Transaction-price indices
The challenge with constructing a constant-quality price index is that potential prices are not observable, and only information on transaction prices and the time when a good sells can be used to calculate an index. There is a problem of missing data. This can be seen by linking transaction prices to potential prices as \[\begin{align*} \rho = \rho(0) + t(\rho(1) - \rho(0)), \end{align*}\] where \(\rho\) is the (log) transaction price, and \(t\) gives the period of sale (either \(t = 1\) or \(t = 0\)). The only information that can be observed from market transactions is \(\rho\) and \(t\)—the price that a good sold for and the time when it actually sold.
Given information on transaction prices, a geometric transaction-price index attempts to mimic the constant-quality index by comparing the average transaction price for the goods that sell in period 1 with the average transaction price for the goods that sell in period 0, and is thus given by \[\begin{align*} \log(I^{T}) &= E(\rho | t = 1) - E(\rho | t = 0) \\ &= E(\rho(1) | t = 1) - E(\rho(0) | t = 0). \end{align*}\]
The transaction-price index generally differs from the constant-quality index, as it compares potential prices for those goods that sell in period 1 to potential prices for those goods that sell in period 0, rather than for all goods. Any change in the composition of goods selling between period 0 and period 1 will contaminate the measurement of a pure price movement, as the same goods are not being compared over time. What the goods selling in period 0 sold for may differ from what the goods selling in period 1 would have sold for in period 0, so comparing average transaction price mixes up a change in price with a change in the composition of goods selling at different points in time.32
Note that this index is not necessarily based on price relatives, as the same number of goods may not sell in each period. If \(n(t)\) denotes the set of goods that sell in period \(t\), the geometric transaction-price index is a ratio of geometric averages, \[\begin{align*} \log(I^{T}) = \frac{\prod_{i \in n(1)} p_{i}(1)^{P_{i}|t = 1}}{\prod_{i \in n(0)} p_{i}(0)^{P_{i}|t = 0}}, \end{align*}\] where \(P_{i} | t\) is the conditional probability of observing good \(i\) in period \(t\). This is just a generalization of the usual geometric price index when different goods sell over time.
Returning to the narrative of an experiment, goods that sell in period 1 are the treatment group and goods that sell in period 0 are the control group, with the transaction-price index giving the average difference in the outcome of the experiment. This may or may not correspond the causal effect of treatment, depending on how the treatment and control groups are formed. For example, suppose the treatment is visiting a hospital and the outcome is a measure of health. Simply comparing health outcomes for those that have recently visited a hospital to those that have not, while ignoring how these groups are formed, would lead one to falsely conclude that visiting a hospital makes people unhealthy as on average those who have recently visited the hospital are in worse health than those who have not. This is because those visiting a hospital were likely to have worse health outcomes anyways, hence an apples-to-orange comparison.↩︎