# Bootstrap¶

Bootstrapping is a method to construct empiric distributions of various statistics (mean, confidence intervals, deviation, etc) by using repreated sampling from an observed dataset.

A little magic is why exactly taking random samples
like `[1,1,2]`

, `[3,2,2]`

, `[2,1,3]`

, etc is a good idea to approximate
properties of a dataset `[1,2,3]`

.

Bootstrap originally proposed by Bradley Efron in 1979. For formal introduction see Horowitz chapter in Handbook of Econometrics (2001) and a usage overview by MacKinnon 2006.

## Toy example¶

Bootstrap confidence intervals by Jeremy Orloff and Jonathan Bloom, pp. 4-6 provides the following basic code example for bootstrap. Their full code for this excercise is here.

```
# Bootstrap
# Adapted from https://math.mit.edu/~dav/05.dir/class24-empiricalbootstrap.r
cat("Example. Empirical boostrap confidence interval for the mean.",'\n')
x = c(30,37,36,43,42,43,43,46,41,42)
n = length(x)
set.seed(1) # for repeatability
# sample mean
xbar = mean(x)
cat("data mean = ",xbar,'\n')
nboot = 20
# Generate 20 bootstrap samples, i.e. an n x 20 array of
# random resamples from x.
tmpdata = sample(x,n*nboot, replace=TRUE)
bootstrapsample = matrix(tmpdata, nrow=n, ncol=nboot)
# Compute the means xbar*
xbarstar = colMeans(bootstrapsample)
# Compute delta* for each bootstrap sample
deltastar = xbarstar - xbar
# Find the 0.1 and 0.9 quantiles for deltastar
d = quantile(deltastar,c(0.1,0.9))
# Calculate the 80\% confidence interval for the mean.
ci = xbar - c(d[2],d[1])
cat('Bootstrap confidence interval: [',ci,']','\n')
```

### Bootstrap do’s and don’ts by Anna Mikusheva¶

If you have a pivotal statistic, bootstrap can give a refinement. So, if you have choice of statistics, bootstrap a pivotal one.

Bootstrap may fix a finite-sample bias, but cannot help if you have inconsistent estimator.

In general, if something does not work with traditional asymptotics, the

bootstrap cannot fix your problem. For example, if we have an inconsistent estimate, the bootstrap bias correction does not fix anything.Bootstrap could not fix the following problems: weak instruments, parameter on a boundary, unit root, persistent regressors.

Bootstrap requires re-centering (the null hypothesis to be true).

Source: MIT lecture notes

## More links (preliminary)¶

## Editor notes¶

Russian article in Wikipedia on bootstrap is much more concise and understandable than English one.