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Re-estimation: Introduction to More Advanced Estimation Methods

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Modeling Markets

Part of the book series: International Series in Quantitative Marketing ((ISQM))

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Abstract

In this chapter we consider methods and procedures for the estimation of model parameters in cases where the basic assumptions of the general linear model are violated. When this is the case we need either other specifications and/or other estimation methods: “re-estimation ”. In Sect. 6.2 we introduce Generalized Least Squares (GLS) estimation methods.

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Notes

  1. 1.

    Since Ω, and thus Ω ∗, is symmetric and positive definite, so are Ω −1 and \(\left (\varOmega ^{{\ast}}\right )^{-1}\), and hence a matrix V satisfying (6.3) exists. See Appendix A.7.

  2. 2.

    To see that the last equality holds, we define the matrix A as: \(A = V \left (V 'V \right )^{-1}V '\). If we multiply A by V, we have that \(AV = V \left (V 'V \right )^{-1}V 'V = V\), which only holds if A = I.

  3. 3.

    First derived by Aitken (1935) , and for that reason also known as the Aitken estimator .

  4. 4.

    Formally both estimates need to be pre-multiplied by \((T - 1)/(T - K)\), but that does not affect the estimate for β.

  5. 5.

    See Judge et al. (1985, pp. 439–441) for a more general expression.

  6. 6.

    It can be shown that \(\sigma _{u}^{2} = \sigma _{\varepsilon }^{2}/(1 -\rho ^{2})\).

  7. 7.

    This is a least squares estimate of ρ. It differs slightly from the maximum likelihood estimator. See Greene (2012, pp. 966–967) for other estimators.

  8. 8.

    Because the N sets of equations in (6.23) do not seem to be related, this structure is referred to as “Seemingly Unrelated Regressions (SUR)”, see Zellner (1962) .

  9. 9.

    See Zellner (1962) ; Kmenta (1971, pp. 517–519) . See also Leeflang (1974, pp. 124–127) .

  10. 10.

    See Clarke (1973) . For an asymmetric, non-hierarchical market share model, Carpenter et al. (1988) use the \(\hat{\sigma }_{ij}\) values to identify potential cross-effects.

  11. 11.

    See McGuire et al. (1968) . See also Hanssens et al. (2001) .

  12. 12.

    See Gaver et al. (1988) .

  13. 13.

    Which means that PF t does not pick up any price discounts in these weeks.

  14. 14.

    We note that this might be due to the small number of observations in the promotional group.

  15. 15.

    Gupta (1991). See also Sect. 8.4.

  16. 16.

    Cameron and Trivedi (2009, pp. 147–149).

  17. 17.

    See e.g. Greene (2012, p. 476) .

  18. 18.

    When θ is a vector of parameters, the difference between the covariance matrix and \(\left (I(\theta )\right )^{-1}\) is a nonnegative matrix.

  19. 19.

    Compare Lindsey (1996, p. 199) .

  20. 20.

    See Scales (1985) ; Eliason (1993) .

  21. 21.

    Based on Cameron and Trivedi (2009, p. 140).

  22. 22.

    We closely follow Cameron and Trivedi (2009, p. 140).

  23. 23.

    We closely follow the study by Van Nierop et al. (2011).

  24. 24.

    See, for example, Fischer et al. (2010) ; Hui et al. (2013) .

  25. 25.

    For applications, see Schultz (1971) ; Carpenter (1987) ; Tellis and Fornell (1988) and Hanssens et al. (2001, p. 90) .

  26. 26.

    Greene (2012, p. 372) .

  27. 27.

    We closely follow Wooldridge (2012, pp. 512–513) .

  28. 28.

    For a much more detailed treatment of Bayesian methods we refer to Rossi et al. (2005) . See also Congdon (2003, 2006) .

  29. 29.

    We assume that prior information is available on the first K′ estimators. The variables can be arranged such that this is the case.

  30. 30.

    See also Sect. 7.3.2.2.

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Authors and Affiliations

  1. Department of Marketing, University of Groningen, Groningen, The Netherlands

    Peter S. H. Leeflang, Jaap E. Wieringa & Tammo H. A. Bijmolt

  2. Aston Business School, Birmingham, UK

    Peter S. H. Leeflang

  3. University of St. Gallen, St. Gallen, Switzerland

    Peter S. H. Leeflang

  4. Department of Marketing, Özyeğin University, Istanbul, Turkey

    Koen H. Pauwels

Authors
  1. Peter S. H. Leeflang
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  2. Jaap E. Wieringa
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  3. Tammo H. A. Bijmolt
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  4. Koen H. Pauwels
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Leeflang, P.S.H., Wieringa, J.E., Bijmolt, T.H.A., Pauwels, K.H. (2015). Re-estimation: Introduction to More Advanced Estimation Methods. In: Modeling Markets. International Series in Quantitative Marketing. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-2086-0_6

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