Palgrave Handbook of Econometrics: Applied Econometrics

David T. Jacho-Chávez and Pravin K. Trivedi 787

Algorithm 15.3.4.0.1 Bootstrap variance estimation – implementation

1. Given data(yi,xi),i=1,...,n, draw (with replacement) a bootstrap sample of
   sizen, denoted(y∗ 1 ,x∗ 1 ),...,(yn∗,x∗n).


2. Calculate the estimatêθ
   ∗
   ofθ.


3. Repeat steps 1–2Bindependent times, whereBis a large number, obtainingB
   bootstrap replications of the statistic of interest, such aŝθ
   ∗
   1 ,...,̂θ

∗

B.

4. Use theseBbootstrap replications to obtain the bootstrap variance VB(̂θ)=

B−^1

∑

(̂θ∗j−̂θ∗)^2 , wherêθ∗= B−^1

∑̂

θ∗j. (It is assumed that the asymptotic

variance of̂θexists.)

In implementing a bootstrap, details generally vary with the specific application.
For example, bootstrap samples may be drawn differently, the value ofBmay vary,
and the target statistic of interest may or may not involve asymptotic refinement.
Section 15.5.1 provides an example that compares standard errors of quantile
regression parameters obtained using an analytical formula with those from a
bootstrap.

15.3.5 Structural models based on dynamic programming

Dynamic programming (DP) models represent a relatively new strand in structural
microeconometric models. The special appeal of the approach comes from the
potential of this class of models to address issues relating to new policies or old
policies in a new environment. Further, the models are dynamic in the sense that
they can incorporate expectational factors and intertemporal dependence between
decisions. From a computational viewpoint these models are an order of magnitude
more complex than most other methods. What follows is merely a bare-bones
sketch of the approach.
The DP approach is rooted in detailed structural specifications derived from
strong theoretical specifications and contrasts sharply with the looser “atheo-
retical” models. The distinctive characteristics of this approach include: a close
integration with underlying theory; an assumption of rational optimizing agents;
extensive use of assumptions and restrictions necessary to support the close
integration of the empirical model with the underlying theory; a high level of
parameterization of the model; concentration on causal parameters that play a
key role in policy simulation and evaluation; and an approach to the estimation
of model parameters that is substantially different from the standard approaches
used in estimating moment condition models.
There are many studies that follow the dynamic programming approach. Rep-
resentative examples are Rust (1987); Hotz and Miller (1993); Keane and Wolpin
(1994). Some key features of DP models can be studied using a model due to Rust
and Phelan (1997), which provides an empirical analysis of how the incentives
and constraints of the US social security and Medicare insurance system affects the
labor supply of older workers.