1. Introduction

The last two decades has seen a major research focus on modelling trends in time series, particularly in those disciplines where trend modelling is of major importance, such a macroeconomics, finance and, in the present context, climatology. The econometrician Peter Phillips, in a sequence of influential papers (Phillips, 2001a,b, 2003, 2005: see also Mills, 2009, for historical perspective), discusses the importance of correctly modelling trends when fundamental changes are taking place. Phillips argues that the conventional alternatives of constant polynomial trends or stochastic trends with constant drift either seem naïve, in the former case, or remain mysterious about the sources of change, as in the latter. His conclusion is that, since ‘no one really understands trends, even though most of us see trends when we look at … data’ (Phillips, 2003, page C35), more flexible trend specifications are needed and that ‘we must learn to do inference about trends in settings where the real trend in the data is far more complex than the coordinates we use to describe it’ (Phillips, 2001b, page 24).

While such views were developed with economic applications in mind, that they are just as applicable to the modelling of temperature time series may be seen from Figure 1, which shows annual observations from 1850 to 2009 on HADCRUT3 global temperatures (see Brohan et al, 2006), with a linear trend superimposed. In possibly the first paper to analyse the trending behaviour of temperature records using modern time series techniques, Bloomfield and Nychka (1992) remark that a key question is whether the trend temperature rise of EMBED Equation.3 across this sample period reflects systematic warming or whether it is simply an effect of natural variability.

As the preceding discussion suggests, simply fitting a linear trend, as has been done in Figure 1, may well be inappropriate since it is clearly of crucial importance to analyse the underlying stochastic process generating the temperature data, as has been emphasised by many authors: see, for example, Galbraith and Green (1992), Seater (1993), Harvey and Mills (2001) and Mills (2010). Indeed, a recurring issue, which is the theme of the latter paper, is that when attempting to model temperature series there are a range of stochastic models to choose from (leading to the use of the old adage ‘there are many ways to skin a cat’) for which comparisons are difficult and for which statistical arguments alone are unlikely to settle any disputes. Hence ‘you pays your money and takes your choice’ is thus an apposite proverb to describe this situation, particularly in view of the ongoing debate concerning the potential costs of combating global warming and climate change!

2. Structural Time Series Models

One particular class of stochastic model that has several potential advantages is the structural time series approach used by Visser and Molenaar (1995), Stern and Kaufmann (2000) and Mills (2004, 2010) to model trends in a variety of temperature series. Only a simple form is required to model the global temperature series of Figure 1 and its underlying hemispheric counterparts of which it is the average.

The structural model represents an observed time series as the sum of a set of ‘unobserved components’ representing key features of the data such as the trend, any cyclical movement around that trend, a seasonal pattern if the data is observed at a frequency higher than annually, and a random noise component. Appropriate models may be selected on the basis of acceptability of statistical fit, parameters may be estimated and accompanied by measures of precision, and the components may be calculated using weighting schemes that depend on the choice of model and which naturally adapt to the information contained in the sample. The entire modelling process can be performed using commercially available software which enables replication and comparison of models to be undertaken, a problem that has bedevilled some temperature research in the past.

For trending, i.e., nonstationary, time series, modelling the trend component is clearly paramount, as it is this component that will capture any long run movement in the series. In its most general specification, the trend takes the form of an n^th order exponentially weighted moving average, more commonly known in the operations research literature as n^th order Brown-Holt-Winters exponential smoothing, a technique that has been used in product inventory control and sales forecasting since the 1950s (see, for example, Gardner, 2006, for a recent survey of the literature). It also has a representation as a particular form of the Box-Jenkins (1962) polynomial predictor and as an ARIMA EMBED Equation.3 process, which is a well known stochastic model for trend projection (see Box and Jenkins, 1970).

The other component needed in the modelling of annual temperatures is found to be a stochastic cycle. This models the cyclical component as a damped cosine wave driven by shocks that, in the most general formulation, are themselves periodic, which can lend a degree of smoothness to the cycle. As it turns out, the shocks driving the damped cosine wave in the temperature series analyzed here are random, so that the cyclical component effectively takes the form of a stationary but correlated series having a mean oscillatory period that can be estimated, along with the damping factor, from the data.

