Abstract
Recall data of visits to recreational sites often contain reported numbers that appear to be rounded to nearby focal points (e.g., the closest 5 or 10). Failure to address this rounding has been shown to produce biased estimates of marginal effects and non-linear combinations of coefficients such as willingness to pay. We investigate the relative performance of three count data models used with data of the kind typically found in survey data. We create a dataset based on observed recreational trip counts and associated trip costs that exhibits substantial rounding. We then conduct a Monte Carlo simulation exercise to compare the estimated parameters, willingness to pay, and the average consumer surplus per trip for three alternative estimators: a standard Poisson model with no adjustment for rounding, a censored Poisson model, a grouped Poisson model, and a latent class Poisson model with rounding and non-rounding classes. The standard Poisson model with no adjustment for rounding exhibits significant and persistent bias, especially in estimates of non-linear effects. The grouped and latent class Poisson models, in contrast, show little to no bias in estimates.
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Notes
We use all three terms interchangeably.
Cameron and Trivedi (1998) also suggest using a truncated model, but we found no instances in which it offered any advantage over a censored model.
Evans and Herriges’ latent class model includes rounding and non-rounding classes while also allowing for the values of β to vary across classes. However, these are two separate empirical issues. Evidence suggests that rounded responses are a universal phenomenon and not specific to any type of respondent. Additionally, if class-varying parameters were also present in our simulations, it would be difficult to disentangle potential bias from rounded responses and determine which empirical strategies are best-suited to handle rounding. We therefore restrict our analysis to instances in which β is class-invariant.
Because the calibration sets half of all observations to the rounding class, up to 40% of all data can be rounded, as depicted by the gray line in Fig. 1.
The instability of the estimates generated by the censored Poisson arises from the censoring. When the share of censored observations is too large, the estimating sample contains too little variation to permit reliable estimation. As the censoring point is raised, the value of \(\beta_{0}\) at which censoring removes too much variation from the sample rises.
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The findings and conclusions in this article are those of the authors and should not be construed to represent any official USDA or U.S. Government determination or policy. This research was supported in part by the U.S. Department of Agriculture, Economic Research Service.
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Page, I.B., Lichtenberg, E. & Saavoss, M. Estimating Willingness to Pay from Count Data When Survey Responses are Rounded. Environ Resource Econ 75, 657–675 (2020). https://doi.org/10.1007/s10640-020-00403-6
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DOI: https://doi.org/10.1007/s10640-020-00403-6