Persistence in UK Historical Data on Life Expectancy

This paper provides estimates of persistence in historical UK data on life expectancy applying fractional integration methods to both an annual series from 1842 to 2019 and a 5-year average from 1543 to 2019. This method is the most appropriate for our purposes since it is more general and flexible than the classical methods based on integer differentiation. The results indicate that the former exhibits an upward trend and is persistent but mean reverting; the same holds for the latter, though its degree of persistence is higher. Similar results are obtained for the logged values. On the whole, this evidence suggests that the effects of shocks to the series are transitory though persistent, which is useful information for policy makers whose task is to take appropriate measures to increase life expectancy.


Introduction
Life expectancy is a useful indicator of a population's health (Roser et al., 2013). Before the Age of Enlightenment, life expectancy was generally very low (around 30 years) across the world. However, during the early 1800s, it began to rise in countries where the Industrial Revolution took place, creating a significant gap in health conditions between wealthy and impoverished nations. Throughout the world, there has been a significant increase in the average lifespan since the early twentieth In particular, the UK government focussed on improving health outcomes through policy interventions, and thus comparing its experience with that other countries can provide insights into the factors accounting for differences in health outcomes across populations. This type of analysis can inform policy decisions concerning health and social care and provide a better understanding of factors that affect health outcomes over time.
The purpose of analysing life expectancy in the UK is to detect time trends, investigate the causes of the recent decline in life expectancy improvements, and evaluate the impact of policy interventions such as public health campaigns, healthcare policy changes, or social welfare programmes, on life expectancy and related health outcomes, in order to identify the most effective policies. In particular, our investigation shed light on the degree of persistence of the series and on whether exogenous shocks have transitory or permanent effects; this is essential information for policy makers to decide whether or not policy actions should be taken.
The rate of progress in life expectancy in the UK has slowed down in recent years and come to a halt between 2011 and 2019 after significant improvement over the past century (Riley, 2001(Riley, , 2005Our World in Data 2023). The Covid-19 pandemic has further affected it and exposed existing health disparities. Factors contributing to the slowdown in life expectancy improvements include rising inequality, changes in lifestyle and diet, and reductions in public health funding (Peltzman, 2009). Various types of models, such as the Chebyshev polynomials-based approach, Bloomfield's exponential model, and Granger and Joyeux's long memory time series, can be used to model the persistence and non-linearities of life expectancy changes over time (Bloomfield, 1973;Cuestas & Gil-Alana, 2016;Granger & Joyeux, 1980;Hosking, 1981). Other statistical models, such as nonstationary hypotheses testing and Poisson common factor models, can be used to identify the factors that affect life expectancy (Gil-Alana & Robinson, 1997;Li, 2013;Robinson, 1994). Ongoing data collection and analysis are crucial to understanding and improving health outcomes in the UK, informing policy decisions and interventions aimed at reducing health disparities and improving overall health (Cutler et al., 2006;Oeppen & Waupel, 2002;Preston, 1975).
Applying fractional integration methods fills a gap in the literature by providing a more general and flexible method for modelling long-range dependence in time series data on life expectancy. Previous studies have examined various aspects of life expectancy, such as time trends and the factors driving it. However, they have not used the long-memory approach followed in the present study, which sheds new light on the stochastic properties of the series of interest and can also yield more accurate forecasts. The layout of the paper is the following: "Data and Methodology" section describes the data and the methodology; "Empirical Results" section discusses the empirical findings; Final section offers some concluding remarks.

