Following discusses some integrated models of climatic time series.
Consider some arbitrary time series, e.g. 6, 3, 1, 4, 5, 3, …. Suppose that we take the cumulative sums: 6, 9, 10, 14, 19, 22, …. A time series obtained via cumulative sums of another series is said to be “integrated” from the other series.
A commonly used class of time series models is the class ARMA (its technical details are not needed here). If a time series that was generated via an ARMA model is integrated, the result is an ARIMA time series (the “I” in the middle stands for integrated).
ARIMA models have been proposed as models for climatic time series. Early research papers that used ARIMA models for climatic time series include the papers of Woodward & Gray  and Zheng & Basher ; those papers now have over a hundred citations. Additionally, some statistics textbooks describe using ARIMA models for climatic time series: e.g. Time Series Analysis And Its Applications by Shumway & Stoffer  and Univariate Time Series In Geosciences by Gilgen .
Are ARIMA models truly appropriate for climatic time series? I do not have an opinion. There seem to be no persuasive arguments for or against using ARIMA models. Rather, studying such models for climatic series seems to be a worthy area of research.
Excursus: Einstein, relativity, and the Nobel Committee
Albert Einstein was one of the greatest scientists who ever lived. Einstein’s greatest achievement was his development of the theory of relativity. Despite that, Einstein did not receive a Nobel Prize for relativity. Einstein was such a great scientist that he did still receive a Nobel Prize, but that was for other, lesser, research.
There was no prize for relativity, because the Nobel Committee was persuaded that relativity was wrong. The particular aspect of relativity that the Committee was persuaded was wrong is time dilation. Time dilation is what causes a clock in a fast-moving vehicle to run more slowly than a clock in a slow-moving vehicle. Time dilation has been measured experimentally: the measurements fit extremely well with the theory of relativity.
Although time dilation is counterintuitive at first, the underlying reason for it is actually easy to understand intuitively. For a nice discussion, see the video “Some Cool Ways of Looking at the Special Theory of Relativity”. (The video is eight minutes long, and was created by a high school student.)
Even though time dilation is easy to understand intuitively, some people refused to accept it. One such person was Henri Bergson. Bergson was apparently too unintelligent to understand time dilation, but seemingly delusional enough about his abilities that he believed he did understand. Bergson managed to convince the Nobel Committee that time dilation does not occur. And so the greatest achievement of one of the greatest scientists who ever lived went unacknowledged by the Nobel Committee.
For more on this topic, see the essay “This philosopher helped ensure there was no Nobel for relativity” by Jimena Canales.
ARIMA and the Met Office
In the U.K., the main institution upon which the government relies for advice on climate science is the Met Office (formerly known as the Meteorological Office). The Met Office has been convinced that ARIMA models are inappropriate for climatic time series. The reason, it seems, is that a blogger, Lucia Liljegren, has managed to convince them. Liljegren is in some ways like Bergson: ignorant, unintelligent, and delusional, but nonetheless influential.
Liljegren’s argument against ARIMA is that ARIMA models have a certain property that the climate system does not have. Specifically, for ARIMA time series, the variance becomes arbitrarily large, over long enough time, whereas for the climate system, the variance does not become arbitrarily large. It is easy to understand why Liljegren’s argument fails.
It is a common aphorism in statistics that “all models are wrong”. In other words, when we consider any statistical model, we will find something wrong with the model. Thus, when considering a model, the question is not whether the model is wrong—because the model is certain to be wrong. Rather, the question is whether the model is useful, for a particular application. This is a fundamental issue that is commonly taught to undergraduates in statistics. Yet Liljegren ignores it.
As an illustration, consider a straight line (with noise) as a model of global temperatures. Such a line will become arbitrarily high, over long enough time: e.g. higher than the temperature at the center of the sun. Global temperatures, however, will not become arbitrarily high. Hence, the model is wrong. And so—by an argument essentially the same as Liljegren’s—we should not use a straight line as a model of temperatures.
In fact, a straight line is commonly used for temperatures, because everyone understands that it is to be used only over a finite time (e.g. a few centuries). Over a finite time, the line cannot become arbitrarily high; so, the argument against using a straight line fails. Similarly, over a finite time, the variance of an ARIMA time series cannot become arbitrarily large; so, Liljegren’s argument fails.
You might ask why Liljegren would be involved in this, if she has such poor understanding. Perhaps the answer lies in psychology. In psychology, there is a well-known syndrome whereby someone is so incompetent at a task that they genuinely do not realize that they are incompetent; the syndrome is known as the Dunning–Kruger effect. Perhaps that is what is happening with Liljegren—and what happened with Bergson. Indeed, I have tried discussing other simple statistical topics with Liljegren, e.g. smoothing; I was unable to communicate in a way that she could understand.
To reiterate, the Met Office has claimed that ARIMA models are inappropriate for climatic time series, on the basis of the argument advanced by Liljegren. So consider the overall situation, with regard to analyzing climatic data. The people of the U.K. rely on the government. The government relies on the Met Office. And the Met Office has eschewed the peer-reviewed literature, statistics textbooks, and reason—and instead relied on Liljegren.
Although the situation has some similarities to what happened with Bergson and the Nobel Committee, there are also some important differences. In particular, the Met Office is a government institution, and as such can be compelled to answer when certain questions are put to it by the government. For that reason, I discussed the situation with a member of the U.K. House of Lords, Lord Donoughue of Ashton.
Lord Donoughue is one of the most numerate people in the U.K. Parliament; e.g. he has a Ph.D. in economics from Oxford and he has also been a visiting scholar at Harvard. Additionally, I have previously served as his statistical advisor on some aspects of climatic data.
Lord Donoughue decided to ask a Parliamentary Question related to ARIMA. The Parliamentary Question that he asked (HL3624) is copied below.
To ask Her Majesty’s Government, further to the Written Answer by Baroness Verma on 26 June 2013 (WA139) and the briefing paper by the Chief Scientist of the Met Office Statistical Models and the Global Temperature Record, cited in the Written Answer, in the light of the use of such models in textbooks, as well as in over a hundred research papers, why they consider integrated models for the global temperature series to be inappropriate; and why the linear trend model that is studied in the briefing paper is not also considered to be inappropriate.
Under the British Parliamentary system, the Parliamentary Question must be answered by the government. The government, in this case, obtains an answer from the Met Office.
In this case, though, the supplied Answer did not really answer the question: rather, the Answer was essentially rhetorical obfuscation. The Met Office has played games like that before. It can take a few tries before they answer properly.
Lord Donoughue then submitted a second Parliamentary Question (HL4936), which is copied below.
To ask Her Majesty’s Government, further to the Written Answer by Baroness Neville-Rolfe on 12 December 2016 (HL3624), whether the criticism set out in the briefing paper Statistical Models and the Global Temperature Record that integrated models are "not consistent with the notion of a climate that is in a steady state" also applies to linear trend models.
The Answer was as follows.
As detailed in the briefing paper Statistical Models and the Global Temperature Record by the Chief Scientist of the Met Office, neither integrated nor linear models are consistent with a climate that is in a steady state. Further, these methods do not explicitly include any description of the physical processes affecting global temperatures and therefore have limited capability in providing information on such processes.
Thus, the Answer acknowledges that the prior criticism, by the Met Office, of integrated (ARIMA) models applies also to a straight line. In other words, the Met Office now agrees with the explanation, given above, about why Liljegren’s argument fails.