Bootstrap Inference

Quick Reference

Class: panelbox.validation.robustness.PanelBootstrap Import: from panelbox.validation.robustness import PanelBootstrap Key method: bootstrap.run() then bootstrap.conf_int() Stata equivalent: bootstrap prefix R equivalent: boot::boot()

Why Bootstrap?

Asymptotic inference relies on assumptions -- normality, correct variance specification, large samples -- that may not hold in practice. Bootstrap inference replaces these assumptions with computation: resample the data, re-estimate the model many times, and let the empirical distribution speak for itself.

Bootstrap is especially valuable when:

• The number of clusters (entities) is small (\(N < 50\)), making clustered SEs unreliable
• The distribution of the test statistic is non-standard
• You want distribution-free confidence intervals
• You suspect heteroskedasticity or serial correlation patterns that analytical SEs may not fully capture

Four Bootstrap Methods

PanelBox implements four bootstrap methods, each suited to different data structures:

Method	Resampling Unit	Preserves	Best For
pairs	Entire entities	Panel structure, within-entity correlation	General purpose (default)
wild	Residuals (Rademacher weights)	Heteroskedasticity pattern	Heteroskedastic errors
block	Blocks of time periods	Temporal dependence	Autocorrelated data
residual	i.i.d. residuals	Nothing special	Homoskedastic i.i.d. errors

Pairs Bootstrap (Default)

Resamples entire entities with replacement. If the original panel has \(N\) entities, draw \(N\) entities randomly (with replacement) and stack their complete time series. This preserves within-entity correlation and is robust to both heteroskedasticity and serial correlation.

bootstrap = PanelBootstrap(results, n_bootstrap=1000, method="pairs", random_state=42)

Wild Bootstrap

Keeps the design matrix \(X\) fixed and perturbs residuals using Rademacher weights \(w_i \in \{-1, +1\}\) with equal probability. The bootstrap outcome is \(y^* = \hat{y} + w \cdot \hat{e}\). Specifically designed for heteroskedasticity but does not preserve serial correlation.

bootstrap = PanelBootstrap(results, n_bootstrap=1000, method="wild", random_state=42)

Block Bootstrap

Resamples blocks of consecutive time periods (moving block bootstrap). Block size defaults to \(T^{1/3}\) or can be set manually. Preserves temporal dependence within blocks while breaking dependence between blocks.

bootstrap = PanelBootstrap(
    results, n_bootstrap=1000, method="block", block_size=3, random_state=42
)

Residual Bootstrap

Resamples centered residuals assuming i.i.d. errors. The algorithm: (1) center residuals \(\tilde{e} = e - \bar{e}\), (2) resample \(\tilde{e}^*\) with replacement, (3) reconstruct \(y^* = \hat{y} + \tilde{e}^*\). Most restrictive assumptions -- use only when confident errors are i.i.d.

bootstrap = PanelBootstrap(results, n_bootstrap=1000, method="residual", random_state=42)

Quick Example

from panelbox import FixedEffects
from panelbox.validation.robustness import PanelBootstrap
from panelbox.datasets import load_grunfeld

data = load_grunfeld()
model = FixedEffects("invest ~ value + capital", data, "firm", "year")
results = model.fit()

# Pairs bootstrap with BCA intervals
bootstrap = PanelBootstrap(
    results=results,
    n_bootstrap=1000,
    method="pairs",
    random_state=42,
    show_progress=True,
)
bootstrap.run()

# Confidence intervals
ci = bootstrap.conf_int(alpha=0.05, method="percentile")
print(ci)

# Compare bootstrap SEs with asymptotic SEs
summary = bootstrap.summary()
print(summary)

# Visualize bootstrap distribution
bootstrap.plot_distribution(param="value")

API Reference

Constructor

PanelBootstrap(
    results=results,         # PanelResults from model.fit()
    n_bootstrap=1000,        # Number of replications
    method="pairs",          # 'pairs', 'wild', 'block', 'residual'
    block_size=None,         # For block bootstrap (default: T^(1/3))
    random_state=42,         # Reproducibility seed
    show_progress=True,      # Display progress bar
    parallel=False,          # Parallel computation (not yet implemented)
)

Backward Compatibility

The model parameter is accepted as an alias for results, and seed as an alias for random_state. Use the preferred names results and random_state for new code.

