 |
| |||||
 |
| |||
Bootstrap sampling and estimation
Stata’s programmability makes performing bootstrap sampling and estimation possible (see Efron 1979, 1982; Efron and Tibshirani 1993; Mooney and Duval 1993). We provide two commands to simplify bootstrap estimation. bsample draws a sample with replacement from a dataset. bsample may be used in user-written programs. It is easier, however, to perform bootstrap estimation using the bootstrap prefix command. bootstrap allows the user to supply an expression that is a function of the saved results of existing commands, or you can write a program to calculate the statistics of interest. bootstrap then can repeatedly draw a sample with replacement, run the user-written program, collect the results into a new dataset, and present the results. The user-written calculation program is easy to write because every Stata command saves the statistics it calculates. For instance, assume that we wish to obtain the bootstrap estimate of the standard error of the median of a variable called mpg. Stata has a built-in command, summarize, that calculates and displays summary statistics; it calculates means, standard deviations, skewness, kurtosis, and various percentiles. Among those percentiles is the 50th percentile—the median. In addition to displaying the calculated results, summarize saves them, and looking in the manual, we discover that the median is saved in r(p50). To get a bootstrap estimate of its standard error, all we need to do is type . bootstrap r(p50), reps(1000): summarize mpg, detail and bootstrap will do all of the work for us. We'll also specify a seed() option so that you can reproduce our results. . webuse auto
(1978 Automobile Data)
. bootstrap r(p50), reps(1000) seed(1234): summarize mpg, detail
(running summarize on estimation sample)
(output omitted)
Bootstrap results Number of obs = 74
Replications = 1000
command: summarize mpg, detail
_bs_1: r(p50)
------------------------------------------------------------------------------
| Observed Bootstrap Normal-based
| Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
_bs_1 | 20 .963566 20.76 0.000 18.11145 21.88855
------------------------------------------------------------------------------
Use the estat bootstrap postestimation command to report a table with alternative confidence intervals and an estimate of bias. . estat bootstrap, all
Bootstrap results Number of obs = 74
Replications = 1000
command: summarize mpg, detail
_bs_1: r(p50)
------------------------------------------------------------------------------
| Observed Bootstrap
| Coef. Bias Std. Err. [95% Conf. Interval]
-------------+----------------------------------------------------------------
_bs_1 | 20 .187 .96356601 18.11145 21.88855 (N)
| 19 22 (P)
| 19 22 (BC)
------------------------------------------------------------------------------
(N) normal confidence interval
(P) percentile confidence interval
(BC) bias-corrected confidence interval
For an example of when we would need to write a program, consider the case of bootstrapping the ratio of two means. We first define the calculation routine, which we can name whatever we wish, program myratio, rclass
version 9
summarize length
local length = r(mean)
summarize turn
local turn = r(mean)
return scalar ratio = `length'/`turn'
end
Our program calls summarize and stores the mean of the variable length in a local macro. The program then repeats this procedure for the second variable turn. Finally, the ratio of the two means is computed and returned by our program in the saved result we call r(ratio). With our program written, we can now obtain the bootstrap estimate by simply typing . bootstrap r(ratio), reps(#): myratio
This means that we will execute bootstrap with our myratio
program for # replications. Below we request 1,000 replications and
specify a random-number seed so you can reproduce our results: . bootstrap r(ratio), reps(1000) seed(4567): myratio
(running myratio on estimation sample)
(output omitted)
Bootstrap results Number of obs = 74
Replications = 1000
command: myratio
_bs_1: r(ratio)
------------------------------------------------------------------------------
| Observed Bootstrap Normal-based
| Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
_bs_1 | 4.739945 .0344786 137.47 0.000 4.672369 4.807522
------------------------------------------------------------------------------
The ratio, calculated over the original sample, is 4.739945; the bootstrap
estimate of the standard error of the ratio is 0.0344786. Had we wanted to
keep the 1,000-observation dataset of bootstrapped results for subsequent
analysis, we would have typed . bootstrap r(ratio), reps(1000) seed(4567) saving(mydata): myratio bootstrap can be used with any Stata estimator or calculation command and even with user-written calculation commands.
We have found bootstrap particularly useful in obtaining estimates of
the standard errors of quantile-regression coefficients. Stata performs
quantile regression and obtains the
standard errors using the method suggested by Koenker and Bassett (1978,
1982). Rogers (1992) reports that these standard errors are satisfactory in
the homoskedastic case but that they appear to be understated in the
presence of heteroskedastic errors. One alternative is to bootstrap the
estimated coefficients to obtain the standard errors. For instance, say that
you wish to estimate a median regression of price on variables
weight, length, and foreign. Typing qreg price
weight length foreign will produce the estimates along with
Koenker–Bassett standard errors. To obtain bootstrap standard errors,
we could issue the command . bootstrap, reps(#): qreg price weight length foreign We recommend this procedure so highly that Gould (1992) wrote a new command in Stata’s programming language to further automate this procedure for quantile regression. Typing bsqreg price weight length foreign will also produce the bootstrapped results. See New in Stata 11 for more about what was added in Stata Release 11. References
|
|||
