Demand system estimation in the absence of price data: an application of Stone-Lewbel price indices

Abstract

This article evaluates the feasibility of estimating a system of demand equations in the absence of price information using the approach developed by Lewbel (1989). Stone-Lewbel (SL) price indices for commodity groups are constructed using information on the budget shares and the Consumer Price Indices (CPIs) of the goods comprising the commodity groups, which allows for household-level prices to be recovered. This study evaluates how susceptible are elasticities and marginal effects estimates from traditional parametric demand systems to the CPI used in the construction of the SL prices. To do this, three alternative regional CPIs are considered for the construction of the SL prices: monthly, quarterly and a constant (unity) price index. Elasticities and marginal effect estimates are computed for eight food commodity groups using the Exact Affine Stone Index (EASI) model as the parametric demand system and data from the United States Consumer Expenditure Survey. The estimates proved to be robust to the alternative regional CPIs considered in the construction of SL price indices, even to the absence of one. Hence, the results suggest that it is possible to accurately estimate a demand system even in the absence of price information.

Keywords:

• food demand
• Exact Affine Stone Index demand model
• Consumer Expenditure Survey
• Consumer Price Index

JEL Classification:

• D12
• Q19

Notes

¹ Though this problem is characteristic of cross-sectional data, it is not endemic to it; Carliner (1973) experienced the same limitation when working with panel data.

² The reference household is the household with average budget shares.

³ Lewbel and Pendakur (2009) conducted an empirical comparison between the actual model and its linear approximation without finding any major differences.

⁴ To analyse the sensitivity of the results to the exclusion of this interaction, we estimated an LA/EASI model with the interaction terms between prices and socio-demographic variables; however, the results were similar to those using the reduced model in Equation 2.

⁵ Observations with values of income and total expenditures below or equal to zero correspond to households with severe ‘missing data’ problems. For example, total zero expenditures correspond to a case where not only all the values of the dependent variables in the system are equal to zero, but also all the prices and the expenditure variable values are ‘missing’. The cases of negative and zero income, as described by the BLS, are believed to be mainly due to nonresponse to questions about income. Following previous studies (e.g., Raper et al., 2002), we deleted these observations from the sample.

⁶ An alternative to the CPI for all expenditure items is the CPI for food at home which is also available at the regional level. The results were robust to the regional CPI used for deflating the national food subgroups CPIs.

⁷ To assess the relevance of SL prices for our data, we also estimated a complete demand system using only monthly CPIs as proxy for prices. Results obtained for this system included positive compensated own-price elasticity for one of the commodity groups.

⁸ To test the sensitivity of our results to the presence of censored observations, we ran a full system of equations using only the uncensored observations. We found our estimates to be robust even when using only households with positive expenditures.

⁹ Different sets of demographic variables were used at the different stages of the estimation process to avoid multicollinearity issues. The variables’ superscripts in Table 2 denote the model in which the variable was used as control.

¹⁰ The reference person is defined by the BLS as the person who owns or rents the home.

¹¹ We also estimated percentage errors for parameter estimates. Mean percentage errors for quarterly and unity CPI-based SL prices were 1.08% and 415%, respectively. The high mean percentage error for unity CPI-based SL prices can be explained by the presence of parameter estimates not statistically different from zero. A table of parameter estimates is available from the authors upon request.

¹² Given space limitations we only report: (1) marginal effects of socio-demographic variables included in the demand equations and (2) marginal effects from models estimated using monthly regional CPI-based SL prices. Marginal effects of variables included only in the sample selection probit models, and marginal effects from the models estimated using quarterly or unity CPI-based SL prices are available from the authors upon request.