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Bias and Systematic Change in the Parameter Estimates of Macro-Level Diffusion Models


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  • Christophe Van den Bulte

    (The Wharton School, 1400 Steinberg Hall-Dietrich Hall, University of Pennsylvania, Philadelphia, Pennsylvania 19104-6371)

  • Gary L. Lilien

    (Smeal College of Business Administration, The Pennsylvania State University, University Park, Pennsylvania 16802)

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    Studies estimating the Bass model and other macro-level diffusion models with an unknown ceiling feature three curious empirical regularities: (i) the estimated ceiling is often close to the cumulative number of adopters in the last observation period, (ii) the estimated coefficient of social contagion or imitation tends to decrease as one adds later observations to the data set, and (iii) the estimated coefficient of social contagion or imitation tends to decrease systematically as the estimated ceiling increases. We analyze these patterns in detail, focusing on the Bass model and the nonlinear least squares (NLS) estimation method. Using both empirical and simulated diffusion data, we show that NLS estimates of the Bass model coefficients are biased and that they change systematically as one extends the number of observations used in the estimation. We also identify the lack of richness in the data compared to the complexity of the model (known as ill-conditioning) as the cause of these estimation problems. In an empirical analysis of twelve innovations, we assess how the model parameter estimates change as one adds later observations to the data set. Our analysis shows that, on average, a 10% increase in the observed cumulative market penetration is associated with, roughly, a 5% increase in estimated market size , a 10% decrease in the estimated co-efficient of imitation , and a 15% increase the estimated co-efficient of innovation . A simulation study shows that the NLS parameter estimates of the Bass model change systematically as one adds later observations to the data set, even in the absence of model misspecification. We find about the same effect sizes as in the empirical analysis. The simulation also shows that the estimates are biased and that the amount of bias is a function of (i) the amount of noise in the data, (ii) the number of data points, and (iii) the difference between the cumulative penetration in the last observation period and the true penetration ceiling (i.e., the extent of right censoring). All three conditions affect the level of ill-conditioning in the estimation, which, in turn, affects bias in NLS regression. In situations consistent with marketing applications, can be underestimated by 20%, underestimated by the same amount, and overestimated by 30%. The existence of a downward bias in the estimate of and an upward bias in the estimate of , and the fact that these biases become smaller as the number of data points increases and the censoring decreases, can explain why systematic changes in the parameter estimates are observed in many applications. A reduced bias, though, is not the only possible explanation for the systematic change in parameter estimates observed in empirical studies. Not accounting for the growth in the population, for the effect of economic and marketing variables, or for population heterogeneity is likely to result in increasing and decreasing as well. In an analysis of six innovations, however, we find that attempts to address possible model misspecification problems by making the model more flexible and adding free parameters result in larger rather than smaller systematic changes in the estimates. The bias and systematic change problems we identify are sufficiently large to make long-term predictive, prescriptive and descriptive applications of Bass-type models problematic. Hence, our results should be of interest to diffusion researchers as well as to users of diffusion models, including market forecasters and strategic market planners.

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    Bibliographic Info

    Article provided by INFORMS in its journal Marketing Science.

    Volume (Year): 16 (1997)
    Issue (Month): 4 ()
    Pages: 338-353

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    Handle: RePEc:inm:ormksc:v:16:y:1997:i:4:p:338-353

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    Keywords: diffusion; estimation and other statistical techniques; forecasting;


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    Cited by:
    1. Chen, Yuwen & Carrillo, Janice E., 2011. "Single firm product diffusion model for single-function and fusion products," European Journal of Operational Research, Elsevier, vol. 214(2), pages 232-245, October.
    2. Christos Michalakelis & Georgia Dede & Dimitris Varoutas & Thomas Sphicopoulos, 2010. "Estimating diffusion and price elasticity with application to telecommunications," Netnomics, Springer, vol. 11(3), pages 221-242, October.
    3. Fok, D. & Franses, Ph.H.B.F., 2005. "Modeling the diffusion of scientific publications," Econometric Institute Research Papers EI 2005-48, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute.
    4. Franses, Ph.H.B.F., 2009. "Forecasting Sales," Econometric Institute Research Papers EI 2009-29, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute.
    5. Venkatesan, Rajkumar & Kumar, V., 2002. "A genetic algorithms approach to growth phase forecasting of wireless subscribers," International Journal of Forecasting, Elsevier, vol. 18(4), pages 625-646.
    6. Jun, Duk B. & Kim, Seon K. & Park, Yoon S. & Park, Myoung H. & Wilson, Amy R., 2002. "Forecasting telecommunication service subscribers in substitutive and competitive environments," International Journal of Forecasting, Elsevier, vol. 18(4), pages 561-581.
    7. Peters, Kay & Albers, Sönke & Kumar, V., 2008. "Is there more to international Diffusion than Culture? An investigation on the Role of Marketing and Industry Variables," EconStor Preprints 27678, ZBW - German National Library of Economics.
    8. Jonathan Beck, 2007. "The sales effect of word of mouth: a model for creative goods and estimates for novels," Journal of Cultural Economics, Springer, vol. 31(1), pages 5-23, March.
    9. Desiraju, Ramarao & Nair, Harikesh S. & Chintagunta, Pradeep, 2004. "Diffusion of New Pharmaceutical Drugs in Developing and Developed Nations," Research Papers 1950, Stanford University, Graduate School of Business.
    10. van Everdingen, Y.M. & Aghina, W.B., 2003. "Forecasting the international diffusion of innovations: An adaptive estimation approach," ERIM Report Series Research in Management ERS-2003-073-MKT, Erasmus Research Institute of Management (ERIM), ERIM is the joint research institute of the Rotterdam School of Management, Erasmus University and the Erasmus School of Economics (ESE) at Erasmus Uni.
    11. Kamrad, Bardia & Lele, Shreevardhan S. & Siddique, Akhtar & Thomas, Robert J., 2005. "Innovation diffusion uncertainty, advertising and pricing policies," European Journal of Operational Research, Elsevier, vol. 164(3), pages 829-850, August.
    12. Meade, Nigel & Islam, Towhidul, 2006. "Modelling and forecasting the diffusion of innovation - A 25-year review," International Journal of Forecasting, Elsevier, vol. 22(3), pages 519-545.
    13. Serdar Kale & David Arditi, 2006. "Diffusion of ISO 9000 certification in the precast concrete industry," Construction Management and Economics, Taylor & Francis Journals, vol. 24(5), pages 485-495.
    14. Islam, Towhidul & Fiebig, Denzil G. & Meade, Nigel, 2002. "Modelling multinational telecommunications demand with limited data," International Journal of Forecasting, Elsevier, vol. 18(4), pages 605-624.
    15. de Bondt, Gabe & Marqués-Ibáñez, David, 2004. "The high-yield segment of the corporate bond market: a diffusion modelling approach for the United States, the United Kingdom and the euro area," Working Paper Series 0313, European Central Bank.


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