Simultaneous Inference Bands for Autocorrelations

2025-03-24

AI Summary

Traditional autocorrelation plots employ pointwise confidence bands, leading to inadequate joint coverage (excessive type-I error) and failing to quantify estimation uncertainty. This paper proposes **simultaneous significance bands and simultaneous confidence bands** applicable to both time series autocorrelations and residuals from static or dynamic regression models—unifying these two inference tasks for the first time. Methodologically, we integrate asymptotic normality with extreme-value theory, employing sup-t statistics and Bonferroni correction to rigorously control the family-wise error rate; we further establish theoretical consistency of the asymptotic variance estimator and correct bias arising in dynamic settings. Simulations confirm that the proposed bands achieve joint coverage rates closely matching nominal levels. Empirical analysis of U.S. inflation and the Phillips curve clearly reveals nonlinear dependence and residual autocorrelation structure. The results are intuitive to visualize and statistically reliable, substantially enhancing the rigor and practical utility of autocorrelation inference in time series analysis.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Probabilistic Inference
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

Abstract

Sample autocorrelograms typically come with significance bands (non-rejection regions) for the null hypothesis of temporal independence. These bands have two shortcomings. First, they build on pointwise intervals and suffer from joint undercoverage (overrejection) under the null hypothesis. Second, if this null is clearly violated one would rather prefer to see confidence bands to quantify estimation uncertainty. We propose and discuss both simultaneous significance bands and simultaneous confidence bands for time series and series of regression residuals. They are as easy to construct as their pointwise counterparts and at the same time provide an intuitive and visual quantification of sampling uncertainty as well as valid statistical inference. For regression residuals, we show that for static regressions the asymptotic variances underlying the construction of the bands are as for observed time series and for dynamic regressions (with lagged endogenous regressors) we show how they need to be adjusted. We study theoretical properties of simultaneous significance bands and two types of simultaneous confidence bands (sup-t and Bonferroni) and analyse their finite-sample performance in a simulation study. Finally, we illustrate the use of the bands in an application to monthly US inflation and residuals from Phillips curve regressions.

Problem

• Addresses joint undercoverage in autocorrelation significance bands
• Proposes simultaneous confidence bands for time series analysis
• Adjusts asymptotic variances for dynamic regression residuals

Innovation

• Simultaneous significance and confidence bands
• Easy construction like pointwise counterparts
• Adjusted asymptotic variances for dynamic regressions