Palgrave Handbook of Econometrics: Applied Econometrics

588 Panel Data Methods

whereh(u)is a nonlinear function of the random vectoru, which has a multivariate
densityf(u)and distribution functionF(u). This kind of expression arises in panel
data models with random effects specifications and with autocorrelated errors and
in multiple equation models with correlated unobservables. The integral can be
approximated using draws fromf(u),ur,r=1,...,R, such that:

∫

[h(u)]dF(u)≈

1

R

∑R

r= 1

[

h(ur)

]

. (12.20)

MSL is a simple extension of classical ML estimation and is useful in many cases
where the log-likelihood function involves high dimensional integrals. The idea is
to replace individual contributions to the sample likelihood function (Li) with an
average overRrandom draws:

li=

1

R

∑R

r= 1

[

l(uir)

]

, (12.21)

wherel(uir)is an unbiased simulator ofLi.The MSL estimates are the parameter
values that maximize:

Lnl=

∑n

i= 1

[

Lnli

]

. (12.22)

For likelihoods derived from the multivariate normal the Geweke–Hajivassilou–
Keane (GHK) simulator is often used. In practice, Halton sequences or antithetics
can be used to reduce the variance of the simulator (see Contoyanniset al.,2004a,
for details).

12.4.3 Bayesian MCMC

In Bayesian analysis a prior density of the parameters of interest, sayπ(θ), is updated
using information from sample data. Given a specified sample likelihood for the
observed data,l(y|θ), the posterior density ofθis given by Bayes’ theorem:

π(θ|y)=

π(θ)l(y|θ)

π(y)

, (12.23)

where:

π(y)=

∫

π(θ)l(y|θ)dθ. (12.24)

The scaling factorπ(y)is known as the predictive likelihood and is used to com-
pare models. It determines the probability that the specified model is correct. The
posterior densityπ(θ|y)reflects updated beliefs about the parameters. Given the
posterior distribution, a 95% credible interval can be constructed that contains the