that it is assumed that the
weighting W j is independent of the period, that is, of time. However, the
weightings or ‘shopping basket’ may in fact change from time to time to reflect
changing patterns of product composition, industrial processes causing pollution,
stocks or securities in a portfolio, or consumer spending.
Substitution of (6.29) into (6.28) gives

Finally, letting

(6.30)
gives

(6.31)

Thus, if n periods of data I , P , j = 1, . . . , n, are available, we can compute the
t
tj
weightings KW . Hence, by assuming
j
(6.32)

that is, that the weights are fractional contributions of each component, we can
find the value of K and each of the W . This involves no more nor less than the
j
solution of a set of linear equations. The work of solving these is, of course,
unnecessary if the person who computes the index publishes his set of weights-as
indeed is the case for several indices published in the Monthly Digest of Staristics † .
Unfortunately, many workers do not deem this a useful or courteous practice
towards their colleagues, and I have on two occasions had to attempt to discover
the weightings. In both cases it was not possible to find a consistent set of weights
over more than n periods, indicating that these were being adjusted over time.
This created some difficulties for my colleagues who brought me the problems,
since they were being asked to use current price data to generate a provisional
estimate of a price index considerably in advance of the publication of the indices
by the agency which normally performed the task. Without the weights, or even
approximate values from the latest period for which they were available, it was
not possible to construct such estimates. In one case the calculation was to have

† Monthly Digest of Statistics UK Central Statistical Office (London: HMSO).