Abstract This report provides an analysis of the wide range of meteorological data for the U.K. that is now publicly available, focusing on the extraction of trends and cycles from monthly observations on various measures of the U.K.’s weather. A brief outline of the report is as follows. After Section 1’s brief introduction, the meteorological data for the U.K. and its districts are presented in Section 2. Section 3 develops the statistical framework, the seemingly unrelated regression (SUR) model, that is used to analyse the various weather measures. Subsequent sections thus analyse temperatures (Section 4), rainfall (Section 5), rain days (Section 6), sunshine hours (Section 7) and frost days (Section 8) using data up to 2022. The measurement of weather volatility is considered in Section 9, while Section 10 focuses on forecasts for the latest available year. 2023. A summary of the various findings are contained in Section 11, from which it is clear that, when attempting to extract the trend and cyclical movements in the weather patterns of the U.K., any analysis must be conducted at the district level, with attention also being focused on seasonal movements. The SUR statistical methodology with flexible Fourier trend-cycle functions proves to be an excellent framework with which to accomplish this.

Analysing the mass of time series data accumulating daily and weekly from the coronavirus pandemic has become ever more important as the pandemic has progressed through its numerous phases. Econometric techniques are particularly suited to analysing this data and research using these techniques is now appearing. Much of this research has focused on short-term forecasting of infections, hospital admissions and deaths, and on generalising to stochastic settings compartmental epidemiological models, such as the well-known "susceptible (S), infected (I) and recovered or deceased (R)", or SIR, model. The focus of the present paper is rather different, however, in that it investigates the changing dynamic relationship between infections, hospital admissions and deaths using daily data from England. It does this using two approaches, balanced growth models and autoregressive distributed lag/error correction models. It is found that there has been a substantial decrease over time in the number of deaths and hospital admissions associated with an increase in infections, with patients being kept alive longer, as clinical practice has improved and the vaccination program rolled out. These responses may be tracked and monitored through time to ascertain whether such improvements have been maintained.

Abstract Since 2019 the U.K. Met Office has provided monthly observations, beginning in 1884, on maximum, minimum and mean monthly air temperatures for the U.K. and its regions. Time series analyses of these data are reported in this paper, using a deterministic monthly trends model. Although the U.K. has seen significant trend temperature increases between 1884 and 2021, these increases differ depending on whether maximum, minimum, or mean temperatures are being considered, with trend minimum temperature increases being greater than trend maximum temperature increases. There is also a difference in the trends exhibited by temperatures in the central and peripheral regions of the U.K., with rather larger trend increases being found in the former than in the latter. The largest trend increases are invariably found during the autumn months, usually October, while the smallest are found in June for maximum temperatures, which are usually insignificant, and in winter for minimum temperatures.

In March 2020 the Rainfall Rescue project was launched on the Zooniverse platform with the aim of transcribing 66,000 sheets of rainfall observations into digital form. The associated website, RainfallRescue.org, received widespread media interest and, within 16 days, the complete set of sheets had been digitised by over 16,000 volunteer ‘citizen scientists’. The project has led to new monthly rainfall estimates for the U.K. and its ten regions being made available to download and time series analyses of these new rainfall data are reported in this paper. Using a seasonal linear trends model fitted to power transformed rainfall data, it is found that over the period 1836 to 2021 winters have become progressively wetter and fluctuations in monthly rainfall more pronounced the further west and north you move in the U.K., with little or no change in summer rainfall.

Current trends in Northern Hemisphere and Central England temperatures are estimated using a variety of statistical signal extraction and filtering techniques and their extrapolations are compared with the predictions from coupled atmospheric-ocean general circulation models.Earlier warming trend epochs are also analysed and compared with the current warming trend, suggesting that the long-run patterns of temperature trends should also be considered alongside the current emphasis on global warming.

Modelling trends and cycles in economic time series has a long history, with the use of linear trends and moving averages forming the basic tool kit of economists until the 1970s. Several developments

We investigate a structural model of demographic-economic interactions for England during 1570 to 1850. We estimate that the annual rate of population growth consistent with constant real wages was 0.4% before 1760 but 1.5% thereafter. We find that exogenous shocks increased population growth dramatically in the early decades of the Industrial Revolution. Simulations of our model show that if these demographic shocks had occurred before the Industrial Revolution the impact on real wages would have been catastrophic and that these shocks were largely responsible for very slow growth of real wages during the Industrial Revolution.

This paper re-examines UK productivity growth in the decades before World War I using a new dataset compiled by Thomas and Dimsdale (2017). We find that the productivity slowdown of the early 20th century was quite modest and does not deserve to be called a climacteric. A more serious slowdown in labour productivity growth occurred in the 1870s. Neither of these episodes should be regarded as a precedent for the current severe deterioration in UK productivity performance. Nor should a late-Victorian productivity slowdown be attributed to the end of the steam age despite the popularity of this belief.

We estimate trend UK labour productivity growth using a Hodrick-Prescott filter method. We use the results to compare downturns where the economy fell below its pre-existing trend. We find that the current productivity slowdown has resulted in productivity being 19.7 per cent below the pre-2008 trend path in 2018. This is nearly double the previous worst productivity shortfall ten years after the start of a downturn. On this criterion the slowdown is unprecedented in the past 250 years. We conjecture that this reflects a combination of adverse circumstances, namely, a financial crisis, a weakening impact of ICT and impending Brexit.

