Statistical inference

In practice the entire distribution of price relatives is usually not known, and an index is calculated using a sample of prices collected from producers or retailers. This means that the index values are calculated with an estimator for either \(I^{A}\) or \(I^{G}\), depending on whether an arithmetic or geometric index is the target index. Although the topics of sampling and statistical inference get complicated fast, and are beyond the scope of this course, it is worth examining how the arithmetic and geometric indices behave when calculated with a random sample of price data.26 One of the benefits of the stochastic approach is that it gives insight into the problem of estimating a price index with a sample.

With random sampling, a natural approach for estimating the arithmetic index is to replace the expected value—a population average—with the sample average. This gives a method-of-moments estimator \(\hat{I}^A = 1 / n_{s} \sum_{i = 1}^{n_{s}} p_{it} / p_{i0}\), where \(n_{s}\) is the sample size, which is just a Carli index. If price relatives are sampled at random, then is it easy to see that \(E(\hat{I}^{A}) = I^{A}\), so the Carli index is an unbiased estimator for the arithmetic index.

A natural estimator for the geometric index is \(\hat{I}^{G} = \prod_{i = 1}^{n_{s}} (p_{it} / p_{i0})^{1 / n_{s}}\), the Jevons index. Unlike the Carli index, however, the Jevons index is a biased estimator of the geometric index. It is again straightforward to show that \(E(\hat{I}^{G}) \geq I^{G}\), with \(E(\hat{I}^{G}) = I^{G}\) only holding in very special circumstances—the Jevons index systematically overestimates the geometric index (i.e., it is biased upwards).27 Although biasedness is a disadvantage of the Jevons index, the bias is not the only important statistical property of an estimator, and the Jevons index has other desirable statistical properties (e.g., it is a consistent estimator of the geometric index, and in certain circumstances achieves the maximum likelihood efficiency bound).28 It is possible to adjust for the upwards bias in the Jevons by dividing the Jevons index by \(1 + \sigma^{2}/(2n)\), where \(\sigma^{2}\) is the variance of the log price relatives (Kennedy 2003, 41), but this is not usually done in practice.

  1. See Balk (2008, chap. 5) and Selvanathan and Rao (1994, chap. 3) for an introduction to sampling and statistical inference for price indices.↩︎

  2. The Jevons index is not biased in cases where the sampling distribution is degenerate, or the variance in price relatives is zero. See Lehmann and Casella (1998, chap. 1 Theorem 7.5).↩︎

  3. The Carli index is also a consistent estimator of the arithmetic index under random sampling, but it cannot be a maximum likelihood estimator as this would require price relatives to have a Gaussian distribution, which means that price relatives can be negative. By contrast, the Jevons index is a maximum likelihood estimator for the geometric index if price relatives have a log-normal distribution, which implies that prices are always positive.↩︎