784 Computational Considerations in Microeconometrics
function subject to linear equality constraints:
min
c$zc$z, subject toAz=y,z≥ 0 ,whereA=[X,−X,In,−In],y=[y 1 ,...,yn]$,z=[βq[^1 ],βq[^2 ],ε[q^1 ],ε[q^2 ]]$,c$=
[ 0 $, 0 $,qι$,( 1 −q)ι$],X=[x 1 ,...,xn]$, 0 is a vector of zeros,ιis a vector of
1s, andInis the identity matrix of ordern.
The classic method for solving the linear program is the simplex method, which
is guaranteed to yield a solution in a finite number of simplex iterations. Whennis
of the order of several thousands, an efficient computational algorithm is essential.
A number of alternative algorithms have been proposed, including a computation-
ally efficient one due to Barrodale and Roberts (1973), thus making QR a suitable
method for practical application in large samples. Point estimation is actually a
lesser computational challenge than the calculation of variances. Bootstrap meth-
ods for variance estimation, often used in preference to analytical expressions,
contribute to computational intensity; an example is given in section 15.2.
15.3.3 Simulation-assisted estimation
The role of simulation in theoretical econometrics is well established (see Doornik,
2006). Its use in empirical microeconometrics is more recent but growing rapidly
(see McFadden and Ruud, 1994; Gouriéroux and Monfort, 1996). First, MCMC
methodology, which is simulation-based, is now standard in modern Bayesian
estimation. Second, estimation of several leading microeconometric models, e.g.,
the multinomial probit (MNP), involve calculation of probability integrals that can
be efficiently estimated using simulation methods. Third, empirical microecono-
metric models often include a latent variable to capture the effects of unobserved
heterogeneity (UH). These too lead naturally to simulation-based estimation. We
consider two leading applications of simulation-based estimation.
15.3.3.1 MNP example
Consider the MNP model withmchoices and with the utility of thejth choice
given by:
Uj=Vj+εj, j=1, 2,...,m, (15.5)where the errorsε=[ε 1 ,...,εm]$are joint normally distributed,ε∼N[ 0 ,].
Usually the linear specificationsVj=x$jβorVj=x$βjare used. This is an additive
random utility model. The covariance matrixis subject to normalization and
identification restrictions becauseUjis latent. The standard practice is to choose
U 1 as the benchmark alternative and place one restriction on.
In estimating the model by ML, a problem is that there is no closed form
expression for the choice probabilities. For anm-choice MNP model the choice
probabilities are (m−1)-fold integrals, e.g.:
p 1 =Pr[y= 1 ]=∫− ̃Vm 1−∞....∫− ̃V 21−∞f( ̃ε 21 ,... ̃εm 1 )d ̃ε 21 ...d ̃εm 1 ,