The time-dummy approach

The hedonic index in the previous section can be simplified to make calculating the index easier with an approach called the time-dummy approach.40 The time-dummy approach simply takes the linear hedonic model in the previous section and restricts it so that \(\beta_{0} = \beta_{1} = \beta\). This doesn’t change anything fundamental about the index, except that it now requires fewer parameters to calculate, as \[\begin{align*} E(\rho | X, t) = \alpha_{0} + t \log(I^{Q}) + X \beta. \end{align*}\]

One interesting feature of the time-dummy index is that, like the general linear hedonic index, it is a type of stratified index when goods are partitioned by their observable characteristics. To see this, note that, with the assumption that \(\beta_{0} = \beta_{1} = \beta\), the hedonic model becomes \(h(X, t) = \alpha_{0} + t (\alpha_{1} - \alpha_{0}) + X\beta\). When goods are stratified according to \(X\), then \(h\) is saturated-in-\(X\) such that there is a unique \(\beta_x\) for each combination of \(x\) for \(X\)—each stratum gets its own parameter in the regression. In this case, the result in Angrist and Pischke (2009, sec. 3.3.1) applies, so that \[\begin{align*} \log(I^{Q}) = \sum_{x}[E(\rho | X = x, t = 1) - E(\rho | X = x, t = 0)] \omega_{x}, \end{align*}\] where \[\begin{align*} \omega_{x} = \frac{\text{var}(t | X = x) P(X = x)}{\sum_{x} \text{var}(t | X = x) P(X = x)}. \end{align*}\]

This index is of the same form as the stratified geometric index in the previous section, except that the weights depend on both the probability of a good belonging to a particular stratum and the variance of sales dates within a stratum. In the special case when each stratum contains a pair of goods, \(\omega_{x} = P(X = x)\), and the time-dummy index reduces to the pure matched-model index.

The point to take away here is that when goods are stratified according to their characteristics, the time-dummy approach for constructing a constant-quality index and the stratified approach are not fundamentally different approaches to operationalize a conditional independence assumption. Both approaches require the overlap condition, and aggregate within-stratum transaction-price indices to produce an overall price index. The usefulness of the time-dummy approach comes when overlap fails—so that goods cannot be stratified along a particular set of characteristics—as the time-dummy approach provides a parametric correction that gives the transaction-price index a constant-quality interpretation at the expense of an extra assumption.

In practice the time-dummy hedonic model is more popular than the hedonic imputation approach, in part because it is simpler, easier to compute, and less data intensive. The disadvantage to the time-dummy approach is that it is more restrictive, and requires an extra assumption. In most cases, however, the two approaches give similar results.

  1. Again, this is a poor name, as the “hedonic imputation” approach recovers the index as the coefficient on a time-dummy variable.↩︎