Estimation and Inference of Linear Trend Slope Ratios with an Application to Global Temperature Data

We focus on estimation and inference of the ratio of trend slopes between two time series where the trending behavior of each series can be well approximated by a simple linear time trend. Our methodological results are motivated by a recent empirical climate literature that seeks to estimate and test hypotheses about the relative rate of warming in the lower-troposphere relative to surface warming - the so-called amplification ratio. We analyze the statistical properties of several estimators and test statistics that are configured to allow serial correlation in the data. The relative merits of the estimators and test statistics depend on the magnitude of the trend slopes relative to the noise in the data. Based on asymptotic theory and finite sample evidence, we make specific and concrete recommendations for practitioners. We apply the recommended estimator and confidence intervals to temperature data from the 1979-2014 period. We find that amplification ratios typically associated with climate models are rejected by the observed temperature data confirming and extending the empirical findings of Klotzbach et al (2009, 2010). Allowing for a structural change at the end of 1998 to account for the so-called "hiatus" in warming gives results similar to Klotzbach et al (2009, 2010).