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

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.