1. Introduction
With the increasing use of household-level data from third-party vendors such as NielsenIQ and Circana to estimate censored response models (e.g., the Tobit model (Zheng et al., Reference Zheng, Dharmasena, Capps and Janakiraman2018) and the Heckman sample selection model (Capps et al., Reference Capps, Cheng, Kee and Priestley2023; Cheng et al., Reference Cheng, Capps and Dharmasena2021a)) as well as demand systems models for various commodities, the issue of price or unit value imputation merits attention. This issue arises from the fact that households are observed to purchase zero amounts of certain products during specific periods. Hence, the ratio of expenditures to quantities purchased, often named unit values as a proxy for retail prices, is unknown. Since previous studies suggest bias associated with missing unit values may occur, apart from the inherent endogeneity issues (Deaton, Reference Deaton1988, Reference Deaton1990, Reference Deaton1997), it is crucial to determine how to impute these unit values when they are missing (Dong et al., Reference Dong, Shonkwiler and Capps1998; Erdem et al., Reference Erdem, Keane and Sun1998).
The literature has extensively explored methods for imputing missing observations (Little and Rubin, Reference Little and Rubin2019; Pigott, Reference Pigott2001; Schafer, Reference Schafer1997). A commonly used approach is ad hoc forward or backward extrapolation (Enders, Reference Enders2022). However, in price imputation, this method has been criticized for introducing selection bias (Erdem et al., Reference Erdem, Keane and Sun1998), especially when missing data are not random.
Imputation methods also have been extensively explored in survey data, primarily focusing on nonresponse (Rubin, Reference Rubin2004). In price imputation, the challenge is most prominent in constructing price indices (Bradley, Reference Bradley2003), where observed data often consists of store-level prices without links to household-level characteristics such as demographics or purchase behavior.
More recently, advanced techniques such as machine learning (Zeng and Rao, Reference Zeng and Rao2024), Markov Chain Monte Carlo methods (Kyureghian et al., Reference Kyureghian, Capps and Nayga2011), and geospatial data integration (Hill and Scholz, Reference Hill and Scholz2018) have gained attention. While these methods offer potential improvements, they are often criticized for their complexity in both modeling and implementation. Given that price imputation is not the primary focus of demand analysis, the choice of method should balance ease of implementation with predictive accuracy.
The most used methods for imputing missing unit values of demand analysis in the literature include regression-based imputation, household mean imputation, and retailer mean imputation. Despite the widespread reliance on imputation techniques in general, there has been limited systematic evaluation of their predictive accuracy and implications for price imputation in demand analysis. To the best of our knowledge, no prior study has rigorously compared these methods to determine which yields the most accurate unit value imputations. By filling this gap, our findings provide new insights into the trade-offs among different imputation strategies, contributing to a more robust foundation for price imputations in empirical demand analysis.
Additionally, our study sheds light on the implications of different imputation methods within a censored QUAIDS demand system framework. This aspect also has been largely unexplored in prior research, and our findings emphasize potential differences that can arise when using various imputation methods. We believe this contribution is valuable for researchers working with scanner data, where missing price information is a persistent challenge.
In a case study, we utilize household purchases of five categories of milk products from the Nielsen Homescan Panel over 2018–2020 to compare the performance of imputed unit values obtained through these three approaches. Furthermore, we assess how the three methods affect the magnitude of compensated own-price and cross-price elasticities as well as expenditure elasticities associated with the estimation of a censored Quadratic Almost Ideal Demand System (QUAIDS) model.
The milk industry serves as a valuable case study due to its widespread consumption, nutritional importance, and evolving market dynamics. As a staple food in many households, milk plays a central role in consumer purchasing behavior. In recent years, this industry has undergone notable transformations, including the rise of plant-based milk alternatives and increased product differentiation within dairy milk categories (e.g., lactose-free, organic, and flavored milk). These developments make milk a representative product for analyzing demand interrelationships using system methods.
Our findings reveal that the differences in predicted prices and estimated price elasticities via these three price imputation methods are not trivial. The predicted values from these three methods were not highly correlated. In our case study, the regression-based method outperforms the household mean and the retailer mean imputation methods for all five milk categories. The retailer mean imputation method generated statistically different estimates of own- and cross-price price elasticities from the other two imputation methods.
