On the Variance/Covariance Structure of the Log Fisher Index, and Implications for Aggregation Techniques

Several aggregation methods, including the EKS, start by calculating bilateral Fisher indices. Prices and quantities are, however, subject to measurement error. This stochastic behavior, which implies both unequal variances, and non-zero correlations, between different Fisher indices, has to be taken into account if optimal estimates of aggregate PPPs are to be derived from the Fisher indices. This paper provides estimates of the variance/covariance structure of the Fisher indices, under two alternative models for stochastic variation at basic heading level: and it applies these formulae to the 1996 OECD data set, illustrating that the Fisher indices for this data set are indeed highly correlated. The paper also establishes a general theoretical result, proving that the EKS is optimal for a particular variance/covariance structure involving non-zero correlations, and hence shows that the standard EKS aggregation method is likely to be near optimal for the 1996 OECD data set