### Data

This paper uses Eurostat data on population structure (Eurostat 2015d) and mortality records by 1-year age groups regions of EU28^{Footnote 1} for the period 2003–2012 (Eurostat 2015a). The data are aggregated at the NUTS-2 level, version of 2010 (Eurostat 2015c); NUTS means Nomenclature of Territorial Units for Statistics. At the moment of data acquisition (March 2015), mortality records covered the period up to 2012. For the majority of regions, data on population structure are available since 2003. Hence, the availability of data limited the observed study period to 2003–2012. We also used Eurostat regional projections (Eurostat 2015b) for three more decades, 2013–2042.

For some regions, data were partially missing. Due to the changes in administrative division at the NUTS-2 level, there were no data for all five regions of Denmark before 2007 (Kashnitsky 2017) and two regions in the eastern part of German, Chemnitz (DED4) and Leipzig (DED5) before 2006. Furthermore, mortality data were missing for Ireland in 2012, and population structure data were missing for Slovenia in 2003–2004. We reconstructed the missings using the data from national statistical offices.

Exploratory data analysis showed inconsistency of population estimates for the regions of Romania. There was a census in Romania in 2011 that registered a large, and previously underestimated, decrease in population size. Evidently, the outmigration from Romania was underreported. Yet no rollback corrections were made, and Eurostat provides non-harmonized data for Romanian regions. Thus, we harmonized the population figures for Romanian regions.^{Footnote 2}

Finally, we excluded all non-European remote territories of France, Portugal, and Spain,^{Footnote 3} which are outliers both in geographical and statistical terms.

The data set used for the analyses contains data for 263 NUTS-2 for the observed (2003–2012) and projected (2013–2042) periods.

### Measuring aging

We measure population aging as a decrease in the ratio of the working-age population to the non-working-age population. In line with Eurostat and UN definitions, we consider ages 15 and 65 as the margins of the working-age population. Thus, the measure of aging that we use is the ratio of population aged 15–64 to the population below 15 years of age and above 65. We call this indicator the total support ratio (TSR), which is in fact the inverse of the widely used total dependency ratio (UN Population Division 2002). There is some confusion around the use of the term support ratio in the literature. Quite often, children are not included in the calculation of the support ratio (Lutz 2006; Lutz et al. 2003; O’Neill et al. 2001). In that case, the indicator only shows the relative burden of the elderly population; UN Population Division (UN Population Division 2002) calls this indicator potential support ratio. In other papers, that deal not only with age structures of population but also with labor force participation and transfer accounts, by support ratio, authors usually mean the ratio of effective labor to effective consumers (Cutler et al. 1990; Lee and Mason 2010; Prskawetz and Sambt 2014). Another definition says that the support ratio is the size of the labor force as a share of the adult population (Börsch-Supan 2003). We prefer to explicitly call the ratio of the working-age to the non-working-age population the total support ratio, in line with the logic of the three versions of dependency ratio: total, youth, and old-age.

### Decomposition of growth in the total support ratio

To explain which demographic factors cause changes in the TSR, we apply a two-step decomposition. First, we examine to what extent changes in the TSR are due to changes in the size of the working-age population and to what extent to changes in the size of the non-working-age population. Second, we examine the demographic causes of changes in the working-age population.

At the *first step*, the overall change in the TSR is decomposed using the formula of Das Gupta (Das Gupta 1991):

$$ {\mathrm{TSR}}_2 - {\mathrm{TSR}}_1 = \frac{W_2}{{\mathrm{NW}}_2}-\frac{W_1}{{\mathrm{NW}}_1} = \left[\frac{1}{2}*\left({W}_2+{W}_1\right)*\left(\frac{1}{{\mathrm{NW}}_2}-\frac{1}{{\mathrm{NW}}_1}\right)\right]+\left[\frac{1}{2}*\left(\frac{1}{{\mathrm{NW}}_2}+\frac{1}{{\mathrm{NW}}_1}\right)*\left({W}_2-{W}_1\right)\right] $$

(1)

where \( W \) is the working-age population, \( \mathrm{N}\mathrm{W} \) is the non-working-age population, and subscripts \( 1 \) and \( 2 \) denote the beginning and the end of the period, respectively. The two right-hand side terms of Eq. 1 represent the effects of changes in non-working-age and working-age populations on the TSR, respectively. Note that changes in \( W \) affect both the first and second terms, but the effect on the first term is very small compared with that on the second term. The average change in the first term due to the changes in the working-age population over all 263 regions was only −0.7% with a standard deviation of 3.3%.

