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Erschienen in:
Building Models for Markets Kapitel lesen Erstes Kapitel lesen

2015 | OriginalPaper | Buchkapitel

6. Re-estimation: Introduction to More Advanced Estimation Methods

verfasst von : Peter S. H. Leeflang, Jaap E. Wieringa, Tammo H. A. Bijmolt, Koen H. Pauwels

Erschienen in: Modeling Markets

Verlag: Springer New York

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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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Vorheriges Kapitel Validation and Testing
Nächstes Kapitel Examples of Models for Aggregate Demand
Fußnoten
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
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
See Zellner (1962); Kmenta (1971, pp. 517–519). See also Leeflang (1974, pp. 124–127).
 
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
See McGuire et al. (1968). See also Hanssens et al. (2001).
 
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
Gupta (1991). See also Sect. 8.​4.
 
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
See Scales (1985); Eliason (1993).
 
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
See, for example, Fischer et al. (2010); Hui et al. (2013).
 
25
For applications, see Schultz (1971); Carpenter (1987); Tellis and Fornell (1988) and Hanssens et al. (2001, p. 90).
 
26
Greene (2012, p. 372).
 
27
We closely follow Wooldridge (2012, pp. 512–513).
 
28
For a much more detailed treatment of Bayesian methods we refer to Rossi et al. (2005). See also Congdon (2003, 2006) .
 
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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Metadaten
Titel
Re-estimation: Introduction to More Advanced Estimation Methods
verfasst von
Peter S. H. Leeflang
Jaap E. Wieringa
Tammo H. A. Bijmolt
Koen H. Pauwels
Copyright-Jahr
2015
Verlag
Springer New York
Buch
Modeling Markets

Print ISBN: 978-1-4939-2085-3

Electronic ISBN: 978-1-4939-2086-0

Copyright-Jahr: 2015

DOI
https://doi.org/10.1007/978-1-4939-2086-0_6