Issues with stratification

A clear benefit of the stratified approach for constructing a constant-quality index is that it relies on only two assumptions to deliver such an index—conditional independence and overlap. These assumptions also have the benefit of being relatively transparent. Both conditional independence and overlap are intuitively simple conditions, although conditional independence is not a testable assumption.

The challenge with a stratified approach is finding the right balance between these two assumptions. If the stratification is too granular, then the overlap condition can fail—a good with a particular combination of characteristics may not sell in a given period, so that the sub-index for that stratum is not defined. When it comes time to aggregate each sub-index, it will not be possible to aggregate over the population of strata, and hence the overall constant-quality index is not identified from transaction prices.34 This is problematic, as a more granular partition of characteristics may give more credibility to the assumption of conditional independence. This is easiest to see when goods are grouped into pairs (the most granular partition), so that the transaction price index is a matched-model index. In this case, overlap fails when a good does not sell in either period 0 or period 1, and so the price relative for a pair of goods cannot be constructed.35

A related issue with stratification is that, if stratification occurs for many characteristics of a good, \(X\) may be of large dimension. In this case, the cell sizes for each strata may be very small, and the stratum-specific indices may be very sensitive to the addition or removal of transactions for a good. Both overlap and dimensionality serve to limit the ability to make homogeneous groups of goods for which it is easier to motivate conditional independence.

  1. One way to mitigate this problem is to construct a price index for a sub-population of goods. For example, a Paasche-like index only requires transactions in period 0 and period 1 for strata in which a good sells in period 1, \(P(t = 1 | X = x) < 1\). This also has the benefit of weakening the conditional independence assumption so that only potential prices in period 0 need to be conditionally independent of time, \(p(0) \perp t | X\).↩︎

  2. One interesting way to relax overlap in a matched-model index is to calculate the index over more than two periods. See Kirby-McGregor and Martin (2019) for an example.↩︎