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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- 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.
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.
First derived by Aitken (1935) , and for that reason also known as the Aitken estimator .
- 4.
Formally both estimates need to be pre-multiplied by \((T - 1)/(T - K)\), but that does not affect the estimate for β.
- 5.
See Judge et al. (1985, pp. 439–441) for a more general expression.
- 6.
It can be shown that \(\sigma _{u}^{2} = \sigma _{\varepsilon }^{2}/(1 -\rho ^{2})\).
- 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.
- 9.
- 10.
- 11.
- 12.
See Gaver et al. (1988) .
- 13.
Which means that PF t does not pick up any price discounts in these weeks.
- 14.
We note that this might be due to the small number of observations in the promotional group.
- 15.
- 16.
Cameron and Trivedi (2009, pp. 147–149).
- 17.
See e.g. Greene (2012, p. 476) .
- 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.
Compare Lindsey (1996, p. 199) .
- 20.
- 21.
Based on Cameron and Trivedi (2009, p. 140).
- 22.
We closely follow Cameron and Trivedi (2009, p. 140).
- 23.
We closely follow the study by Van Nierop et al. (2011).
- 24.
- 25.
- 26.
Greene (2012, p. 372) .
- 27.
We closely follow Wooldridge (2012, pp. 512–513) .
- 28.
- 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.
See also Sect. 7.3.2.2.
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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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