2. Unit value imputation methods
Using the ratio of dollar sales to quantities purchased, we derive unit values and proxies for retail prices. The construction of unit values is consistent with the methodology proposed by Deaton (Reference Deaton1987). Indeed, as pointed out by Deaton, bias associated with the use of unit values may occur (Deaton, Reference Deaton1988, Reference Deaton1990, Reference Deaton1997). The bias is attributed to quality variation and reporting errors in expenditures and/or quantities (measurement errors). Deaton (Reference Deaton1988) suggested that the bias associated with quality variation makes the demand for a commodity appear to be more elastic, overstating the response of quantity to changes in price.
Gibson and Rozelle (Reference Gibson and Rozelle2011) suggested that two types of measurement error bias are evident: (1) attenuation bias because unit values are noisy measures of market prices; and (2) bias due to correlated errors in measuring expenditures and/or quantities. In the case of attenuation bias, they noted that the bias was in the opposite direction to that attributed to quality variation. If so, then the bias due to quality variation and the bias due to attenuation are offsetting to some degree. However, Gibson and Rozelle (Reference Gibson and Rozelle2011) also pointed out that the bias due to correlated errors operated in the opposite direction to attenuation bias. Consequently, the bias due to correlated errors reinforces the bias due to quality effects. Importantly, Gibson and Rozelle (Reference Gibson and Rozelle2011) documented that the bias associated with quality variation was relatively minor, also consistent with the finding of Deaton (Reference Deaton1997).
2.1. Regression-based imputation
The regression-based imputation method utilizes demographic information from purchasing households to infer unit values for non-purchasing households. This method has been widely used for unit value imputation in the economic literature (Alviola and Capps, Reference Alviola and Capps2010; Bakhtavoryan et al., Reference Bakhtavoryan, Capps and Dharmasena2022; Capps et al., Reference Capps, Cheng, Kee and Priestley2023, Cheng et al., Reference Cheng, Capps and Dharmasena2021a, Reference Cheng, Capps and Dharmasena2021b; Dharmasena and Capps, Reference Dharmasena and Capps2012, Reference Dharmasena and Capps2014; Kyureghian et al., Reference Kyureghian, Capps and Nayga2011; Lopez et al., Reference Lopez, Malaga, Chidmi, Belasco and Surles2012). In Alviola and Capps (Reference Alviola and Capps2010), Dharmasena and Capps (Reference Dharmasena and Capps2012, Reference Dharmasena and Capps2014), Cheng et al. (Reference Cheng, Capps and Dharmasena2021a, Reference Cheng, Capps and Dharmasena2021b), and Capps et al. (Reference Capps, Cheng, Kee and Priestley2023). Missing imputed values for households who did not purchase the products in question were generated via auxiliary regressions in which observed unit values for each of the respective products were regressed as a function of demographic factors, typically household income, household size, and region as well as dummy variables pertaining to time period. These instrument variables have been used in these prior studies to not only obtain values of missing prices but also to mitigate price endogeneity issues. Notably, the predicted unit values using a regression-based method are specific to the household, particularly household income, household size, geographic region, and to a particular period.
2.2. Household mean imputation
Household mean imputation, also known as group mean imputation and cell mean imputation (Lopez, Reference Lopez2014), replaces missing unit values of non-purchasing households with mean unit values based on purchasing households according to various criteria. For example, Ackerberg (Reference Ackerberg2001) used observed unit values obtained in the same week and in the same store from purchasing households to replace missing unit values for non-purchasing households. Additionally, Dong et al. (Reference Dong, Gould and Kaiser2004) and Golan et al. (Reference Golan, Perloff and Shen2001) replaced missing prices for non-purchasing households with the mean price of purchasing households located in the same state and in the same area of urbanization. This imputation method assumes that both non-purchasing and purchasing households face the same average price level for a specific product in a particular geographic location and during a particular time. Household income and household size do not play any role in predicting unit values based on household mean imputation.
2.3. Retailer mean imputation
Unlike the regression-based and household mean imputation methods, which use data from household purchasing records (e.g., the Nielsen Homescan Panel), the retailer mean imputation method utilizes actual retail price information based on purchases that occur at stores located in various geographic markets affiliated with third-party vendors like NielsenIQ and Circana. The respective vendors themselves impute prices using the average price of the Universal Product Code (UPC) during a particular time by retail outlet. Hence, the retailer mean imputation method relies on average prices common to the same geographic area(s) to represent the unobserved prices of products related to non-purchasing households (Zhen et al., Reference Zhen, Finkelstein, Nonnemaker, Karns and Todd2014). Importantly, these price imputations do not vary across households within the same period. The variability of unit values based on the household imputation method and the retailer imputation method typically is much less than the variability of unit values based on the regression-based imputation method. Additionally, like the household mean imputation method, household income and household size do not play any role in predicting unit values based on retailer mean imputation.
