Testing Hypotheses About Ratios of Linear Trend Slopes in Systems of Equations with a Focus on Tests of Equal Trend Ratios

2026-02-26

AI Summary

This study addresses the problem of testing equality of slope ratios across multiple pairs of time series exhibiting linear trends. To mitigate finite-sample bias arising from small trend slopes, the authors propose a statistical inference approach that reparameterizes the ratio as a product of slopes. Extending the single-pair framework of Vogelsang and Nawaz (2017) to a multi-pair setting, the method develops corresponding asymptotic theory and a critical value computation scheme. Monte Carlo simulations demonstrate its superior finite-sample performance. The approach is applied empirically to compare amplification ratios across five pairs of global observed temperature series, offering a reliable statistical tool for evaluating climate models.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Machine Learning: Time-Series/Data Streams
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Security and Privacy: Large-scale security measurements

Abstract

This paper develops inference methods for ratios of deterministic trend slopes in systems of pairs of time series. Hypotheses based on linear cross-equation restrictions are considered with particular interest in tests that trend ratios are equal across pairs of trending series. Tests of equal ratios can be used for the empirical assessment of climate models through comparisons of trend ratios (amplification ratios) of model generated temperature series and observed temperature series. The analysis in this paper builds on the estimation and inference methods developed by Vogelsang and Nawaz (2017, Journal of Time Series Analysis) for a single pair of trending time series. Because estimators of ratios can have poor finite sample properties when the trend slope are small relative to variation around the trends, tests of equal trend ratios are restated in terms of products of trend slopes leading to inference that is less affected by small trend slopes. Asymptotic theory is developed that can be used to generate critical values. For tests of equal trend ratios, finite sample performance is assessed using simulations. Practical advice is provided for empirical practitioners. An empirical application compares amplification ratios (trend ratios) across a set of five groups of observed global temperature series.

Problem

• trend ratios
• hypothesis testing
• time series
• climate models
• amplification ratios

Innovation

• trend ratio
• equal trend ratios
• systems of equations
• finite sample inference
• amplification ratio