Using compositional and Dirichlet models for market share regression

ABSTRACT

When the aim is to model market shares, the marketing literature proposes some regression models which can be qualified as attraction models. They are generally derived from an aggregated version of the multinomial logit model. But aggregated multinomial logit models (MNL) and the so-called generalized multiplicative competitive interaction models (GMCI) present some limitations: in their simpler version they do not specify brand-specific and cross effect parameters. In this paper, we consider alternative models: the Dirichlet model (DIR) and the compositional model (CODA). DIR allows to introduce brand-specific parameters and CODA allows additionally to consider cross effect parameters. We show that these two models can be written in a similar fashion, called attraction form, as the MNL and the GMCI models. As market share models are usually interpreted in terms of elasticities, we also use this notion to interpret the DIR and CODA models. We compare the properties of the models in order to explain why CODA and DIR models can outperform traditional market share models. An application to the automobile market is presented where we model brands market shares as a function of media investments, controlling for the brands price and scrapping incentive. We compare the quality of the models using measures adapted to shares.

KEYWORDS:

• Multinomial logit
• market shares models
• compositional data analysis
• Dirichlet regression
• marketing

Acknowledgments

We would like to thank BVA and the Marketing Direction of Renault for sharing valuable data with us, and for their support during the elaboration of the data base and the model specification. We would also like to thank V. Pawlowsky-Glahn, M. Vives-Mestres, J. J. Egozcue, J. A. Martin Fernandez and K. Hron for their help to get familiar with the compositional data analysis.

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Joanna Morais http://orcid.org/0000-0002-3604-6698

Notes

1. Note that this work is based on a previously available working paper available on HAL (see Morais et al. [17]).

2. For details on IWLS algorithm, see for example Green  [8].

3. The alternative parametrization uses the parameters ${\mu }_j=\mathbb{E}\left(S_j\right)$ to account for the expected values of the shares, and $\phi ={\alpha }_0$ to account for the precision (see Hijazi and Jernigan [10]).

4. Note that the dependent composition and the explanatory compositions could be of different dimensions.

5. Under these conditions, $\mathbf{B}⊡\mathbf{Z}$ is an endomorphism of the simplex ${\mathcal{S}}^D$ (see Kynclova et al. [11]). Thus model (6(6) ${\mathbf{S}}_t=\mathbf{a}\underset{k=1}{\overset{K}{\oplus }}{\mathbf{B}}_{\mathbf{k}}⊡{{\mathbf{Z}}_{\mathbf{k}}}_t\oplus {\mathit{\epsilon }}_t$(6) ) is a linear model in the simplex.

6. The orthonormality of coordinates allows us to estimate the D−1 models separately.

7. Here the GMCI model is presented with the MCI specification. Note that if X is replaced by $\exp X$, it corresponds to the MNL specification.

8. The market share $S_{jt}$ is here expressed as a function of $X_{klt}$ directly and not as a function of $Z_{klt}$ because $S_{jt}$ is obtained by a closure operation (dividing by the denominator), thus it can be shown that the explanatory variables can be used in volume as they are closed at the end.

9. Segments of the automobile market are determined according to the size of the chassis. Segment B corresponds to small mainstream vehicles like the Renault Clio which is the most famous of this segment in France.

10. A scrapping incentive is an incentive given by a government to promote the replacement of old vehicles with modern vehicles.

11. The reason of that is linked to the shape of the elasticity of market shares to the price. Moreover, to keep the market shares equal, the logged variables have to increase in the same proportion while the non-logged variables have to increase by the same amount.

12. In forthcoming work, we consider using an adstock function, which is a cumulative function of actual and past investments.

13. Here we want to have an efficient model all along the studied period, the aim is not to have a good predictive model for the future. Moreover the presented models are not taking into account the potential auto-correlation of error terms. That is the reason why the cross-validation can be made on randomly drawn dates and not on a split of the studied period according to time.

Additional information

Funding

This work was supported by the market research agency BVA and the Association Nationale de la Recherche et de la Technologie (ANRT).