738 Microeconometrics: Methods and Developments
wherêf(·)is the Monte Carlo estimate or simulator:
̂f(yi|xi,u(S)
i ,θ)=1
S∑Ss= 1̃f(yi|xi,usi,θ), (14.12)whereu(iS)=(u^1 i,...,uSi)denotesSdraws with marginal densityg(ui|θ 2 ), and ̃f(·)
is a subsimulator such asf(y|x,us,θ 1 ). Many possible simulators may be used – the
essential requirement is that̂fi
p
→fiasS→∞. The MSL estimator is consistent and
asymptotically equivalent to the ML estimator, provided thatS→∞, in addition
to the usual assumption thatN→∞, with
√
N/S→∞so thatSgrows at a rate
slower thanN.
The MSL estimator opens up the possibility of using a much wider range of para-
metric models, such as richer models for unobserved heterogeneity that may be
more robust to model misspecification. At the same time the method can be com-
putationally demanding. An early application of MSL was by Lerman and Manski
(1981) for the multinomial probit model. ThenI×N×Sdraws ofusiare made if
analytical derivatives are used, whereIis the number of iterations, and even more
draws are needed if numerical derivatives are used. A currently popular application
is to the random parameters logit or mixed logit model.
For models with unobserved heterogeneity, an alternative is to treat heterogene-
ity as being discretely distributed. Such finite mixture or latent class models are
especially popular in the duration and count (number of health services) literatures
(see Meyer, 1990; Deb and Trivedi, 2002). These models do not require simulation
methods, and can be more easily estimated using quasi-Newton methods or the
expectation maximization algorithm. Often a heterogeneity distribution with just
two or three points of support is sufficient.
The MSL can be extended to MM and GMM estimation. In that case, theory leads
to a moment condition E[m(yi|xi,θ)]=0, wherem(·)is a scalar for simplicity, but
there is no closed form expression form(y,x,θ). Insteadm(y,x,θ)is an integral:
m(yi|xi,θ)=∫
h(yi|xi,ui,θ 1 )g(ui|θ 2 )dui, (14.13)for some functions h(·)andg(·), wherem(·) has no closed form. Let̂mi =
m̂(yi|xi,u(iS),θ) be a simulator form(yi,xi,θ). Then the method of simulated
moments (MSM) estimator usesm̂iin place ofmiin GMM estimation. A key result,
due to McFadden (1989) and Pakes and Pollard (1989), is that the MSM estimator
is consistent forθasN→∞even ifSis very small, provided that an unbiased
simulator is used, meaning E[̂mi]=mi. Furthermore, smallSmay lead to little
loss of precision. In the special case that̂m(·)is the frequency simulator, the MSM
estimator has variance( 1 +( 1 /S))times that of the MM estimator.
There are several subtleties in the use of MSL and related estimators. Book refer-
ences are Gouriéroux and Monfort (1996), who also discuss indirect inference, and
Train (2003), who focuses on applications to multinomial choice. First, because
the simulated likelihood is usually maximized by iterative gradient methods, the
simulator̂fishould be differentiable (or smooth) inθ. For example, for limited