548 Discrete Choice Modeling
are well established. Ongoing contemporary theoretical research is largely focused
on less parametric approaches and on panel data. The parametric models developed
here still overwhelmingly dominate the received applications.
Notes
- For a lengthy and detailed development of these ideas, see Daniel McFadden’s Nobel
Prize Lecture (McFadden, 2001).
- See, as well, Samuelson (1947) and Goldberger (1987).
- Some formulations of the models, such as models of heteroskedasticity and the random
parameters, will also involve additional parameters. These will be introduced later. They
are omitted at this point to avoid cluttering the notation.
- The formulation assumes that theTichoices made by individualiare unconditionally
independent. This assumption may be inappropriate. In one of our applications, the
assumption is testable.
- See, e.g., the documentation for LIMDEP (Econometric Software, Inc., 2007) or Stata
(Stata, Inc., 2007).
- We are assuming that the data are “well behaved” so that the conditions underlying the
standard optimality properties of MLEs are met here. The conditions and the properties
are discussed in Greene (2008a). We will take them as given in what follows.
- The sign of the result for the logistic distribution is obvious. See, e.g., Maddala (1983, p.
366) for a proof of the result for the normal distribution.
- There are data configurations, in addition to simple multicollinarity, that can produce
singularities. Another possibility is that of a variable inxiorzithat can predictdiperfectly
based on a specific cut point in the range of that variable.
- Recall that the average predicted probability,Pˆ, equals the average outcome in the binary
choice model,P 1. To a fair approximation, the standard deviation of the predicted prob-
abilities will equal[P 1 ( 1 −P 1 )]0.5. If the sample is highly unbalanced, sayP 1 <0.05
orP 1 >0.95, then a predicted probability as large as (or as small as) 0.5 may become
unlikely. It is common in unbalanced panels for the simple prediction rule always to
predict the same value.
- A symposium on the subject is Hardle and Manski (1993).
- See Manski (1975, 1985, 1986) and Manski and Thompson (1986). For extensions of this
model, see Horowitz (1992), Charlier, Melenberg and van Soest (1995) and Kyriazidou
(1997).
- Bootstrapping has been used to estimate the asymptotic covariance matrix for the maxi-
mum score estimator. However, Abrevaya and Huang (2005) have recently cast doubt on
the validity of that approach. No other strategy is available for statistical inference in this
model.
- One would proceed in precisely this fashion if the central specification were a linear
probability model (LPM) to begin with. See, e.g., Eisenberg and Rowe (2006) or Angrist
(2001) for an application and some analysis of this case.
- This is precisely the platform that underlies the generalized linear models/generalized
estimating equations (GLIM/GEE) treatment of binary choice models in, e.g., the widely
used programs SAS and Stata.
- Much of the recent research in semiparametric and nonparametric analysis of discrete
choice and limited dependent variable models has focused on how to accommodate
individual heterogeneity in panel data models while avoiding the incidental parameters
problem.
- The requirement does not state how largeRmust be, only that it “increase” faster than
n^1 /^2. In practice, analysts typically use several hundred, perhaps up to 1,000, random
draws for their simulations.
