Palgrave Handbook of Econometrics: Applied Econometrics

804 Computational Considerations in Microeconometrics

Algorithm 15.5.1.0.1 QR–bootstrap standard errors

1. Draw(ybi,x$ib),b=1,...,B,i=1,...,n, a (paired) bootstrap sample drawn
   from the empirical distribution of(yi,x$i).


2. Estimate the conditional quantile functionx$ib̂β
   b
   q, wherêβ

b

qis the bootstrap

estimate ofβq.

3. The bootstrap estimate of the variance, var(̂βq), is given by:

̂var[̂βq]=

n

B

∑

(̂β

b

q−β

b

q)(̂β

b

q−β

b

q)

$,

whereβbq=B−^1

∑̂

βbq.

are derived from the Medical Expenditure Panel Survey, and consists of 2,873 obser-
vations. The dependent variable is log(TOTEXP), so that zero values are omitted.
The explanatory variables are supplementary private insurance (SUPPINS), which
is a dummy variable, the number of chronic conditions (TOTCHR) as a measure
of health status, and two demographic variables (AGE, FEMALE), and log(income)
(LINC).
The numerical results are displayed in Table 15.4, which reports the OLS results
with Eicker–White heteroskedasticity robust standard errors, and QR estimates for
three values ofq, 0.25, 0.50, 0.75. The OLS results in column 1 can be compared
with the median regression results in column 3; the two should not be too dif-
ferent if the conditional distribution of the dependent variable is symmetric. The
standard errors were computed using a paired bootstrap with 499 bootstrap repli-
cations. Because of the relatively small number of regressors in the model, this
computation is manageable even on a desktop PC. Figure 15.4 displays the graphs
of the quantile regression coefficients at different values ofq. Such a visual repre-
sentation is potentially very informative as it reveals the heterogeneity in response
to variables at different quantiles of expenditure. For example, the impact of SUP-
PINS atq=0.25 is nearly three times as large as atq=0.75. The graphs also include
the constant least squares coefficient as a benchmark.

Table 15.4 OLS and bootstrapped quantile regressions

(1) OLS (2) QR: q=0.25 (3) QR: q=0.50 (4) QR: q=0.75

Coef. Std. error Coef. Std. error Coef. Std. error Coef. Std. error

SUPPINS 0.224 0.0484 0.336 0.0631 0.246 0.0550 0.118 0.0658

AGE 0.0150 0.00364 0.0190 0.00481 0.0178 0.00420 0.0212 0.00500

FEMALE −0.0512 0.0469 0.0252 0.0565 −0.0493 0.0507 −0.145 0.0548

TOTCHR 0.445 0.0176 0.461 0.0239 0.391 0.0195 0.371 0.0208

LINC 0.0593 0.0275 0.0774 0.0304 0.0716 0.0329 0.0566 0.0391

Intercept 5.876 0.298 4.609 0.408 5.726 0.347 6.445 0.415