Hedonic Imputation versus Time Dummy Hedonic Indexes

1. Hedonic Imputation versus Time Dummy Hedonic Indexes Erwin Diewert, Saeed Heravi and Mick Silver Paper presented at the 10th Meeting of the Ottawa Group, Ottawa, Canada, October 10-12, 2007

2. The problem • Statistical offices try to match item models when measuring inflation between two periods. However, for product areas with a high turnover of differentiated models, there are “new” unmatched models available in the current period , but not the base, and “old” unmatched models available in the base and not current. These models are excluded in matching, but their prices are atypical. • The use of hedonic indexes is more appropriate since they can include the prices and quantities of unmatched new and old models with a correction for quality differences.

3. Two main approaches • The two main approaches to hedonic indexes are hedonic imputation (HI) indexes and dummy time hedonic (HD) indexes. • Both approaches • use hedonic regressions to correct prices for quality; • allow the indexes to incorporate matched and unmatched models; • can incorporate different weighting systems, be formulated as a geometric, harmonic or arithmetic means, and as chained or fixed base indices. • But can give quite different results.

4. The paper • Develops an exact expression for the difference in constant quality log price change between the hedonic time dummy and imputation measures – improves on previous work in the area (Silver and Heravi). • Does so for the unweighted and weighted case. • Provides an example • In Appendix 1 reconciles different approaches • In Appendix 2 demonstrates how approximate standard errors for hedonic imputation indices may be derived

5. Time dummy method • (using Jan’s paper for ease of exposition) Coefficient on dummy variable for time used as estimate of quality-adjusted price change

6. Hedonic imputation • The predicted prices in period 1 for unmatched old period 0 models • are derived from a hedonic regression estimated using period 1 data. • Period 0 characteristics are inserted into the equation. • Similarly the predicted prices in period 0 for unmatched new period 1 • models are derived from a hedonic regression estimated using period 0 • data. Period 1 characteristics are inserted into the equation. • For matched could use predicted or actual – predicted here

7. Findings for unweighted It is found that hedonic time dummy and hedonic imputation measures of log price change will be identical if any of the following three conditions are satisfied: • the average amount of each characteristic across models in each period stays the same or • the model characteristics total variance covariance matrix is the same across periods or • separate (unweighted) hedonic regressions in each period give rise to the same characteristics quality adjustment factors. NB: it is the product of these factors that counts

8. For weighted Weighted hedonic time dummy and the weighted hedonic imputation measures of log price change will be identical if any of the following three conditions are satisfied: • the period expenditure share weighted amount of each characteristic across models in each period stays the same or • the expenditure share weighted model characteristics variance covariance matrix is the same in the two periods or • separate (weighted) hedonic regressions in each period give rise to the same characteristics quality adjustment factors. NB: it is the product of these factors that counts

9. Empirical illustration: desktop personal computers (PCs) • The empirical study is quality-adjusted monthly prices of British desktop PCs in 1998. • The data are monthly scanner data: 7,387 observations (a particular make and model of a PC sold in a given month in an either specialized or non-specialized PC store-type) representing a sales volume of 1.5 million models worth £1.57 billion. • For the January to December price comparison only 161 matched models were available with 509 unmatched “old” models (available in January, but unmatched in December) and 436 unmatched “new” models (available in December but unavailable in January for matching).

10. Product of four matrices 0.1182= ½[139.593, 180.140, 14.405]

11. Which method? • The main concern with the use of the hedonic time dummy index approach is that, by construction, it constrains the parameters on the characteristic variables to be the same. • The hedonic imputation method is inherently more flexible (in that it can deal with changes in purchasers’ valuations of characteristics over the periods being compared).

12. Parameter stability • The difference between the two approaches has been found to depend on three change factors: the change in the mean characteristics, relative variance-covariance characteristics matrix, and parameter estimates. • More specifically it was found that the difference depends on the product of such changes. As such, parameter instability by itself need not be a cause for concern. Even if parameters were unstable, the difference between the indexes may be compounded or mitigated by a small change in any of the other components.

13. Why are parameters restrictions imposed on time dummy? • To conserve degrees of freedom. • To give an unambiguous estimate of the amount of price change going from period 0 to 1. • To minimize the influence of outliers, particularly in situations where degrees of freedom are small. On balance: • HI is “better” because it allows for changing characteristics prices over time; i.e., it is more “flexible” – it falls into normal index number theory: but at the cost of: • Using up more degrees of freedom • Thus all things considered, we favour HI methods unless degrees of freedom are very limited.

14. But... • ...there are different ways of formulating hedonic imputation indices and expressions for the differences between them as derived by Jan de Haan. We have used but one formulation.

15. Jan’s equations (1) (2) (3) (4)

18. unweighted

19. Weighted