Overlap

In order to aggregate the sub-indices for each stratum into an overall price index, it must be that \(0 < P(t = 1 | X = x) < 1\)—this means that, for each stratum, some goods must sell in each period. This is the overlap (or common support) condition that ensures a price index can be constructed for each stratum. Unlike conditional independence, overlap is verifiable.

When the overlap condition holds, along with conditional independence, \[\begin{align*} \log(I^{Q}) &= E(\rho(1)) - E(\rho(0)) \\ &= E[E(\rho(1) | X) - E(\rho(0) | X)] \\ &= E[E(\rho(1) | X, t = 1) - E(\rho(0) | X, t = 0)] \\ &= E[E(\rho | X, t = 1) - E(\rho | X, t = 0)], \end{align*}\] so that \[\begin{align*} I^{Q} = \prod_{x} (I^{T}_{x})^{P(X = x)}. \end{align*}\] The constant-quality index is simply a weighted product of each sub-index, calculated with transaction prices, where the weights correspond to the likelihood of a good belonging to a particular stratum. All this information is observable, and hence the constant-quality index is identified.