# Bootstrap Confidence Intervals#

We use the notation and formulations provided in chapter 10 of [Han20].

The first supported confidence interval type is the **“percentile”** confidence
interval, as discussed in section 10.10 of the Hansen textbook. Let
\(\{ \hat{\theta}_1^*, ..., \hat{\theta}_B^*\}\) denote the estimates of estimator
\(\hat{\theta}\) for the B bootstrap samples. The idea of the percentile confidence
interval is to simply take the empirical quantiles \(q_{p}^*\) of this distributions, so
we have

The second supported confidence interval **“normal”** is based on a normal approximation
and discussed in Hansen’s section 10.9. Let \(s_{boot}\) be the sample standard error of
the distribution of bootstrap estimators, \(z_q\) the q-quantile of a standard normal
distribution and \(\hat{\theta}\) be the full sample estimate of \(\theta\). Then, the
asymptotic normal confidence interval is given by

The bias-corrected **“bc”** bootstrap confidence interval addresses the issue of biased
estimators. This problem is often present when estimating nonlinear models. Econometric
details are discussed in section 10.17 of Hansen. Let

and define \(z_0^* = \Phi^{-1} (p^*)\), where \(\Phi\) is the standard normal cdf. The bias correction works via correcting the significance level. Define \(x(\alpha) = \Phi(z_\alpha + 2 z_0^*)\) as the corrected significance level for a target significant level of \(\alpha\). Then, the bias-corrected confidence interval is given by

A further refined version of the bias-corrected confidence interval is the
bias-corrected and accelerated interval, short **“bca”**, as discussed in section 10.20
of Hansen. The general idea is to correct for skewness sampling distribution. Downsides
of this confidence interval are that it takes quite a lot of time to compute, since it
features calculating leave-one-out estimates of the original sample. Formally, again,
the significance levels are adjusted. Define

where \(\bar{\theta}=\frac{1}{n} \sum_{i=1}^{n} \widehat{\theta}_{(-i)}\). This is an estimator for the skewness of \(\hat{\theta}\). Then, the corrected significance level is given by

and the bias-corrected and accelerated confidence interval is given by

The studentized confidence interval, here called **“t”** type confidence interval first
studentizes the bootstrap parameter distribution, i.e. applies the transformation
\(\frac{\hat{\theta}_b-\hat{\theta}}{s_{boot}}\), and then builds the confidence interval
based on the estimated quantile function of the studentized data \(\hat{G}\):

The final supported confidence interval method is the **“basic”** bootstrap confidence
interval, which is derived in section 3.4 of [Was06], where it is called
the pivotal confidence interval. It is given by

where \(\hat{\theta}_{u}^{\star}\) denotes the \(1-\alpha/2\) empirical quantile of the bootstrap estimate distribution for parameter \(\theta\) and \(\hat{\theta}_{l}^{\star}\) denotes the \(\alpha/2\) quantile.