3. Structural Time Series Models for Global and Hemispheric Temperatures

Parameter estimates being provided in Table 1. In this section attention is concentrated on graphical representations of the output from the structural time series models fitted to hemispheric temperatures and their average, global temperatures. Figure 2 shows the structural trend superimposed on global temperatures, while Figure 3 shows the trend and cyclical components individually.

It is recognized that there have undoubtedly been periods in which the general direction of the trend has been upwards, most notably in the last quarter of the 20th century. Both NASA and the U.S. National Oceanic and Atmospheric Administration (NOAA), have confirmed that the decade 2000 through 2009 is the warmest decade since 1880. Since 1980, there has also been upward trend in surface temperature, of about 0.36 degrees Fahrenheit (0.2 degrees Celsius) per decade, as based on data obtained from NASA's Goddard Institute of Space Studies. However, 2005 was the warmest year, and no significant difference were found when comparing surface temperatures for 1998, 2002, 2003, 2006, 2007 and 2009.

Under this model these "trends" are simply the accumulation of uncorrelated increments and are just a manifestation of stochastic behaviour that has been well known to time series analysts since Working (1934) and Slutzky (1937). The random walk nature of the trend is clear from Figures 1, 2, and 3, and Table 1. The rather volatile behaviour of the cyclical component is also apparent , with the random innovations continually shocking the component away from a smooth oscillation, although a stochastic cycle with an average period of approximately six years is still evident, perhaps reflecting El Niño induced temperature movements.

Figure 4 shows the weight functions for computing the trend component at 1929, the centre of the sample, and at the end of the sample, 2009. At the centre of the sample the weights used in computing the trend are two-sided, symmetric and monotonically declining, being essentially zero after a displacement of ten years. At the end of the sample the weight function is, of course, one-sided and backward-looking as future observations are unavailable. There is now a ripple effect in the weights, which are effectively zero after a displacement of eight years, and a comparison of the two functions shows how the weights adapt as the trend calculation runs through the sample period.

Figures 5 and 6 show the structural trends fitted to the two hemispheric temperature series superimposed on the temperatures themselves and again the familiar pattern of a random walk is in evidence. The weight functions for these trends are not shown as they are very similar to those depicted in Figure 4 for global temperatures. Neither are the cyclical components shown since they display similar fluctuations to the global cycle.

4 Discussion

The most notable implication of these structural models is that concerning the presence of a global warming signal. The absence of a significant drift in the trend component, making this a simple random walk, thus precludes the presence of a warming – or, indeed, a cooling – trend. Long term forecasts of trend temperatures are given by the current value of the trend component and future trend temperatures have as much chance of going down as they have of going up. This may seem surprising given the general upward movement of the trend component over the last thirty years or so but such departures, which would be attributable to natural variation, are entirely possible for random walks and are a consequence of the arcsine law of probability (see Feller, 1971). As a consequence, forecast bounds for the trend component quickly become large as forecast error variances increase linearly with the forecast horizon for a random walk. Consequently, while the one-year ahead trend forecast has 68% (one standard error) bounds of EMBED Equation.3, twenty year forecasts have bounds of EMBED Equation.3 , so that there is considerable uncertainty built into the future behaviour of temperatures under this model.

How does the structural model relate to other trend models fitted to temperature data? Polynomial time trends are a special case of the structural model. As only a first order stochastic trend is found to be required, this would reduce to a linear trend if the innovation variances While the latter is indeed zero, the results presented in Table 1 show that the former is certainly positive, hence ruling out a linear trend component.

What cannot be ruled out by the structural models, however, is the possibility that temperatures are driven by underlying nonlinear trend functions, perhaps of the smooth transition variety studied by Harvey and Mills (2001) or the breaking linear trends of Gay-Garcia et al (2009). Interestingly, the two traditional statistics for testing whether a time series contains a random walk component or is better characterised as stationary fluctuations around a linear trend – the ADF test, which has the random walk as the null, and the KPSS test, which has the linear trend as the null – both reject their respective nulls for global temperatures at the 1% level of significance, thus adding to the uncertainty concerning the stochastic process driving temperatures. What is thus abundantly clear, then, is that analyzing the trend behaviour of temperature series such as these will remain an active research area for the foreseeable future.