I have attempted to reconcile the Karl trends and the CIs for those trends without success and will now have to contact Karl, as the lead author of Karl (2015), about the details of their calculations used in this paper.
I obtain substantially larger CIs and somewhat smaller trends using either a linear regression or a spline smooth for determining the trend and using that trend to subtract from the ERSST V4 series to determine the residuals. From these residuals I obtain an ARMA model from which I can do Monte Carlo simulations to estimate CIs for the trends. For the spline smooth (df=7 and spar=0.75) I can use the entire 1854-2014 time period while for the linear regression trend I use the most linear part of the series and that being 1951-2014. The trends and CIs agree well using either method and mainly because the period analyzed follows a reasonably linear trajectory. I prefer, as a general approach to problems such as this one, to use the spline smooth method or alternately the Singular Spectrum Analyses decomposition and reconstruction method which do not assume the secular trends are linear.
What I have found by simulating a long series of 161 years using an ARMA model – similar to what I found by the above described methods for the residuals – and then finding the best fitting ARMA model using the last 15 and 17 years of that series is that the shorter 15 and 17 years are too short to see most of the autocorrelation effects of the 161 years series and thus the best fitting model in the shorter series lies closer to a white noise domain, i.e. ARMA(0,0). This is what I found when modeling the last 15 and 17 years of the ERSST V4 series.
If I restricted the ARMA model to ar1, as was the stated case for the Karl method of determining the trends CIs, the CIs would be substantially smaller if Karl used the 1998-2014 and 2000-2014 time periods for determining the ar1 than if the longer periods were used as I prescribed above. Even when I attempt to estimate CIs by what I judge to be the method used in Karl (2015) the CIs are larger than those reported by Karl.
If indeed the ARMA models that best fit the longer and shorter ERSST V4 series were as different as one would measure using the longer and shorter time periods one would have to suspect something very different was affecting these 2 periods – like the adjustments made. I do not think that is the case, but rather the limitation of ARMA modeling short time series – as my simulations showed.