We investigate a structural model of demographic-economic interactions for England during 1570 to 1850. We estimate that the annual rate of population growth consistent with constant real wages was 0.4 per cent before 1760 but 1.5 per cent thereafter. We find that exogenous shocks increased population growth dramatically in the early decades of the Industrial Revolution. Simulations of our model show that if these demographic shocks had occurred before the Industrial Revolution the impact on real wages would have been catastrophic and that these shocks were largely responsible for very slow growth of real wages during the Industrial Revolution.

The autoregressive-moving average (ARMA) process is the basic model for analyzing a stationary time series. First, though, stationarity has to be defined formally in terms of the behavior of the autocorrelation function (ACF) through Wold's decomposition. Several simple cases of the ARMA model are then introduced and analyzed, with the partial autocorrelation function (PACF) also being defined, before the general model is introduced. ARMA model building and estimation may then be developed, and this is done via a sequence of examples designed to demonstrate some of the intricacies of selecting an appropriate model to explain the evolution of an observed time series.

The univariate models studied so far may be generalized to include one or more “input variables,” leading to the traditional transfer function-noise model. A simplified version of this, the autoregressive distributed lag, or ARDL, model has become a very popular framework for modeling a stationary output series as a linear function of current and lagged values of a set of stationary input series, not least because an automatic model selection procedure is available for finding the most appropriate dynamic specification. An important representation of the ARDL model is one in which the long-run relationship between the output and input variables is separated out from the short-run effects.

The ARDL model may be reexpressed in "error correction" form, which separates out short-run effects from the discrepancy from long-run equilibrium—the error correction. This form is especially interesting when the variables entering the ARDL are allowed to be nonstationary. Regression of nonstationary time series is shown to be fraught with difficulties, and generally leads to the "spurious regression" problem. However, it is possible for nonstationary time series to "cointegrate," where a linear combination of, say, I(1) processes is I(0) rather than I(1), as might be expected. Cointegration is shown to be intimately related to error correction and, if it exists, makes empirical modeling much simpler and interpretable. Tests for cointegration are therefore essential, as are methods for estimating the cointegration regressions that exist in its presence.

In the context of a time series, volatility is generally taken to be a period in the evolution of the series that is associated with high variability or increasing variance. This may be modeled by allowing the variance or, more typically, the conditional variance, of the process generating the time series to change continuously or at certain discrete points in time. A popular model for doing this is the autoregressive conditional heteroskedastic, or ARCH, process and this is introduced and discussed. There are many variants of the basic ARCH model and several of the most useful are introduced. Ignoring the presence of ARCH can lead to serious statistical problems, so testing for its presence is clearly important. Estimating the conditional variance is an integral feature of ARCH models, as this provides a direct measure of volatility.

This paper analyses a recently created continuous 305-year (1711–2016) monthly rainfall series for the island of Ireland. The findings are as follows. The excess skewness in the monthly series may be eradicated by using a Box-Cox transformation with parameter equal to 0.6: a value very similar to that found for the U.K. and its regions. There is no evidence of either an overall stochastic trend or of evolving monthly seasonal patterns, but positive linear trends are found for January, March, and December and a negative linear trend is found for July. Analysis of the seasonal and annual series (which require no transformation) confirms the implication from the monthly data that winters have become progressively wetter and summers progressively drier, with the positive linear trend for winter being twice the size of the negative summer trend. Since there is no trend in either spring or autumn rainfall, annual rainfall shows a positive linear trend. Given that the rainfall series exists for over three centuries, breaks and structural shifts in the model were investigated. Five breaks were identified, three of which occurred in the early portion of the series during the eighteenth century. However, trends were found to be much more stable from the middle of the nineteenth century. For the seasonal series, only a single break, at 1790 for the winter series, was found: it was only after this break that winters became wetter; before then, winter rainfall had a negative trend. In terms of predictability, predictions from the model were found to be more volatile during the second half of the eighteenth century and again from 1976 onwards.

The ARDL model of the previous chapter may be extended to allow for several output series in a framework known as the multivariate dynamic regression model. Because it is often difficult to decide whether an input series is truly exogenous, in that there is no feedback to it from an output/endogenous variable, a multivariate model that automatically allows for such feedbacks has become popular. This is the vector autoregression, or VAR, and it allows feedback to be assessed via a concept known as Granger causality. Methods are introduced for determining the lag order of a VAR and, because VARs often contain many coefficients and are thus difficult to interpret directly, the techniques of innovation accounting and variance decomposition are developed which provide indirect ways of interpreting the dynamic links between the series making up the VAR.

Many series have natural constraints placed upon them which should be adhered to in both modeling and forecasting. Compositional time series are proportions or shares of a whole and must therefore be fractions that sum to one across the series for each observation. Thus, for example, any forecasts of future proportions must obey these constraints. An approach which transforms the series using log-ratios is recommended which naturally imposes these constraints. Some time series are recorded as, typically small, integers or counts and thus cannot be treated as continuous variables, as has been implicitly assumed throughout the book. To model and forecast such series "coherently", integer-autoregressive (IN-AR) models, typically with Poisson distributed innovations, can be used. Related time series are "intermittent"—those that contain long sequences of zeros—and nonnegative time series.