Methods

Method	Returns	Description
run()	PanelBootstrap	Execute bootstrap (returns self for chaining)
conf_int(alpha, method)	pd.DataFrame	Confidence intervals (lower/upper columns)
summary()	pd.DataFrame	Comparison of original and bootstrap estimates
plot_distribution(param)	--	Histogram of bootstrap distribution with CI bands

Result Attributes (after run())

Attribute	Type	Description
bootstrap_estimates_	np.ndarray	Bootstrap coefficient estimates (\(B \times K\))
bootstrap_se_	np.ndarray	Bootstrap standard errors
bootstrap_t_stats_	np.ndarray	Studentized bootstrap t-statistics
n_failed_	int	Number of failed replications

Confidence Interval Methods

# Percentile method (simplest, recommended)
ci = bootstrap.conf_int(alpha=0.05, method="percentile")

# Basic (reflection) method
ci = bootstrap.conf_int(alpha=0.05, method="basic")

# Bias-corrected accelerated (most accurate)
ci = bootstrap.conf_int(alpha=0.05, method="bca")

# Studentized (requires nested bootstrap)
ci = bootstrap.conf_int(alpha=0.05, method="studentized")

Method	Formula	Properties
percentile	\([\theta^*_{\alpha/2}, \theta^*_{1-\alpha/2}]\)	Simple, range-preserving
basic	\([2\hat\theta - \theta^*_{1-\alpha/2}, 2\hat\theta - \theta^*_{\alpha/2}]\)	Bias-corrected by reflection
bca	Bias-corrected and accelerated	Most accurate; adjusts for bias and skewness
studentized	Uses bootstrap t-distribution	Asymptotically optimal; computationally intensive

BCA and Studentized

The bca and studentized methods currently fall back to the percentile method with a warning. The percentile method is adequate for most applications.

Rules of Thumb

Goal	Minimum n_bootstrap
Standard errors	500
Confidence intervals	1,000
Hypothesis testing	2,000
Percentile precision	Use odd numbers (e.g., 999, 1999)

Handling Failures

If more than 10% of replications fail, PanelBox issues a warning. If more than 50% fail, an error is raised. Many failures indicate problems with the model specification or insufficient data within resampled subsets. Try a different bootstrap method or simplify the model.

Choosing a Bootstrap Method

Is serial correlation a concern?
├── Yes → Is heteroskedasticity also present?
│        ├── Yes → pairs (preserves both)
│        └── No  → block (preserves time dependence)
└── No  → Is heteroskedasticity present?
         ├── Yes → wild (specifically designed for het.)
         └── No  → residual (most efficient under i.i.d.)

When in doubt → pairs (safest default)

Comparing Bootstrap and Asymptotic Inference

import pandas as pd

# Run bootstrap
bootstrap = PanelBootstrap(results, n_bootstrap=1000, method="pairs", random_state=42)
bootstrap.run()

# Summary table: Original vs Bootstrap
summary = bootstrap.summary()
print(summary)
# Columns: Original, Bootstrap Mean, Bootstrap Bias, Original SE, Bootstrap SE, SE Ratio

# If SE Ratio >> 1: asymptotic SEs are too small (liberal inference)
# If SE Ratio << 1: asymptotic SEs are too large (conservative inference)
# If SE Ratio ≈ 1: asymptotic inference is reliable

Visualization

# Plot distribution for a single parameter
bootstrap.plot_distribution(param="value")

# Plot all parameters
bootstrap.plot_distribution()

The plot shows a histogram of bootstrap estimates with a red dashed line at the original point estimate.

Common Pitfalls

Common Issues

1. Too few replications: Using \(B < 500\) gives noisy SE estimates. Always use at least 1,000 for CIs.
2. Ignoring failures: Check bootstrap.n_failed_ after run(). High failure rates invalidate results.
3. Wrong method for the DGP: Using residual bootstrap when errors are heteroskedastic will produce incorrect CIs.
4. Block size too large: For block bootstrap, a block size near \(T\) effectively resamples the entire time series, defeating the purpose.

See Also

• Jackknife -- Deterministic leave-one-out alternative to bootstrap
• Sensitivity Analysis -- Subsample stability assessment
• Robustness Overview -- Full robustness toolkit

References

• Cameron, A. C., & Trivedi, P. K. (2005). Microeconometrics: Methods and Applications. Cambridge University Press, Chapter 11.
• Efron, B., & Tibshirani, R. J. (1994). An Introduction to the Bootstrap. Chapman and Hall/CRC.
• Cameron, A. C., Gelbach, J. B., & Miller, D. L. (2008). Bootstrap-based improvements for inference with clustered errors. Review of Economics and Statistics, 90(3), 414-427.
• Liu, R. Y. (1988). Bootstrap procedures under some non-i.i.d. models. The Annals of Statistics, 16(4), 1696-1708.
• Kunsch, H. R. (1989). The jackknife and the bootstrap for general stationary observations. The Annals of Statistics, 17(3), 1217-1241.