I’ve had a few references sent in to me on applications of arima to surface temperature series.
I made it clear that I didn’t know the literature in this area and was just musing. One reader sent in Woodward and Gray, 1992. "Global Warming and the Problem of Testing for Trend in Time Series Data" J Climate 6, 953-962. This is a very specific application of ARIMA models to the CRU and Hansen-Lebedeff data sets, which noted the problems with residuals in a trend regression. They discussed prior efforts to estimate trend where the error structure is more general than white noise, referring to Grenander [1954 Ann Math Stat 29, 252], Brillinger [1989, Biometrika 76, 23], Cryer [1986, Time Series Analysis], Kuo et al [1990 Nature 343, 709], Bloomfield [1992 Clim Change 21, 1] and Bloomfield and Nychka [1992, Clim Change 21, 275], the latter three all dealing with the specific issue of global temperature data.
Woodward and Grey do not opine on the presence/absence of physically-based global warming, but note that:
"we do not claim that our best predictions, using the temperature data alone, are that there will be no increase in the temperature, or that the fixed-mean ARIMA models fit to the temperature series are preferable to fits using model (1.1) [simple trend]. We do claim, however, that these ARIMA models are at least plausible models for the temperature series. Realizations from these ARIMA models have random trends that cause the trend tests to detect a long-term trend in a high percentage of the cases…
We have shown here however that based solely on the available temperature data, there is no conclusive evidence that the trend should be predicted to continue. This is primarily due to the difficulty in distinguishing between data with long-term trends and those with random trends for series of the lengths of the temperature series. Consequently tests based on the [simple trend] model for the purpose of prediction or inference concerning future behavior should be used with caution"
Ross referred me to McKitrick , Inference About Trends in Temperature Data After Controlling for Serial Correlation and Heteroskedastic Variance, Invited Paper, Proceedings of the Russian Geographical Society 164(3), 16-24, here, also applying econometric methods of dealing with serially correlated data to surface temperature data, referring to Zheng, Xiaogu and Reid E. Basher (1999). "Structural Time Series Models and Trend Detection in Global and Regional Temperature Series." Journal of Climate 12, 2347-2358. McKitrick (who is very experienced with matters ARMA) concluded:
Moving eventually to an ARMA model with correction for ARCH residuals resolves the specification problems, reduces the post-1979 measured surface warming rate to 0.13 oC/decade. A Wald test suggests significance may be maintained but the upper bound is in the insignificant range. A likelihood ratio test does not reject the hypothesis of no trend in the surface data pre- or post-1979. That is, any observed upward movement is consistent with random noise in the temperature data. An important conclusion of this paper is that properties of time-series variables must be taken into account when testing for trends. Inference based on naàƒ⭠ve modeling strategies can easily lead to unreliable conclusions. In the case of surface temperatures analyzed here, the conventional methods based on OLS lead to high trend estimates and artificially small standard errors. Improved model specification and corrected statistical modeling yields a lower trend magnitude and evidence against statistical significance. This echoes the concerns of Zheng and Basher (1999) who have also warned against the dangers of erroneous trend detection based on improper use of time series climatological data.
The moral of the story: probably only that it would be worthwhile for someone more knowledgeable than me to do an updated ARMA analysis of the satellite data: if it was publishable on several occasions for the surface data, it would presumably be publishable for the satellite data. Maybe it would even be possible to use ARMA approaches to glean a little more from the two data sets considered in combination.