Addressing Homoskedasticity in Invariant Random Utility Models
V Marzano, Università degli Studi di Napoli ?Federico II?, IT; A Daly, ITS, University of Leeds and RAND Europe, UK
This paper aims to provide a critical review of the literature and a better understanding of the homoskedasticity properties of invariant random utility models.
Invariant (or additive) random utility models (IRUM), that is models whose error term distribution does not depend on any component of the alternatives? systematic utility, play an important role in the literature of discrete choice models, thanks to their theoretical and operational simple properties. The objective of this paper is to explore the extent of applicability of these simple models, with an ultimate view to providing simpler and more readily applicable tools to represent complex choice.behaviour.
For practical purposes, IRUM can be classified in different ways. Regarding the hypotheses underlying the cumulative distribution function of the utilities of the alternatives, IRUM can be classified as GEV models, if utilities follow a multivariate extreme value distribution derived from the generating functions complying with McFadden (1978) theorem requirements, and non-GEV models, such as the Probit model. Furthermore, with reference to the variance of the perceived utilities, IRUM can be classified as homoskedastic, that is equal variance utilities, or heteroskedastic (obviously, not all the forms of heteroskedasticity which can take place can be covered by IRUM). Moreover, IRUM can be classified as closed-form or non closed-form models, depending on the availability of a closed-form probability statement, i.e. not requiring simulation for the calculation of choice probabilities. Those classifications are mutually integrated in well-known tenets in the literature: for instance, GEV models are expected to be homoskedastic and closed-form, while heteroskedastic models are expected to be non-GEV and not closed-form.
Actually, in contrast with this state of the art, some contributions in the literature (Smith 1984, Dagsvik 1995, Joe 2001, Daly 2005, Ibáñez, 2006) seem to provide evidence that the class of GEV model is wider than expected with respect to the class of IRUM. The relationship between GEV and IRUM is explored in these papers following different approaches and reaching different, even if not contrasting, results. In that respect, this paper aims firstly to provide for a better understanding of this issue, in order to find a consistency among these results and verify their compliance with McFadden (1978) theorem.
Obviously, such a result implies looking at again the preceding classifications. Firstly, if all IRUM are equivalent to (or can be approximated by) a GEV model, then non-closed GEV models can exist: indeed, this result can be shown to be consistent with McFadden?s theorem, and can also be found in Daly (2005) and Karlstrom (2003). Moreover, the equivalence between GEV and IRUM implies that either all IRUM are homoskedastic or the existence of heteroskedastic GEV models. The latter would raise serious.issues, since the homoskedasticity of GEV models is a direct consequence of McFadden?s theorem hypotheses. Therefore, the paper proposes a critical literature review, aiming at a systematisation of the literature and supporting the conclusion that any IRUM is in fact equivalent to a GEV model (that is to a homoskedastic model).
The dichotomy between homoskedastic and heteroskedastic IRUM is also addressed from a different perspective. In more detail, starting from IRUM properties that depend on the covariance matrix of utility differences, Daly (2001) argued (homoskedasticity lemma) that any heteroskedastic covariance matrix (underlying an IRUM) can be transformed into an equivalent homoskedastic one, even if the underlying homoskedastic choice model is not guaranteed to be real. This paper presents a further investigation of the homoskedasticity lemma, which is reformulated by means of a more direct and useful set of constraints assuring the underlying homoskedastic choice model to be proper. At the current stage of the work, a theoretical proof of this lemma has not been achieved yet, but positive empirical evidence is presented of its validity in contexts with up to ten alternatives.
A concluding section indicates how these results could be used in practice to speed up model applications for all kinds of heteroskedastic choice models, including those that are not IRUM.
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