Interpretation of explanatory variables impacts in compositional regression models

Autor: Michel Simioni, Joanna Morais, Christine Thomas-Agnan
Přispěvatelé: Toulouse School of Economics (TSE), Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-École des hautes études en sciences sociales (EHESS)-Centre National de la Recherche Scientifique (CNRS)-Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement (INRAE), BVA, Marchés, Organisations, Institutions et Stratégies d'Acteurs (UMR MOISA), Institut national d’études supérieures agronomiques de Montpellier (Montpellier SupAgro)-Centre International de Hautes Etudes Agronomiques Méditerranéennes - Institut Agronomique Méditerranéen de Montpellier (CIHEAM-IAMM), Centre International de Hautes Études Agronomiques Méditerranéennes (CIHEAM)-Centre International de Hautes Études Agronomiques Méditerranéennes (CIHEAM)-Institut National de la Recherche Agronomique (INRA)-Centre de Coopération Internationale en Recherche Agronomique pour le Développement (Cirad)-Centre international d'études supérieures en sciences agronomiques (Montpellier SupAgro), École des hautes études en sciences sociales (EHESS)-Institut National de la Recherche Agronomique (INRA)-Centre National de la Recherche Scientifique (CNRS)-Université Toulouse 1 Capitole (UT1), Centre de Coopération Internationale en Recherche Agronomique pour le Développement (Cirad)-Institut National de la Recherche Agronomique (INRA)-Centre international d'études supérieures en sciences agronomiques (Montpellier SupAgro)-Centre International de Hautes Etudes Agronomiques Méditerranéennes - Institut Agronomique Méditerranéen de Montpellier (CIHEAM-IAMM), Centre International de Hautes Études Agronomiques Méditerranéennes (CIHEAM)-Centre International de Hautes Études Agronomiques Méditerranéennes (CIHEAM)-Institut national d’études supérieures agronomiques de Montpellier (Montpellier SupAgro), CIFRE, Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Institut National de la Recherche Agronomique (INRA)-École des hautes études en sciences sociales (EHESS)-Centre National de la Recherche Scientifique (CNRS), Institut national d’études supérieures agronomiques de Montpellier (Montpellier SupAgro), Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Centre International de Hautes Etudes Agronomiques Méditerranéennes - Institut Agronomique Méditerranéen de Montpellier (CIHEAM-IAMM)
Jazyk: angličtina
Rok vydání: 2018
Předmět:
Statistics and Probability
model selection
Computer science
media_common.quotation_subject
Automotive industry
compositional model
01 natural sciences
modèle de régression
QA273-280
Economies et finances
010104 statistics & probability
Methods and statistics
elasticity
odds ratio
marginal effect
compositional differential calculus
market-shares
media investments impact
[MATH.MATH-ST]Mathematics [math]/Statistics [math.ST]
0502 economics and business
Econometrics
050207 economics
0101 mathematics
Market share
B- ECONOMIE ET FINANCE
media_common
modélisation
[SHS.STAT]Humanities and Social Sciences/Methods and statistics
Variables
business.industry
Applied Mathematics
Model selection
Statistics
05 social sciences
sciences humaines et sociales
Regression analysis
[SHS.ECO]Humanities and Social Sciences/Economics and Finance
Méthodes et statistiques
HA1-4737
Economies and finances
probabilité
Statistics
Probability and Uncertainty
business
Compositional data
Probabilities. Mathematical statistics
Zdroj: Austrian Journal of Statistics 5 (47), 1-25. (2018)
Austrian Journal of Statistics
Austrian Journal of Statistics, 2018, 47 (5), pp.1-25. ⟨10.17713/ajs.v47i5.718⟩
Austrian Journal of Statistics, Vol 47, Iss 5 (2018)
ISSN: 1026-597X
DOI: 10.17713/ajs.v47i5.718⟩
Popis: Regression models have been developed for the case where the dependent variable is a vector of shares. Some of them, from the marketing literature, are easy to interpret but they are quite simple and can only be complexified at the expense of a very large number of parameters to estimate. Other models, from the mathematical literature, are called compositional regression models and are based on the simplicial geometry (a vector of shares is called a composition, shares are components, and a composition lies in the simplex). These models are transformation models: they use a log-ratio transformation of shares. They are very flexible in terms of explanatory variables and complexity (component-specific and cross-effect parameters), but their interpretation is not straightforward, due to the fact that shares add up to one. This paper combines both literatures in order to obtain a performing market-share model allowing to get relevant and appropriate interpretations, which can be used for decision making in practical cases. For example, we are interested in modeling the impact of media investments on automobile manufacturers sales. In order to take into account the competition, we model the brands market-shares as a function of (relative) media investments. We furthermore focus on compositional models where some explanatory variables are also compositional. Two specifications are possible: in Model A, a unique coefficient is associated to each compositional explanatory variable, whereas in Model B a compositional explanatory variable is associated to component-specific and cross-effect coefficients. Model A and Model B are estimated for our application in the B segment of the French automobile market, from 2003 to 2015. In order to enhance the interpretability of these models, we present different types of impact assessment measures (marginal effects, elasticities and odds ratios) and we show that elasticities are particularly useful to isolate the impact of an explanatory variable on a particular share. We show that elasticities can be equivalently computed from the transformed model and from the model in the simplex and that they are linked to directional C-derivatives of simplex-valued function of a simplex variable. Direct and cross effects of media investments are computed for both models. Model B shows interesting non-symmetric synergies between brands, and Renault seems to be the most elastic brand to its own media investments. In order to determine if component-specific and cross-effect parameters are needed to improve the quality of the model (Model B) or if a global parameter is reasonable (Model A), we compare the goodness-of-fit of the two models using (out-of-sample) quality measures adapted for share data.
Databáze: OpenAIRE
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