Hedonic prices for cars: an application to the Spanish car market, 1981–2005

Abstract

In this article we provide a comparison of different formulations for hedonic regression analysis in order to construct a quality adjusted price index for the Spanish car market over the period 1981 to 2005. Specifically, we address the issue of instability of coefficients over time, and propose two alternative estimation procedures based, firstly, on a moving sample of observations and, secondly, on a moving average of estimated coefficients in single period equations. The statistical tests applied support the proposed methodologies. On empirical grounds two conclusions can be emphasized. Firstly, our study concludes that, taking quality changes into account, car prices in Spain deflated by consumer price index (CPI) declined by 40% between 1981 and 2005. This result is robust to the alternative estimation procedures employed in the study. Secondly, an analysis of σ-convergence shows that for quality adjusted prices a clear trend in σ-convergence emerges between 1986 and 1992, whereas such a trend does not exists for observed prices. This result has to be related to Spain's integration into the European Community (EC).

Acknowledgements

This work has benefited from a research grant from the Spanish Ministerio de Educación y Ciencia (SEJ 2006-14849). The authors thank A.I. Guerra for her support in collecting the data and the Instituto de Estudios de Automoción for providing the information on automobile sales and to J. Asensio for his comments.

Notes

¹ Rosen (1974) developed a theoretical framework for hedonic prices as equilibrium prices for supply and demand functions, both defined on the characteristics of products. Under this framework, the characteristic coefficients in the hedonic equation are the result of the interaction between the consumer's marginal valuation and the producer's marginal cost. As has been demonstrated by many authors, only under very restrictive assumptions it is possible to identify estimated coefficients as consumers’ preferences or producers’ costs. However, this problem does not affect the possibility of computing the hedonic index as an approximation to the exact hedonic index (Feenstra, 1995). Moreover, several publications by Pakes (2003, 2004) have recently revived the interest in the economic rationale of hedonic prices. This author derives the hedonic framework from the industrial organization theory in the context of technological change, differentiated product markets and heterogeneous consumers. Under his approach producers’ mark-ups are a function of the characteristics and costs of all goods and of the distribution of consumer attributes, and vary over time and depending on products. Under this approach, the coefficients in the hedonic equation may vary quickly over time and do not necessarily have to take the expected sign. Pakes's formulation has received great attention in hedonic literature. It has however generated some still unresolved controversies (see, for instance, Triplett, 2004, p. 159).

² See Diewert (2003) and Triplett (2004, Chapters V and VI).

³ Empirical evidence shows that the estimated coefficients in yearly equations are highly volatile. As long as these coefficients approximate implicit prices for characteristics, it seems desirable for both their magnitude and sign to behave according to the expected values. In our view, the volatility in the estimates can be largely explained by econometric problems. The main problems when estimating hedonic equations are multicollinearity and the reduced number of degrees of freedom which result in large variance of some of the estimated coefficients which in turn produce instability in their value. This view has however been recently disputed by Pakes in several provocative articles. In the context defined in footnote 2, this author argues that the coefficients in the hedonic function do not necessarily obey any of the restrictions associated with utility or costs functions (Pakes, 2004). As has been recognized (Berndt and Rappaport, 2003; Triplett, 2004, pp. 236–38), although some of the issues in Pakes's work are still controversial, it has made remarkable contributions to the hedonic theory. The issue we address in this article is the econometric problems that arise when estimating yearly equations with limited sample size.

⁴ In this point we acknowledge the suggestion of an anonymous referee.

⁵ Granger and Newbold (1986) made a similar proposal for combining individual forecasts derived from separate models.

⁶ The information on sales volumes was provided by the Instituto de Estudios de Automocion (Spanish motor vehicle manufacturers’ association). Unfortunately, no data was available for the different model versions. Our choice was to select a middle-range model.

⁷ We draw heavily on Raff and Trajtenberg (1995).

⁸ For instance, number of gears, maximum speed, number of airbags and central locking.

⁹ Rosen (1974) established that the hedonic functional form is an empirical issue. Subsequent studies showed that only under very restrictive assumptions is it possible to restrict the functional form.

¹⁰ It should be noted that the same result was reached by the single year equations approach. Nevertheless, it would be too cumbersome to present all the estimations year-by-year.

¹¹ The semi-log functional form has been widely used with automobile data; see, for instance, Bajic (1993), Murray and Sarantis (1999), van Dalen and Bodie (2004) and Requena-Silvente and Walker (2006).

¹² A clear distinction should be made between weighting the hedonic regression and weighting the hedonic price index (see, Triplett, 2004, pp. 189–193).

¹³ A frequently discussed issue in the literature is whether in order to estimate the hedonic price equations the variables must be weighted or not. In fact, under the usual regression model hypothesis, weighting the observations by the market shares of car models has not a conceptual justification. As it is well known, in this case weighting must lead to a loss of efficiency meanwhile not weighting must lead to a Best Linear Unbiased (BLU) estimation. Also, under these hypotheses, both weighted least squares and unweighted least squares must be consistent. So, if the model specification satisfies the standard hypothesis, it could be expected that weighted and unweighted least squares will lead to similar values for the estimated coefficients. Nonetheless, for the exposed reasons, the not weighted least squares is the preferred alternative. In our case, using a sample for the period 1987 to 2004 for which data on sales by car model are available, the estimated parameters in the weighted and not weighted alternatives are rather similar. Computing a 95% confidence interval for the valuation of the characteristics that results from the two options the two intervals always overlap, and taking as a reference the unweighted estimates, the percentage of overlapping is 47% for the variable displacement, 74% for horsepower/weight, 92% for size, 84% for fuel consumption, 84% for diesel, 54% for minivan, 67% for number of doors, 94% for air conditioned, 87% for climate control, 88% for Antilock Braking System (ABS), 63% for assisted steering, 94% for electric windows and 100% for automatic gear.

¹⁴ The process of tariff reduction started in 1986 and finished in 1992. Over those years taxes on European cars fell from 65% to zero.

¹⁵ We estimated 93 different equations. Since presentation of all equations would be too cumbersome, the results are summarized in Table 3.

¹⁶ A normal distribution has been generated for each estimated elasticity, the mean being the average of the estimated elasticities for each year in the sample and the SD the average of the estimated SEs. It is interesting to remark that the mean value of the estimated elasticities under the three estimation approaches are very similar but the SDs substantially differ.

¹⁷ See Triplett (2004, pp. 60–61).

¹⁸ When computing the index for the single year equation between years t and t − 1, it is necessary to choose the reference year. A common option suggested in the literature (Diewert, 2003) is to use the arithmetic average of the coefficients estimated in t and t − 1. This procedure offers very similar results to those obtained when estimating according to adjacent year method. In fact, as Triplett (2004, p. 63) argues, one of the earliest empirical regularities found in the hedonic literature is that the adjacent period regression often yields coefficients that are approximately the average of the coefficients estimated from a separate regression in the two periods. This is the reason why the single year approach is not presented.

¹⁹ The moving sample and moving average formulations do not allow the index for the latest years in the sample to be calculated. These values have been forecast by using an Auto-Regressive (AR) model.

²⁰ In the original specification these variables were limited to the period 1981 to 1992.