Weighted-Average Least Squares Estimation of Multinomial and Mixed Logit Models

Weighted-average least squares estimation of multinomial and mixed logit models

Antti M. Liski1 and Erkki P. Liski2

1Dain Studios, Finland 2University of Tampere, Finland

Abstract We consider the multinomial logit (MNL) model [4] that is often used in nominal polytomous data. The most popular MNL extension is the mixed multinomial logit (MIXL) model which allows parameters to vary randomly over individuals. MNL and MIXL have a solid basis in the theory of discrete choice models where the central idea lies in the ”utility maximization principle”, see [5]. These quantitative techniques are used in a range of disciplines nowdays. A model averaging (MA) procedure oﬀers a smooth compromise across a set of candidate models, thus taking the model uncertainty into account and alleviating the instability associated with selecting a single model. Bayesian MA (BMA) has proven to be a successful tool, and has given rise to a large body of literature, see eg. [1]. Many of the frequentist MA (FMA) strategies are formed using scores of information criteria. Wan, Zhang and Wang [6] develop an alternative FMA strategy for the MNL model. Magnus et al. [3] introduced a new model averaging (MA) tech- nique, called weighted-average least squares (WALS), in the context of Gaussian linear models. They proved that WALS enjoys important advantages, both theoretical and practical, over BMA and the tradi- tional FMA. De Luca et al. [2] show that many of the advantages of the WALS approach to Gaussian linear model continue to hold in the wider class of the generalized linear models (GLM). The aim of this paper is to extend WALS to multivariate outcomes (like MNL and MIXL) and to mixed GLM’s (like MIXL).

Keywords Model uncertainty, Model selection and averaging, Mixed model.

References

[1] Burda, M., Harding, M., and Hausman, J. (2008). A Bayesian mixed logit-probit for multinomial choice. Journal of Econometrics 147, 232– 246.

1 [2] De Luca, G., Magnus, J.R., and Peracchi, F. (2018). Weighted-average least squares estimation of generalized linear models. Journal of Econo- metrics 204, 1–17. [3] Magnus, J.R., Powell, O., and Pr¨ufer,P. (2010). A comparison of two model averaging techniques with an application to growth empirics. Jour- nal of Econometrics 154, 139–153. [4] McFadden, D. (1974). Conditional Logit Analysis of Qualitative Choice Behaviour. In: P. Zarebka (ed.), Frontiers in Econometrics, (pp. 105– 142). Academic Press.

[5] Train, K. (2009). Discrete Choice Methods with Simulation (2nd ed). Cambridge University Press. [6] Wan, A.T.K., Zhang, X., and Wang, S. (2014). Frequentist model averaging for multinomial and ordered logit models. International Journal of Forecasting 30, 118–128.