# ConfidenceBands.jl

Confidence bands for simultaneous statistical inference

ConfidenceBands.jl is a lightweight Julia package for computing confidence bands that are useful for simultaneous statistical inference. In contrast to pointwise confidence intervals computed for each parameter separately, a confidence band treats the entire vector of parameters as a single object and is more suitable for comparisons involving multiple parameters.

## Example Usage

ConfidenceBands.jl extends the confint function for computing confidence bands. Accepted arguments may vary depending on the type of confidence band. Details may be found from docstrings in the help mode of Julia REPL.

#### Plug-In Confidence Bands

Computation of a plug-in confidence band is based on a critical value:

using ConfidenceBands
# Compute the critical value for Bonferroni bands with 90% confidence level
# when there are five parameters
criticalvalue(BonferroniBand(), 0.9, 5)
# A variance-covariance matrix Σ is required for sup-t bands
criticalvalue(SuptBand(), 0.9, Σ)


To obtain confidence bands:

# First obtain point estimates θ as a vector and variance-covariance matrix Σ
lb, ub = confint(SuptBand(), θ, Σ, level=0.95)


#### Bootstrap Confidence Bands

Some types of confidence bands are designed for a valid bootstrap sample provided by users. A bootstrap sample of point estimates needs to be collected in a matrix with each column being a vector of point estimates from the same draw. Currently, quantile-based and critical-value-based bootstrap implementation of sup-t bands (SuptQuantileBootBand and SuptCVBootBand) are implemented following Montiel Olea and Plagborg-Møller (2019):

lb, ub, pwlevel = confint(SuptQuantileBootBand(), draws)
lb, ub, cv = confint(SuptCVBootBand(), θ, draws)


The former additionally returns the confidence level when the intervals from the confidence band are viewed as pointwise confidence intervals. The latter additionally returns the critical value.

## References

Montiel Olea, José Luis and Mikkel Plagborg-Møller. 2019. "Simultaneous Confidence Bands: Theory, Implementation, and an Application to SVARs." Journal of Applied Econometrics 34 (1): 1-17.