Palgrave Handbook of Econometrics: Applied Econometrics

David T. Jacho-Chávez and Pravin K. Trivedi 785

where the∼denotes normalization relative to the first alternative. Whenm>3,
this integral is difficult to evaluate numerically. An alternative is to use simulation
methods. One well-known simulator is the GHK simulator due to Geweke (1992),
Hajivassiliou, McFadden and Ruud (1994) and Keane (1994) (see Train, 2003,
for details). In this context the maximum simulated likelihood (MSL) estimator
maximizes:

L̂n(β,)=

∑n

i= 1

∑m

j= 1

yijln̂pij,

where thêpijare obtained using the GHK or other simulator. For consistent esti-
mation we require that the number of draws in the simulatorS→∞as well as
n→∞. Because the method is computationally burdensome, a great deal of work
has been done on improving the simulator and on developing other models that
are good substitute specifications but are easier to estimate (e.g., McFadden and
Train, 2000).

15.3.3.2 Heterogeneity example

Suppose the conditional densityh(yi|xi,θ,ui)for an observation involves a contin-
uous separable UH termui, assumed to be independent ofxi, with densityg(ui).
Then the marginal density is defined by the integral:

f(yi|xi,θ)=

∫

h(yi|xi,θ,ui)g(ui)dui, (15.6)

which needs to be estimated numerically if there is no closed form solution.
An unbiased and consistent estimator̂fioffiis the direct Monte Carlo simulator:

̂f(yi|xi,uiS,θ)=^1

S

∑S

s= 1

h(yi|xi,θ,usi), (15.7)

which averagesh(yi|xi,θ,usi)over theSdraws, whereusi,s=1,...,S, are inde-
pendent draws fromg(ui). Estimators superior to this direct estimator are also
available.
The MSL estimator̂θMSLmaximizes:

̂Ln(θ)=

∑n

i= 1

ln̂f(yi|xi,θ,uiS). (15.8)

If̂f(·)is differentiable inθthen̂θMSLcan be computed using the standard gradient
methods mentioned above. This MSL estimator is asymptotically equivalent to the
ML estimator ifS,n→∞and

√
n/S→0, and has a limit normal distribution.
The MSL method is one of the simplest simulation-based estimators. Just as the
MSL method parallels ML estimation, the method of simulated moments (MSM)
parallels the method of moments. For space reasons we do not elaborate on the
distinctions between these methods.
A direct application of the MSL method sometimes leads to very slow conver-
gence of the simulated likelihood function. Simulation-acceleration techniques are