Comparison of continuous and discrete representations of unobserved heterogeneity in logit models
Comparison of continuous and discrete representations of unobserved heterogeneity in logit models
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Logit 模型中未观察到的异质性的连续表示和离散表示的比较
DOI：
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发表时间：
2014
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影响因子：
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通讯作者：
F. Koppelman
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作者：
Xiaojing Dong;F. Koppelman
代表营销分析中未观察到的异质性或品味变化在消费者选择模型的估计中越来越受到关注。为此，混合 Logit (MXL) 模型将随机系数合并到多项 Logit 模型中，已被广泛采用。在这种情况下最常用的方法是假设随机系数遵循连续的单峰分布，并且可以使用最大模拟似然估计来获得分布的参数以及模型的其他参数。在本文中，我们将此方法称为连续混合 Logit (CMXL) 模型。该方法需要先验假设随机系数的分布是连续的，并且通常是单峰的。放宽这一假设的一种方法是通过假设具有有限支持的离散分布来非参数估计分布。我们将此方法称为离散混合 Logit (DMXL) 模型。基于 DMXL 模型，我们提出质点 MXL 模型作为连续分布假设的一种替代方案，并将其性能与潜在类 Logit 模型 (LCLM)（也是 DMXL 系列的一部分）进行比较。任一模型都可用于表示参数空间中具有离散分布的未观察到的异质性。在本文中，我们使用具有已知参数的模拟数据和具有离散选择的真实数据进行实证分析，并比较 Logit 模型中未观察到的异质性的连续和离散表示。通过模拟数据进行分析，可以深入了解区分连续参数分布和离散参数分布的能力，并更好地理解用于评估真实数据模型性能的拟合优度度量。通过仿真研究，我们发现当数据由正态分布生成时，具有单峰分布假设的 CMXL 模型优于 DMXL 模型。从实际数据分析中，我们发现CMXL模型无法恢复DMXL模型所识别的异质性。总之，我们建议在估计随机系数 MXL 模型时，应该从 CMXL 模型开始，但在未使用质点 MXL 模型或具有不同起始值的 LCLM 估计一系列 DMXL 模型的情况下，不应接受“无异质性”结论。
Representing unobserved heterogeneity or taste variations in Marketing Analytic behavioral-choice analysis is receiving increasing attention in the estimation of consumer-choice modeling. The mixed logit (MXL) model, which incorporates random coefficients into the multinomial logit model, has been widely adopted for this purpose. The most commonly adopted method in this context is to assume that the random coefficient follows a continuous, unimodal distribution, and the parameters of the distribution as well as the other parameters for the model can be obtained using maximum simulated likelihood estimation. In this article, we refer to this method as the continuous mixed logit (CMXL) model. This method requires the a priori assumption that the distribution of the random coefficient is continuous and, usually, unimodal. One way to relax this assumption is to estimate the distribution nonparametrically, by assuming a discrete distribution with finite support. We refer to this approach as the discrete mixed logit (DMXL) model. Based on the DMXL model, we propose the mass-point MXL model as one alternative to the continuous-distribution assumption and compare its performance with the latent class logit model (LCLM), also part of the DMXL family. Either model can be used to represent unobserved heterogeneity with a discrete distribution in the parameter space. In this article, we conduct empirical analyses and compare the continuous and discrete representations of unobserved heterogeneity in logit models using simulated data with known parameters and real data with discrete choices. Analysis with simulated data provides insights on the ability to distinguish between continuous and discrete parameter distributions and a better understanding of the goodness-of-fit measures used in evaluating model performance with real data. From the simulation study, we find that when the data is generated from a normal distribution, the CMXL model with the unimodal-distribution assumption is preferred to the DMXL mode. From the real data analysis, we find that the CMXL model fails to recover heterogeneity that is identified by the DMXL model. In conclusion, we suggest that when estimating a random-coefficient MXL model, one should start with a CMXL model, but should not accept a ‘no heterogeneity’ conclusion without estimating a series of DMXL models using either the mass-point MXL model or the LCLM with different starting values.