HAC robust trend comparisons among climate series with possible level shifts

Abstract

Comparisons of trends across climatic data sets are complicated by the presence of serial correlation and possible step-changes in the mean. We build on heteroskedasticity and autocorrelation robust methods, specifically the Vogelsang-Franses (VF) nonparametric testing approach, to allow for a step-change in the mean (level shift) at a known or unknown date. The VF method provides a powerful multivariate trend estimator robust to unknown serial correlation up to but not including unit roots. We show that the critical values change when the level shift occurs at a known or unknown date. We derive an asymptotic approximation that can be used to simulate critical values, and we outline a simple bootstrap procedure that generates valid critical values and p-values. Our application builds on the literature comparing simulated and observed trends in the tropical lower troposphere and mid-troposphere since 1958. The method identifies a shift in observations around 1977, coinciding with the Pacific Climate Shift. Allowing for a level shift causes apparently significant observed trends to become statistically insignificant. Model overestimation of warming is significant whether or not we account for a level shift, although null rejections are much stronger when the level shift is included.

A Test of the Tropical 200‐ to 300‐hPa Warming Rate in Climate Models

“…In the restricted model the latter parameter is set equal to zero and equation reduces to a simple linear trend. Since a break term was identified by McKitrick and Vogelsang () at 1979, which coincides with prior information from other sources regarding the PCS, we impose λ = 0.35 based on the length of our sample.…”

“…A test that the average model trend equals the average observed trend would be formed by setting each of the first 102 elements of R to 1/102 and the final three each to −1/3, and r = 0. The test statistic VF , based on Vogelsang and Franses (), is where ̂ denotes a least‐squares estimator, is the residual vector from a regression of t on a i + d i D ( λ ), and is the heteroskedasticity‐ and autocorrelation‐consistent variance‐covariance matrix for as derived in McKitrick and Vogelsang (). The VF test statistic has attractive size and power characteristics for tests of the kind we are doing here.…”

“…If λ = 0 the critical values are as given in Vogelsang and Franses (). In our application λ = 0.35 and to generate appropriate critical values we apply the bootstrap methodology outlined in McKitrick and Vogelsang ().…”

“…The model output was obtained and used as‐is from the Koninklijk Nederlands Meteorologisch instituut Climate Data Explorer site (van Oldenborg, 2016). Since autocorrelation structures in climate data can be complex and may differ among data types (see, e.g., Varotsos et al, ) we use a variance estimation methodology robust to general forms of heteroskedasticity and autocorrelation (McKitrick & Vogelsang, ; Vogelsang & Franses, ). We also allow for a possible break term at 1979 associated with the Pacific climate shift (see Seidel & Lanzante, ; Tsonis et al, ; Powell & Xu, , and extensive references therein).…”

Autoregressive wild bootstrap inference for nonparametric trends

“…In addition, when modeling trends in temperature and emission data, researchers also need to take into account that serial dependence and heteroskedasticity may be present in the data, see for example Franses and Vogelsang (2005) and McKitrick and Vogelsang (2014) who study parametric trend modeling in temperature series in the presence of serial dependence. Bootstrap methods provide an easy and powerful way to account for heteroskedasticity and autocorrelation.…”