Impacts of sample size and quality-adjusted imputed prices on own-price elasticities estimated using cross-sectional data

Journal of Agricultural and Applied Economics, Aug 2003 by Stockton, Matthew C

Cross-sectional data sets containing expenditure and quantity information are typically used to calculate quality-adjusted imputed prices. Do sample size and quality adjustment of price statistically alter estimates for own-price elasticities? This paper employs a data set pertaining to three food categories-pork, cheese, and food away from home-with four sample sizes for each food category. Twelve sample sizes were used for both adjusted and unadjusted prices to derive elasticities. No statistical differences were found between own-price elasticities among sample sizes. However, elasticities that were based on adjusted price imputations were significantly different from those that were based on unadjusted prices.

Key Words: cross-sectional data, imputed prices, quality-adjusted prices

The ability to conduct sound empirical work in economics relies on the abundance and reliability of the data, but data collection is a time-consuming and costly process. Efficient use of information requires employing the smallest sample and doing the least amount of manipulation possible without compromising outcome. This desire for efficiency is reflected by researchers' endeavors to extract as much knowledge from as small a set of data as possible while maintaining the integrity of the results, which modern business culture refers to as "data mining."

One output of cross-sectional data with quantity and expenditure information is the calculation of imputed price. However, this calculation does not come without a cost. Typically, cross-sectional surveys have a wide variety of respondents who consume many different commodities as well as different qualities of the same type of good. This variation in qualities is referred to as "quality difference." Quality difference can be illustrated by an example of two consumers who purchase selected cuts of the aggregate commodity beef. Consumer 1 purchases 5 lbs. of ground beef at $2.00/lb. and 1 lb. of T-bone steak at $6.00/lb., while consumer 2 purchases 2 lbs. of ground beef at $2.00/lb. and 10 lbs. of T-bone steak at $6.00/lb. Both consumers purchase beef, but the beef each consumer buys is of different quality. For consumer 1, the imputed price of beef is $2.67/lb., whereas for consumer 2, the imputed price is $5.33/lb.

Imputed prices are in turn used to derive elasticities of demand. If the differences in quality are not adjusted for, this omission may lead to estimations of own-price and/or cross-price elasticities that misrepresent the true nature of the demand relationship. To compensate for quality differences, it is generally accepted that quality adjustments on imputed prices are necessary (Cox and Wohlgenant). Adjusted prices then are used in estimating own-price and possibly cross-price elasticities of the aggregate good. To use limited time and resources most efficiently, it is essential to understand whether to use adjusted or unadjusted prices in the estimation of these elasticities.

A second issue of concern when using cross-sectional data is sample size. What effect does sample size have on the estimation of elasticities? Assuming that research resources are limited, spending the budget on an oversized sample would be unwise; resources would be better used for more precise and complete data collection. This paper will investigate two issues: (1) is the extra work associated with quality-adjusted imputed prices necessary in calculating own-price elasticities from cross-sectional data, and (2) what effect does sample size have on the estimation of the own-price elasticities?

Literature Review

We first conducted a study of existing research related to cross-sectional survey information. As mentioned in the introductory section, cross-sectional survey information with quantity and expenditure data can be used to estimate imputed prices (Allen and Bowley; George and King; Prais and Houthakker). However, Griliches found quality variation among brands of a single commodity. Furthermore, Nelson states: "The simple sum of physical quantities is found to be theoretically arbitrary and a potentially misleading measure of demand when goods are heterogeneous." These works have led to the widespread practice of applying price adjustment processes to most cross-sectional studies.

The primary reasons for price adjustment are demographic differences in consumers and quality differences in products. Demographic factors add information to demand analysis (Cox and Wohlgenant).

Deaton further justifies the need for a quality adjustment of price through an examination of geographically clustered household data. Polinsky shows that the failure to adequately specify cross-sectional price effects could result in biased and misleading demand elasticities. Hence, quality differences in goods purchased by demographically varied consumers suggest the need to use adjusted prices in elasticity calculations.

Methodology

One of the difficulties encountered in using survey data containing both household expenditures for food items and quantities purchased is that some respondents report zero expenditures and purchases for the respective products. Using a censored model estimation procedure such as the Heckman two-step method solves this zero observation dilemma (Nelson). In the probit component of the Heckman procedure, a variable is estimated accounting for the bias created by the nonparticipation of consumers, which is known as the inverse Mills ratio or IMR, which is unique for every observation. Probit analysis requires that the dependent variable be either 0 or 1. The value for nonparticipation observations is 0, and for all positive observations, the value is 1. In the second step of the Heckman procedure, the demand equation is estimated using the IMR as an additional regressor. The Heckman procedure is not the only way to address the nonparticipation response bias. Alternatives to this procedure include a maximum likelihood approach, which simultaneously estimates both the probit and demand equations.