Commensurability in Hedonic Price Indices : Axiomatic Foundations and The Necessity of Log Transformations

Accurate measurement of price change over time and across countries is central to macroeconomic analysis and welfare assessment. The increasing availability of scanner and web-scraped data means that price comparisons must be constructed from unbalanced data where the set of goods varies across time and space. A central requirement for meaningful price comparisons is commensurability: bilateral comparisons should remain invariant to changes in units of measurement. This paper develops an axiomatic foundation for regression-based price comparisons under incomplete commodity coverage and establishes an analytical framework for examining the commensurability requirement in time-product-dummy and time-dummy hedonic models. We show that the commensurability axiom imposes strong restrictions on the functional form of hedonic price models. In particular, the logarithmic transformation is necessary for commensurability. In CPD/TPD-type models, and more generally in pooled hedonic regressions with commodity fixed effects, the log specification is both necessary and sufficient, as rescaling shocks are absorbed by item effects. By contrast, in hedonic models without commodity fixed effects, the log transformation remains necessary but is generally insufficient in unbalanced panels, changes in unit of measurement tend to induce distortions. The analysis extends to imputation methods. Pooled first-stage imputation can preserve commensurability when a common item-specific absorption channel is present, whereas period-by-period imputation typically does not. This creates a trade-off: time varying specifications enhance flexibility but weaken unit invariance. Closed-form expressions are derived to characterise the role of unbalancedness in time dummy hedonic models. Empirical results using Japanese beverage scanner data show that time-dummy hedonic indices can exhibit substantial departures from commensurability even in direct temporal comparisons, while imputation-based Jevons indices display smaller bilateral effects that can nevertheless accumulate into chain drift. These findings suggest that hedonic and imputation-based methods are best suited to relatively homogeneous product groups where unit comparability can be controlled, and they highlight the need for explicit commensurability checks in applied price measurement.