Random Parameters Logit Model

A RPL model may be more appropriate because it allows for correlation between choices and individuals (Nilsson, 2005). More specifically, the parameters in the CE are assumed to be individual specific and taste is assumed to vary randomly across the population according to a continuous distributional function rather than being fixed as it is in the MNL framework. Thus, researchers can identify how preferences for various attributes might vary in a population (Lusk and Hudson, 2004). Since the RPL model allows coefficients to vary randomly and does not exhibit the restrictive ―IIA‖ property, it is considered an improvement over the MNL model.⁷⁸ Thus, the IID property of the MNL model has to be relaxed (for a derivation of the model see Ben-Akiva and Lerman, 1984; Revelt and Train, 1998; Hensher and Green, 2003; Hensher et al., 2005). According to Hensher and Green (2003), one way to do this is to partition the stochastic component of the indirect utility function U (see expression 4.2), which becomes: _in





nit n

in x

U ^'    (4.8)

where _inis a random term with zero mean whose distribution over individuals and alternatives depends on underlying parameters and observed data related to alternative i and individual n. εnit

78 West et al. (2002), Liljenstolpe (2008a), and Tonsor et al. (2008c) are authors that employed the RPL model in analyzing data from choice experiment surveys that elicited consumer preference for livestock products with FAW attributes and revealed the RPL model superiority over the MNL model.

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is a random term with zero mean that is IID over alternatives and does not depend on underlying parameters or data (Hensher and Green, 2003, p.135).

The RPL model assumes a general distribution for _inand an IID extreme value type 1 distribution for the error term.⁷⁹ That is, _incan take on a number of distributional forms such as normal, lognormal, and triangular (Hensher and Green, 2003; Hensher et al., 2005). For any given value of _in, the conditional probability for choice i is logit since the remaining error term

is IID extreme value:

 

Thus, the probability P_ni is labelled as unconditional choice as it is still conditional on observable characteristics of pork chops and demographic information of the sample captured in _n^', but it is not conditional on the unobservable _in (Greene et al., 2006). Hensher et al. (2005) assert that the concept of ―conditional choice‖ tells us that a specific choice is conditional on something else. For instance, the choice of a dish for a dinner (i.e., marinated pork chops without antibiotics) is conditional on a prior choice to eat or not to eat. It may also be conditional on the

79 ―Extreme value type 1 (EV1) is a commonly used distribution in discrete choice analysis. The phrase ―extreme value‖ arises relative to the normal distribution. The essential difference between the EV1 and the normal distributions is in the tails of the distribution where the extreme values reside. With a small choice set such as two alternatives this may make little difference because the resulting differences in the choice probabilities between the normal and EV1 is usually negligible. When there are more than two alternatives, however, there can be a number of very small choice probabilities. As a result, differences between distributions can be quite noticeable‖ (see Hensher et al. 2005 citing Jones and Hensher, 2004, p.1016).

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prior choice to eat away from home versus eat at home (conditional on the decision to eat). An unconditional choice is one that is not conditioned on any prior choice. It is only when we have taken into account all of these prior (or in some cases joint) conditions that we can refer to individual (unconditional) demand (Hensher et al., 2005, p.70).

Unlike the simple MNL model that has a closed form solution and guarantees a unique globally optimal set of parameters estimates, the RPL model can produce a wide range of solutions, only one of which is globally optimum (Jones and Hensher, 2004, p.1017).⁸⁰ As shown in Hensher and Green (2003) and Hensher et al. (2005), the concern that one may not know the location of each individual‘s preferences on the distribution can be accommodated by retrieving estimates of individual-specific preferences by deriving the individual‘s conditional distribution based (within sample) on their choices (prior knowledge). This is made using Bayes‘ rule, a procedure which is described in detail by Hensher and Green (2003) and Train (2003). As well, the choice probability from equation (4.10) cannot be calculated exactly because the integral does not have a closed form and instead is approximated through simulation (Hensher and Green, 2003). RPL estimation results are presented in Chapter 5.

Although the RPL model has a major strength over the MNL since it includes the source of preference heterogeneity in its procedure, it imposes some constraints on the number of parameters that may be estimated in regressions and thus cannot explain very well all the reasons that individual parameters vary. Work by Lusk and Hudson (2004), Nilsson et al. (2006) and

80 ―The mixed logit model has a likelihood surface that is capable of producing local optima in contrast to a single unique global optimum from the MNL model. Using the MNL parameter estimates as starting values produces a global solution since it begins the gradient search at a location of the nonlinear surface that tends to be the best starting location for determining the global optimum‖ (Jones and Hensher, 2004, p.1017).

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Tonsor et al. (2008c) showed that incorporating socio-demographic characteristics and attitudinal information failed to improve the statistical performance of the RPL model. As a consequence, the RPL estimates contain unexplained heterogeneity. In other words, when reported as a mean for the population, the estimates may hide important variations in preference across the population. In this respect, the Latent Class Logit (LCL) model is an alternative to the RPL model as it incorporates unobserved preference heterogeneity into the estimation procedure.