Best Log-Linear Index Numbers of Prices and Quantities

By Blankmeyer, Eric | Atlantic Economic Journal, June 1990 | Go to article overview

Best Log-Linear Index Numbers of Prices and Quantities


Blankmeyer, Eric, Atlantic Economic Journal


Best Log-Linear Index Numbers of Prices and Quantities

I. Introduction

In many areas of economic research, aggregation of commodities into index numbers is essential to develop models of manageable size and to obtain reliable estimates by reducing the number of parameters and avoiding collinearity among related products. However, conventional index numbers compare prices or quantities two periods at a time, a procedure that is equivalent to fitting a line to a pair of observations instead of using the entire sample. As a result, the indices are sensitive to the often arbitrary choice of the base period. Moreover, they fail the circularity test, which has to do with the consistent computation of rates of growth over three or more periods.

At the same time, the absence of a workable probability model for index numbers largely precludes the tests of hypotheses available in other areas of quantitative economics. As a recent paper emphasizes, "Research on consumer price indices has been concentrated on methodological rather than statistical issues ...Apparently, nobody bothers about the accuracy of the published indices and the question whether, with respect of their accuracy, the indices are suitable to be used for the tasks for which they are actually used." [Balk and Kersten, 1986, p. 19].

Balk and Kersten use Dutch survey data to illustrate the effects of sampling error on Laspeyres, Paasche, and Tornqvist consumer price indices. They note that the nonlinear weighting schemes of the latter two formulas substantially complicate the analysis of the indices' sampling variances. Moreover, it will often be the case that the price and quantity data have been obtained without recourse to a formal sampling procedure of the type employed in the Dutch indices. In that event, any estimate of the indices' sampling errors probably will have to be derived from a set of strong probability assumptions of the kind familiar in econometric models.

In this paper, price and quantity indices are developed in terms of the linear regression model - specifically, in the format of a two-way analysis of variance. In a departure from the standard two-period approach, the indices for each period are derived jointly from all the sample data. Evaluation of the estimators and tests of hypotheses follow directly from the well-known properties of the general linear model. The technique is used to analyze Third World commodity exports in terms of price and quantity variations between 1971 and 1981. The variations are found to differ in magnitude and timing from indices published by the World Bank.

The approach to index numbers taken here attempts to synthesize the ideas of certain researchers, especially Banerjee and Theil, whose contributions are cited in the last section of the paper.

II. A Linear Statistical Model

When the design of index numbers is considered as a problem in aggregation, the central hypothesis is one of proportional variation. If, between two periods, all the commodity prices under consideration were to change in the same ratio, then any commodity's price would be an exact index of all the prices. Quantities that varied equiproportionally could likewise be indexed by a single commodity. For T joint observations on K commodities, it would be true that: [V.sub.rt] = [P.sub.r] + [P.sub.t] + [zeta]; r,t = 1, ..., T.

Here: exp [Mathematical Expression Omitted] is the aggregate value obtained when the [Kappa] quantities of period [theta] are evaluated at their respective prices in period [rho], exp ([p.sub.r]) is the price level in period [rho], exp ([phi].sub.[theta]) is the quantity level in period [theta]; and exp [zeta] is a constant of proportionality.

Since equation (1) is an hypothesis, it is necessary to allow for exceptions to strict equiproportionality. These discrepancies, [e.sub.rt], are supposed to enter equation (1) additively and to be independently distributed with zero means and a common variance. …

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