More on stochastic index numbers

Abstract

There are three major approaches used to estimate index numbers. The first is Fisher's test approach whereby indexes are judged on their ability to satisfy certain criteria. The economic theory of index numbers is the second approach and this deals with their foundations in utility theory. The third approach is a less well-known methodology, but one which is now attracting considerable attention, the stochastic approach which is a new way of viewing index numbers in which uncertainty and statistical ideas play a central role. While providing a point estimate for the index number like the other two approaches, the stochastic approach additionally provides the SE of the point estimate. This article enhances understanding of stochastic index numbers by showing that they are formally equivalent to the familiar optimal combination of forecasts with the individual prices playing the role of n forecasts of the overall rate of inflation. This leads to new analytical results on the impact of adding additional information within the stochastic approach framework. We provide two concrete examples of the sources of such additional information: (i) the quantity theory of money; and (ii) the use of quantity data in addition to price data. We also illustrate some of these theoretical results using real data.

Acknowledgement

We would like to acknowledge the research support of the Australian Research Council and the help of Yihui Lan.

Notes

¹ The main literature on the SA is Balk (1980), Clements and Izan (1981, 1987), Crompton (2000), Diewert (1995), Giles and McCann (1994), Miller (1984), Ogwang (1995), Ong et al . (1999), Prasada Rao and Selvanathan (1992a,b), Selvananthan (1989, 1991, 1993), Selvanathan and Prasado Rao (1992, 1994, 1999), Selvanathan et al . (1997), Selvanathan and Selvanathan, 2005.

² The above has been extended to allow for more general specifications for the distribution of the disturbance terms. Crompton (2000) analyses White-type heteroscedasticity and derives analytical scalar expressions for the standard error of inflation under this formulation. Selvanathan and Rao (1999) also consider a more general error covariance structure. Even with these extensions, the basic insight remains unchanged, namely, the variance of the estimate of the rate of inflation increases with the degree of variability of relative prices.

³ Essentially, this idea is Fisher's (1927, pp. 72–6) ‘factor reversal test’ whereby the product of the price index in period t with base period 0 and the corresponding quantity index equals the ratio of observed total expenditure in the two periods. That is, if denote the product of the price and quantity indexes, then according to the factor reversal test  

where p_it , q_it are the price and quantity consumed of good i (i = 1, … , n) in period t. The right-hand side of the above equation is what Fisher called the ‘value ratio’, which ‘unlike an index number of prices or quantities, is not an estimate but a fact … [It], therefore, afford [s] a fixed rock of truth, by which we may reckon the drifting courses of the various index numbers of prices and quantities.’ (Fisher, 1927, pp. 74–5). The equation in the previous sentence of the text is (F1) above in the form of logarithmic changes over time.