Data and Methodology
The series used for the analysis are UK life expectancy from 1842 to 2019 at an annual frequency and its 5-year average from 1543 to 2018. They have been constructed by 'OurWorldinData', which is a project of the Global Change Data Lab, a non-profit organisation based in the UK (Registered Charity Number 1186433), and available from Roser et al. (2013) at https:// ourwo rldin data. org/ life-expec tancy (online resource). The annual figures are the average number of years a newborn would live if age-specific mortality rates in the c rrent year were to stay the same throughout its life, whilst the 5-year average is calculated in each case as the arithmetic mean over 5 years. The 5-year average is used to filter out short-run fluctuations.
The chosen, univariate, fractional integration approach is ideal for our purposes given its ability to capture long-term dependencies and persistence in the data; it has previously been used to estimate time trends in life expectancy in sub-Saharan Africa (Caporale & Gil-Alana, 2016). Multivariate methods would instead be required to analyse the relationship between life expectancy and variables such as economic development, mortality inequality, and changing levels of economic development (Katz et al., 1983;Murray and Lopez 1996;Oeppen & Waupel, 2002;Peltzman, 2009;Preston, 1975;Roser et al., 2013;World Bank Group, 2018), which is beyond the scope of the present study.
Figures 1 and 2 display the two series of interest, namely the annual one (1842-2019) and the 5-year average (1543-2018), respectively. While negative shocks such as World War I and II (see Fig. 1) and the Anglo-Spanish Wars in the XVI and XVIII centuries (see Fig. 2) have had an impact on life expectancy, an overall upward trend is visible. This trend can be attributed to various factors, including improvements in healthcare, sanitation, immunizations, access to clean water, and better nutrition, which are highlighted in Riley (2005). Measuring active life expectancy, or the number of years a person can expect to live without disability or limitations in daily activities, as in Katz et al. (1983) can provide a more comprehensive understanding of health outcomes. The World Bank's data (2018) on global life expectancy trends further support the notion of an overall upward trend. Furthermore, Oeppen and Waupel (2002) challenge the idea that there is a biological limit to the human lifespan, suggesting that continued improvements in health outcomes and life expectancy are possible, and Murray and Lopez (1996) emphasize the need for continued efforts to improve health outcomes and reduce mortality and disability from diseases, injuries, and risk factors.
The degree of persistence of these series is examined by estimating the following regression model: where y t stands for the series of interest, β 0 and β 1 denote the intercept and the coefficient on a linear trend, respectively, and x t is the error term, which is assumed to be integrated of order d: Using a Binomial expansion, the left-hand side of (2) can be expanded, with B being the backshift operator, such that B k x t = x t-k , and u t is I(0) (as in Granger &Joyeux, 1980 andHosking, 1981), to obtain the following expression: (1) where the parameter d is a measure of persistence and sheds light on the properties of the process being modelled. Specifically, if d = 0 this exhibits short memory, whilst d > 0 implies long memory; if d < 0.5, it is covariance stationary and mean reverting; if 0.5 ≤ d < 1 it is nonstationary but mean reversion still occurs; if d ≥ 1, the process is explosive. By allowing d to be any real value, including fractional ones, this framework encompasses the classical unit root case as a special one when d = 1, and is preferable to the simple approach of estimating linear trends in Eq. (1) under the assumption that d = 0 in (2). We employ the testing procedure proposed by Robinson (1994) that is based on the Lagrange Multiplier (LM) principle and includes a version of the Whittle function in the frequency domain, where the null is the following: Note that in Eqs. (1) and (2) d can be any real number, including decimals from the nonstationary range (d ≥ 0.5), but the limit distribution of the test statistc is standard N(0, 1) (for its functional form see Gil-Alana & Robinson, 1997).

