Andrew M. Jones 591following the suggestion of Mundlak (1978) and Chamberlain (1984). The vectors
of parametersθ 1 ,...,θC,γ 1 ,...,γC− 1 are estimated jointly by ML.
After estimating the model, it is possible to calculate the posterior probability that
each individual belongs to a given class. The posterior probability of membership
of classjdepends on the relative contribution of that class to the individual’s
likelihood function. This is given by:
P[i∈j]=πijfj(
yi|xi;θj)∑C
k= 1πikfk(
yi|xi;θk). (12.28)
Each individual can then be assigned to the class that has the highest posterior
probability for them.
12.4.4.2 Finite density estimators and discrete factor models
In LCMs class membership has a discrete distribution with a fixed number of
mass points. The models are very flexible in that all of the parameters can be
allowed to vary across classes. A special case of this general model assumes that
the slope coefficients are fixed across classes and that only the intercepts vary. This
case is widely used to model unobserved heterogeneity, without imposing para-
metric assumptions on the distribution of the heterogeneity. The specification has
a dual interpretation: the population may truly fall into a discrete set of classes or
types or, alternatively, the mass points can be viewed as an approximation of some
underlying continuous distribution – the finite density estimator. The finite density
estimator was introduced into the econometrics literature by Heckman and Singer
(1984) in the context of hazard models (see, e.g., Van Ours, 2004, 2006). More
recently, the finite density estimator has been widely used in multiple equation
models where a common factor structure is assumed, as in equation (12.9) above.
This is often called the discrete factor model (DFM).
Since the DFM includes an intercept for each equation, the location of the dis-
tribution of the common factorηis arbitrary; also the scale ofη is arbitrary and
undetermined (Mroz, 1999). Therefore, identification of the DFM requires some
normalizations. The existing literature on the DFM offers a range of equivalent
strategies to identify the additional parameters of the discrete distribution by fix-
ing the scale and the location of the distribution. If both are fixed, one of the
factor loadings is set to 1 and either one of theηjis set to 0 (see Mroz, 1999) or
the mean of the discrete distribution is restricted to be 0, so that one of theηj
can be expressed as a function of the others (Kanet al.,2003). If only the location
is fixed, the first and last mass points are set to 0 and 1 (this strategy is used by
Mroz, 1999, whenC>2). Other applications also impose that the remaining mass
points follow a logistic distribution such thatηk∈(0,1) (see Melloet al., 2002, Picone
et al., 2003b). Theπkcan be parameterized using various distributions, such as the
logistic, normal or the sine function, such that eachπkis between 0 and 1 and
they sum to 1.