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

\[ CI^{percentile} = [q_{\alpha/2}^*, q_{1-\alpha/2}^*]. \]

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

\[ CI^{normal} = [\hat{\theta} - z_{1- \alpha/2} s_{boot}, \hat{\theta} + z_{1- \alpha/2} s_{boot}]. \]

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

\[ p^* = \frac{1}{B} \sum_{b=1}^B 1(\hat{\theta}_b^* \leq \hat{\theta}) \]

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

\[ CI^{bc} = [q_{x(\alpha/2)}^*, q_{x(1-\alpha/2)}^*]. \]

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

\[ \hat{a}=\frac{\sum_{i=1}^{n}\left(\bar{\theta}-\hat{\theta}_{(-i)}\right)^{3}} {6\left(\sum_{i=1}^{n}\left(\bar{\theta}-\hat{\theta}_{(-i)}\right)^{2} \right)^{3 / 2}}, \]

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

\[ x(\alpha)=\Phi(z_{0}+\frac{z_{\alpha}+z_{0}}{1-a(z_{\alpha}+z_{0})}) \]

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

\[ CI^{bca} = [q_{x(\alpha/2)}^*, q_{x(1-\alpha/2)}^*]. \]

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 \(\hat{G}\):

\[ CI^{t} = \left[\hat{\theta}+\hat{\sigma} \hat{G}^{-1}(\alpha / 2), \hat{\theta}+\hat{\sigma} \hat{G}^{-1}(1-\alpha / 2)\right] \]

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

\[ CI^{basic} = \left[\hat{\theta}+\left(\hat{\theta}-\hat{\theta}_{u}^{\star}\right), \hat{\theta}+\left(\hat{\theta}-\hat{\theta}_{l}^{\star}\right)\right], \]

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.

Han20

Bruce E. Hansen. Econometrics. Unpublished, https://www.ssc.wisc.edu/ bhansen/econometrics/, 2020.

Was06

Larry Wasserman. All of nonparametric statistics. Springer Science & Business Media, 2006.