Empirical Results
Tables 1, 2, 3, 4, 5, 6, 7, and 8 report the estimated values of d alongside their 95% confidence intervals for three different model specifications: (i) without deterministic terms, (ii) with an intercept only, and (iii) with an intercept as well as a   Table 4 Estimated coefficients from the selected model in Table 3 In bold in columns 3 and 4 the t-values of the corresponding coefficients   Table 6 Estimated coefficients from the selected model in Table 5 In bold in columns 3 and 4 the t-values of the corresponding coefficients linear time trend. The coefficients in bold are those from the models selected on the basis of the statistical significane of the regressors. Tables 1, 2, 3, and 4 display the results for the original data, while Tables 5, 6, 7, and 8 show the corresponding ones for the logged transformed series, in both cases under the alternative assumptions of white noise and autocorrelated residuals; the latter are modelled using the exponential spectral approach of Bloomfield (1973) that approximates well AR structures in the context of Robinson's (1994) tests (see Gil-Alana, 2004). Concerning the raw data, with white noise residuals the time trend is significant and positive for both series (Table 2); as for d, the estimated values are 0.65 and 0.73, respectively, for the annual and 5-year average series (Table 1). Moreover, the confidence intervals do not include 1, which supports the hypothesis of mean reversion. When allowing for autocorrelated residuals (Tables 3 and 4) the time trend is again statistically significant and positive for both series, and the estimates of d are now 0.82 and 1.00 (i.e. they are much higher than in the previous case) and mean reversion (d < 1) is only found in the case of the annual series.
Concerning the logged values (Tables 5, 6, 7, and 8), the time trend is significant for both series regardless of the error term specification, and the estimated values of d are now 0.61 (annual) and 0.57 (5-years) with white noise errors, and 0.77 and 0.78 with autocorrelation, whilst the hypothesis of mean reversion cannot be rejected in any single case, which implies that the effects of shocks will gradually die away. In addition, in all cases the hypothesis of short memory or I(0) behaviour is rejected, which implies that estimating the linear trends using only the model given by Eq. (1) will produce misleading results.

Conclusions
This paper examines persistence in historical UK data on life expectancy using a fractional integration approach. The advantage of this method is that it is a very general and flexible framework encompassing a variety of cases including those when the data are nonstationary but still display mean reversion; moreover, the estimated fractional differencing parameter provides a direct estimate of the degree of persistence of the series under examination. The data analysed are annual observations for the period from 1842 to 2019 and the 5-year average from 1543 to 2018. The results indicate that the former series exhibits a statistically significant and positive time trend as well as a high degree of persistence but nevertheless it is mean reverting, and thus shocks only have transitory effects; the findings are Table 8 Estimated coefficients from the selected model in Table 7 In bold in columns 3 and 4 the t-values of the corresponding coefficients similar for the latter series, though its degree of persistence is higher. Nonstationary (though mean reverting) behaviour is also found in the case of the logged series. This evidence represents useful information for policy makers whose task is to take appropriate measures to increase life expectancy. Policy makers can implement several measures to increase life expectancy. These include improving healthcare access and quality, investing in preventive healthcare, addressing socioeconomic determinants of health, promoting healthy behaviours (focussing on mental health where necessary), fostering social connections, ensuring healthy ageing, supporting relevant research with data collection and fostering collaboration across sectors. These measures can enhance healthcare services, reduce risk factors, tackle social inequalities, promote healthy lifestyles, and create supportive environments that have a positive impact on health outcomes and overall life expectancy.
One limitation of the current study is that it focuses exclusively on the UK case. Unfortunately, extending the analysis to other countries is constrained by data availability issues; also, different countries have different data collection methods and reporting systems, which can affect the accuracy and comparability of the results. The findings for the UK are robust to using alternative methods, including both parametric (Sowell, 1992) and semiparametric (Geweke & Porter-Hudak, 1983) methods as, for instance, in Caporale et al. (2016) (these results are not reported for reasons of space). Future work could also investigate possible non-linearities by including non-linear deterministic components such as Chebyshev polynomials in time (Cuestas & Gil-Alana, 2016), Fourier functions in time (Caporale et al., 2022), or neural network approximations (Yaya et al., 2021), all within a fractional integration framework, and assess the empirical adequacy of more complex models of life expectancy (Caporale & Gil-Alana, 2016;Cutler et al., 2006;Katz et al., 1983;Peltzman, 2009;Riley, 2001;Roser et al., 2013).
While each country has its unique context and challenges, the similarities in lifestyle and development between the UK and other developed countries make the conclusions from this study relevant and applicable to a wider set of countries. Policymakers can use the insights provided by our analysis to tailor policies that are suitable for their own populations, promoting healthier lifestyles, improving healthcare systems, and ultimately increasing life expectancy. Overall, while the immediate focus of the present study is on the UK, its findings contribute more generally to the existing knowledge body of knowledge on life expectancy and provide new insights that can inform discussions and policy decisions in other regions and countries around the world.