502 Discrete Choice Modeling
wherewhandzhare the weights and nodes of the quadrature (see Abramovitz and
Stegun, 1971) andHis the number of nodes chosen (typically 20, 32 or 64). An
alternative approach to the approximation is suggested by noting that:
[∫∞
−∞(∏
Ti
t= 1 #(
qit(x′itβ+z′iγ+σuui)))
φ(ui)dui]=Eui[∏
Ti
t= 1
#(
qit(x′itβ+z′iγ+σuui))]
.The expected value can be approximated satisfactorily by simulation by using a
sufficiently large sample of random draws from the population ofui:
[∫∞
−∞(∏
Ti
t= 1 #(
qit(x′itβ+z′iγ+σuui)))
φ(ui)dui]≈
1
R∑R
r= 1∏Ti
t= 1
#(
qit(x′itβ+z′iγ+σuuir))
.Sampling from the standard normal population is straightforward using modern
software (see Greene, 2008a, Ch. 17). The right-hand side converges to the left-
hand side asRincreases (so long as
√
n/R →0; see Gourieroux and Monfort,
1996).^16 Thesimulated log-likelihoodto be maximized is:
lnLS=∑n
i= 1 ln1
R∑R
r= 1∏Ti
t= 1 #(
qit(x′itβ+z′iγ+σuuir))
.Recent research in numerical methods has revealed alternative approaches to
random sampling to speed up the rate of convergence in the integration. Hal-
ton sequences (see Bhat, 1999) are often used to produce approximations which
provide comparable accuracy with far fewer draws than the simulation approach.
11.3.6.4 Dynamic models
An important extension of the panel data treatment in the previous section is the
dynamic model:
d∗it=x′itβ+z′iγ+λdi,t− 1 +αi+εit
dit=1ifdit∗>0 and 0 otherwise. (11.8)Recent applications include Hyslop’s (1999) analysis of labor force participation,
Wooldridge’s (2005) study of union membership and Contoyannis, Jones and Rice’s
(2004) analysis of self-reported health status in the BHPS.^17 In these and other
applications, the central feature isstate dependence, or theinitial conditions problem:
individuals tend to “stick” with their previous position. Wooldridge (2002b) lays
out conditions under which an appropriate treatment is to model the individual
effect as being determined by the initial value in:
αi=α 0 +α 1 di 0 +x′iπ+σuui, ui∼N[0, 1]. (11.9)This is the Mundlak treatment suggested earlier with the addition of the initial state
in the projection.^18 Inserting equation (11.9) in (11.8) produces an augmented