3. Data
We utilize household purchase data concerning various milk products from the Nielsen Homescan Panel for price imputation using regression-based and household mean methods. These datasets are aggregated by quarter and by year.Footnote 1 Additionally, we categorize these products into five categories: traditional white milk, traditional flavored milk, lactose-free milk, organic milk, and the aggregate of plant-based milk alternatives (PBMA).Footnote 2 Our dataset contains quarterly milk purchase data of 43,310 households from 2018 to 2020.
For the regression-based method, we used an out-of-sample validation approach. Specifically, we regressed observed unit values for each of the five products for calendar years 2018 and 2019 (serving as the training period), where observed unit values for each of the five product categories were regressed on household income, household size, DMA fixed effects, and quarter and year indicators.Footnote 3 For all five categories considered, heteroscedasticity was detected using the Breusch-Pagan test in each of the regression-based imputation equations. We address heteroscedasticity by calculating robust standard errors (White, Reference White1980). We then applied the estimated models to predict unit values for calendar year 2020 (the testing period) and evaluated the prediction accuracy by comparing imputed values against the observed 2020 values. For the household mean method,Footnote 4 we took the average of the observed unit values by DMA and quarter to obtain the predicted values for each of the five products for the calendar year 2020. For the retailer mean method, we matched households based on retail prices reported by Nielsen from retail outlets in the same DMA and obtained the average of observed DMA unit values per quarter for the calendar year 2020.Footnote 5
Table 1 shows summary statistics of the observed values and the missing rates of unit values for each product category over the period 2018–2020. The missing rate for the price of a specific product is calculated as the number of observations with zero purchases divided by the total number of observations. Given the rather sizeable missing rates associated with the milk-related products, the issue of unit value imputations warrants attention.
Table 1. Average unit values and missing rates for each milk category, 2018–2020
Note: Standard errors are in parentheses.
4. Empirical results
Mean predicted unit values vary across imputation methods, as shown in Table 2. In Table 3, we examine the correlations among predicted unit values from the three imputation methods to assess their consistency. High correlations indicate similar imputed prices across methods, suggesting minimal impact on demand estimates. Lower correlations, however, highlight discrepancies that may influence price elasticity estimates. The respective predicted unit values associated with these three methods were not highly correlated. These results imply that the use of these imputations may yield different magnitudes of own-price elasticities, cross-price elasticities, and total expenditure elasticities.
Table 2. Means of observed and predicted unit values for calendar year 2020
Table 3. Correlations among predicted unit values based on the three imputation methods
To measure the precision of the predicted unit values against the observed unit values, we used three conventional metrics associated with forecasting: root mean square error (RMSE), mean absolute error (MAE), and mean absolute percent error (MAPE). These metrics, presented in Table 4, revealed that unit values predicted via the regression-based method had the smallest RMSE, MAE, and MAPE for all five product categories. Hence, among the three methods considered, the regression-based method outperformed the household mean and the retailer mean methods regarding prediction accuracy. Notably, most MAPE values exceeded 25%, indicating disparities between predicted and observed unit values, especially for traditional flavored milk.
Table 4. Evaluations of predictions based on the three imputation methods with observed values for calendar year 2020
Finally, we compared the compensated own-price and cross-price elasticities derived from the estimation of a household-level censored QUAIDS model (Banks et al., Reference Banks, Blundell and Lewbel1997), based on imputed values using the three methods. Specifically, we adopted and re-estimated the QUAIDS model of Capps and Wang (Reference Capps and Wang2024) using the imputed values associated with each of the three methods in analyzing interrelationships among dairy milk and plant-based milk alternatives for U.S. households from 2018 to 2020.
In Figure 1, we show the estimates of compensated own-price and cross-price elasticities with 95% confidence intervals based on the three imputations associated with missing unit values. In Figure 2, we compare the estimates of expenditure elasticities with 95% confidence intervals based on these unit value imputations. In most cases, the compensated price elasticities estimated via the regression-based and the household mean methods for missing unit values were relatively consistent with each other. But these compensated price elasticities were statistically different from those obtained using the retailer mean method. For example, from Figure 1, the compensated own-price elasticity for traditional white milk, calculated using the regression-based and household mean method for missing unit values, was less than 1 in absolute value, indicative of inelastic demand. In contrast, the compensated own-price elasticity for traditional white milk based on missing unit values imputed using the retailer mean method was calculated to be greater than 1 in absolute value, indicative of elastic demand.
