Research note the impact of heterogeneity and ill-conditioning on diffusion model parameter estimates
Research note the impact of heterogeneity and ill-conditioning on diffusion model parameter estimates
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DOI：
10.1287/mksc.21.2.209.151
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发表时间：
2002-03-01
影响因子：
5
通讯作者：
Lee, J
中科院分区：
文献类型：
--
作者：
Bemmaor, AC;Lee, J
准确的市场规模和早期采用模式的评估对于参与新产品发布的管理者的战略决策至关重要。本文提出的方法论通过数据中“超低音”偏斜的方向，或者同等地，通过总体的潜在异质性来解释 Bass 模型的参数估计、p（创新系数）、q（模仿系数）和 c（市场渗透率）的变化。这项研究显示了这些参数估计的显着相反的模式，具体取决于广义模型检测到的扩散曲线的偏斜，即伽马/移位 Gompertz (G/SG) 模型，该模型嵌入了 Bass 模型作为特例。 Bemmaor (1994) 最初提出的 G/SG 模型基于两个假设：(1) 个人层面的首次购买时间分布为平移 Gompertz；(2) 个人层面的购买倾向遵循整个人群的伽马分布。我们假设平移 Gompertz 分布的尺度参数在消费者中是恒定的。 G/SG 模型相对于非均匀影响模型等其他扩散模型的优势在于其累积分布函数采用封闭式表达式。根据 Van den Bulte 和 Lilien (1997) 的观点，我们使用 G/SG 模型作为真实模型和 12 个真实采用数据集，从模拟数据中分析了这些相反的模式。模式是（1）随着审查水平的降低，当数据表现出比 Bass 模型更多的右偏时，p 和 c 的估计值减少，而 q 的估计值增加；（2）当数据表现出比 Bass 模型更多的左偏时，p 和 c 的估计值增加，而 q 的估计值减少。对于模拟数据，我们操纵了四个维度：(1) 数据中的“超低音”倾斜，(2) q/p 比率，(3) 扩散速度，以及 (4) 误差方差。模拟数据和实际采用数据集的结果都证实了 Bass 模型依赖于“extra-Bass”偏斜的参数估计的两种相反模式的存在。当用模拟数据正确指定模型时，Bass 和 G/SG 模型的 c 估计值增加，q 估计值减少。 p 的估计值随着仅为 G/SG 模型添加数据点而增加。 Bass 模型中没有检测到 p 参数估计的显着趋势。对于病态问题，即使数据是以 G/SG 模型为真实数据生成的，在某些情况下，G/SG 模型参数估计的系统变化可能比 Bass 模型获得的系统变化大得多。因此，模型的复杂性会随着数据点的增加而加剧参数系统性变化的趋势。模拟数据的预测结果显示了G/SG模型的优越性。它在提前一步、提前两步和提前三步预测方面提供了比 Bass 模型更准确的结果。在真实数据集下，G/SG模型比Bass模型提供了更准确的提前一步预测，但当转向提前两步和提前三步预测时，该模型的预测性能比Bass模型恶化得更快。模型越复杂，参数估计的系统变化就越大。我们的研究表明，G/SG模型是一种灵活的模型，用于分析当规格误差和病态发生时参数估计的系统变化。由于我们的研究结果包含两种可能类型的参数估计偏差，与之前的单向视图相比，它们可以提供重要信息，以利用新技术创新提高产品和服务的预测准确性。我们对模拟和实际采用数据的预测结果提出了一个关于应用灵活扩散模型的最佳预测范围的问题。 G/SG模型还为联合研究“起飞速度”和“起飞后扩散速度”提供了依据。
Assessment of accurate market size and early adoption patterns is essential to strategic decision making of managers involved in new-product launches. This article proposes methodology that explains changes in parameter estimates of the Bass model, p (coefficient of innovation), q (coefficient of imitation), and c (market penetration rate) by direction of "extra-Bass" skew in the data, or equivalently, by underlying heterogeneity of the population. This research shows significantly opposite patterns of these parameter estimates, depending on skew of the diffusion curve detected by a generalized model, i.e., the gamma/shifted Gompertz (G/SG) model, which embeds the Bass model as a special case. The G/SG model originally presented in Bemmaor (1994) is based on two assumptions: (1) Individual-level times to first purchase are distributed shifted Gompertz and (2) individual-level propensity to buy follows a gamma distribution across the population. We assume that the scale parameter of the shifted Gompertz distribution is constant across consumers. The advantage the G/SG model has over alternative diffusion models such as the nonuniform influence model is that its cumulative distribution function takes a closed-form expression.In line with Van den Bulte and Lilien (1997), we analyze these opposite patterns from simulated data using the G/SG model as the true model and 12 real adoption data sets. The patterns am (1) as the level of censoring decreases, the estimates of p and c decrease and those of q increase when data exhibit more right skew than the Bass model and (2) the estimates of p and c-increase and those of q decrease when data exhibit more left skew than the Bass model. For the simulated data, we manipulated four dimensions: (1) "extra-Bass" skew-in the-data, (2) ratio q/p, (3) speed of diffusion, and (4) error variance. Both results of the simulated data and the real-adoption data sets confirm the existence of two opposite patterns of parameter estimates of the Bass model depending on "extra-Bass" skew.When the model is correctly specified with simulated data, estimates of c increase and those of q decrease for both the Bass and the G/SG models. The estimates of p increase as one adds data points only for the G/SG model. No significant tendency in parameter estimates of p was detected for the Bass model. As for ill-conditioning issues, systematic changes in the parameter estimates of the G/SG model can be substantially larger in some cases than those obtained with the Bass model, even though the data were generated by taking the G/SG model as the true one. Therefore, model complexity can aggravate the tendency for parameters to change systematically as one adds data points.The forecasting results from the simulated data show the supremacy of the G/SG model. It provides more accurate results than the Bass model in the one-step ahead, two-step ahead, and three-step ahead forecasts. With the real data set, the G/SG model provides more accurate one-step ahead forecasts than the Bass model, but the model's forecasting performance deteriorates more rapidly than the Bass model when one shifts to two-step ahead and three-step ahead forecasts. The systematic changes in parameter estimates are larger for the more complex model.Our research shows that the G/SG model is a flexible model used to analyze the systematic changes in parameter estimates when specification error and ill-conditioning occur. As our findings incorporate two possible types of parameter estimate bias, compared to the previous single-direction view, they can provide essential information to enhance forecasting accuracy of products and services using new technological innovations. Our forecasting results of simulated and real adoption data raise a question about the optimal horizon of forecasting in applying flexible models of diffusion. The G/SG model also provides grounds to investigate jointly "the speed of takeoff" and "the diffusion speed after takeoff".