Palgrave Handbook of Econometrics: Applied Econometrics

788 Computational Considerations in Microeconometrics

The main components of the DP model are as follows. State variables are denoted
byst, control variables bydt.βis the intertemporal discount factor. In implementa-
tion all continuous state variables are discretized – a step which greatly expands the
dimension of the problem. Hence all continuous choices become discrete choices,
dtis a discrete choice sequence, and the choice set is finite. There is a single period
utility functionut(st,dt,θu)andpt(st+ 1 |st,dt,θp,α)denotes the probability den-
sity of transitions fromsttost+ 1. The optimal decision sequence is denoted by

δ=(δ 0 ,...,δT)$, wheredt=δt(st), and is the optimal solution that maximizes the
expected discounted utility:

Vt(s)=max

δ

Eδ{

∑T

j=tβ

j−tu

j(sj,dj,θu)|st=s}. (15.9)

Estimation of the model typically uses the likelihood function:

L(θ)=L(β,θu,θp)

=

∏n

i= 1

∏T

t= 1 Pt(d

i

t|x

i

t,θu)pt(x

i

t|x

i

t− 1 ,d

i

t− 1 ,θp), (15.10)

whereθ=(θ$u,θ$p)$, and assumptions are invoked to separate the parameters
in the transition probabilitypt(st+ 1 |st,dt,θp,α)from those in the utility function
ut(st,dt,θu).
DP models are typically high dimensional. To handle the dimensionality issue a
popular estimation strategy has two steps: (1) estimateθpusing a first-stage partial
likelihood function involving only products of theptterms; (2) estimateθpby
solving the DP problem numerically, using a nested fixed point (NFXP) algorithm,
applied to the partial likelihood function consisting of only products ofpt; (see,
e.g., Rust, 1986, 1987, 1994, 1997). Given the enormous computational burden of
a high-dimensional dynamic discrete choice model, a great deal of recent research
seeks ways of making it more manageable; e.g., Aguirregabiria and Mira (2002)
provide survey methods which avoid repeated full solution of the structural model
in estimation, and the application of simulation and approximation methods (see,
e.g., Aguirregabiria and Mira, 2007).

15.4 Non/semiparametric methods

In many situations of interest in economics, we are interested in identifying and
estimating particular aspects of the joint distribution of a random vector[y,x$]

based on a sample{yi,x$i},i=1,...,n, wherey∈Randxis a mixture of contin-

uous variables,xc=[x^1 ,...,xq^1 ]∈Rq^1 , and discrete,xd=

[

xq^1 +^1 ,...,xq

]$

∈Sd,

whereSdis the support ofxd, andq 2 =q−q 1. For example, our aim could be to
analyze the relationship betweenyandxencapsulated in the conditional mean
functionE[y|x]=m(x), the conditional density functionf

(

y|x

)
, or on a finite-
dimensional vector of parametersβ∈Rq, leaving other aspects of the distribution
unspecified. The available methods, such as kernels, rely on averaging observations
which are closer to those we want to make inference about. A weighting function,