Figure 1. Compensated own-price and cross-price elasticity estimates and 95% confidence intervals using three unit value imputation methods.
Figure 2. Total expenditure elasticity estimates and 95% confidence intervals using three unit value imputation methods.
However, regarding total expenditure elasticities, as presented in Figure 2, the estimates from all three methods displayed relative consistency. That said, realize in demand system analysis that due to the homogeneity condition, the sum of the unconditional own-price and cross-price elasticities along with the total expenditure elasticity for each category must sum to zero. Hence, if differences across imputation methods give rise to differences in own-price and cross-price elasticities, then these differences may translate into differences in total expenditure elasticities.
5. Concluding remarks
Regression-based, household mean, and retailer mean imputation methods are commonly used to address missing unit values in estimating censored response and demand systems models. This study compared these imputation methods using data from household purchases of five milk products from 2018 to 2020, finding that predicted unit values for 2020 were not highly correlated across methods. In our case study, the regression-based method was preferred based on RMSE, MAE, and MAPE metrics. The study also assessed the impact of these imputation methods on the magnitude and significance of compensated own-price, cross-price, and expenditure elasticities from a censored QUAIDS model. While expenditure elasticities were unaffected by the imputation method, the type of imputation significantly influenced compensated price elasticities, with those from the retailer mean method differing statistically from the others. All these results suggest that the choice of price imputation method plays a non-trivial role in estimating price elasticities using household-level scanner data.
The observed differences in imputation outcomes can be attributed to how each method handles missing price data, particularly in relation to the extent and pattern of missingness. Household mean imputation assumes stable purchasing patterns within households, making it appropriate when missing prices occur among regular buyers. In contrast, regression-based imputation leverages observable household and market characteristics, which may be more effective when price variation is driven by demographics or regional differences. Retailer mean imputation, on the other hand, assumes uniform store-level pricing; however, if prices vary significantly across retailers, this method may introduce bias. Given these distinctions, selecting an imputation method that aligns with the data structure is critical, as it can influence demand estimation results.
In this study, we employ a linear model to impute missing prices using the regression-based approach, consistent with standard approaches in the literature. While this method provides a straightforward and interpretable framework, we acknowledge that alternative regression specifications, including non-linear models or additional predictor variables, could enhance imputation accuracy. In addition, as is common in studies using scanner data, if a household does not record a purchase of a particular item in a given period, it is not possible to determine whether the household chose not to buy the item (true zero demand), did not encounter the product, or failed to scan the item due to recording error (Einav et al., Reference Einav, Leibtag and Nevo2010).
Additionally, while our analysis focuses on a specific set of products and time periods, replicating this approach across different product categories and extended time frames would further assess the robustness of our findings. The primary objective of this paper is to provide a practical reference for commonly used price imputation methods in demand estimation. Future research could explore more complex models, including machine-learning techniques, to refine prediction accuracy. Going forward, we recommend replicating this analysis across different products and time periods to further validate and refine our conclusions.
Acknowledgements
Researchers’ own analyses calculated (or derived) based in part on data from Nielsen Consumer LLC and marketing databases provided through the NielsenIQ Datasets at the Kilts Center for Marketing Data Center at The University of Chicago Booth School of Business. The conclusions drawn from the NielsenIQ data are those of the researchers and do not reflect the views of NielsenIQ. NielsenIQ is not responsible for, had no role in, and was not involved in analyzing and preparing the results reported herein.
Author contribution
Conceptualization, OCJ; Methodology, LW, OCJ; Formal Analysis, LW; Data Curation, LW; Writing – Original Draft, LW, Writing – Review and Editing, OCJ, LW; Supervision, OCJ; Funding Acquisition, NA.
Financial support
This research received no specific grant from any funding agency, commercial or non-profit sectors.
Data availability statement
Researcher’s own analyses calculated (or derived) based in part on data from The Nielsen Company (US), LLC, and marketing databases provided through the Nielsen Datasets at the Kilts Center for Marketing Data Center at the University of Chicago Booth School of Business. The conclusions drawn from the Nielsen data are those of the researcher and do not reflect the views of Nielsen. Nielsen is not responsible for, had no role in, and was not involved in analyzing and preparing the results reported herein. Because of contractual stipulations, we are not at liberty to share the data publicly.
Competing interests
Authors declare no conflict of interests.
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