At the *second step*, the working-age term in the second term of the right-hand side of Eq. 1 is decomposed further into changes due to the three components of the demographic balance at working ages: cohort turnover, migration, and mortality.

To estimate the components of change in working-age population, we use the demographic balance formula:

$$ {W}_2={W}_1+\mathrm{C}\mathrm{T}+{M}_W-{D}_W $$

(2)

where \( \mathrm{C}\mathrm{T} \) is the cohort turnover between periods 1 and 2, \( {M}_W \) is the net migration at working ages, and \( {D}_W \) is the number of deaths at working ages. As the accuracy of migration records is always a problematic issue, following De Beer, Erf, and Huisman (2012), we derive net migration at working ages indirectly from Eq. 2 for the observed period, 2003–2012. For the projected period, 2013–2042, the migration data are provided by Eurostat, so we derive the numbers of deaths using the demographic balance formula. Cohort turnover is calculated as the difference between people entering working ages, aged 14, and people leaving working ages, aged 64.

Replacing the \( {W}_2-{W}_1 \) part of the working-age term in Eq. 1 using the demographic balance formula, Eq. 2, yields

$$ \frac{1}{2}*\left(\frac{1}{{\mathrm{NW}}_2}+\frac{1}{{\mathrm{NW}}_1}\right)*\left({W}_2-{W}_1\right)=\left[\frac{1}{2}*\left(\frac{1}{{\mathrm{NW}}_2}+\frac{1}{{\mathrm{NW}}_1}\right)*\mathrm{CT}\right] + \left[\frac{1}{2}*\left(\frac{1}{{\mathrm{NW}}_2}+\frac{1}{{\mathrm{NW}}_1}\right)*{M}_W\right]-\left[\frac{1}{2}*\left(\frac{1}{{\mathrm{NW}}_2}+\frac{1}{{\mathrm{NW}}_1}\right)*{D}_W\right] $$

(3)

The three right-hand side terms of Eq. 3 denote the effects of cohort turnover, migration at working ages, and mortality at working ages on TSR, respectively.

### Beta-convergence aproach to aging

To estimate beta convergence, we use the classical linear regression model specification, where change in a variable (in our case, total support ratio) over some period is regressed on the initial level. The specification looks as follows

$$ {\mathrm{TSR}}_2-{\mathrm{TSR}}_1 = \alpha + \beta\ {\mathrm{TSR}}_1 + \varepsilon $$

(4)

where \( T S R \) is the total support ratio, \( \alpha \) is the intercept of the regression line, \( \beta \) is the regression coefficient, and \( \varepsilon \) is the error term. If the regression coefficient is negative, then beta convergence is observed between years 1 and 2, meaning that the change in TSR is negatively correlated with the initial level of the TSR. Thus, beta convergence implies that a region with a relatively high TSR experiences less growth in the TSR than a region with a low TSR.

In convergence analysis, weights reflecting population sizes are often used (Dorius 2008; Goesling and Firebaugh 2004; Milanovic 2005; Theil 1989). Population-weighted convergence analysis shows whether inequality in the population becomes smaller; unit-weighted (in fact, non-weighted, as all units receive equal weights) convergence analysis tests whether the differences between units (countries/regions/districts) decrease. In this study, we are interested in the development of European regions as statistical units; thus, we choose the unit-weighted convergence analysis. Our choice is driven by the fact that European cohesion policy is aimed at regions, irrespective of their population sizes.^{Footnote 4}

The specification of the regression model allows to perform a decomposition of convergence (the beta coefficient) into various separate effects. To understand how each of the demographic factors contributed to beta convergence in aging, we decompose the dependent variable, the change in TSR (see the previous subsection), and run separate regressions for each partial change in TSR keeping the explanatory variable, the initial value of TSR, constant. A partial regression model shows the beta convergence of regions taking into account only the change in TSR due to the component under consideration. As the components of change in TSR add up to total change, and all the partial models have the same regressor, beta coefficients of the partial models add up to the total effect. That means, beta coefficients from convergence models for the change in TSR due to the dynamics of non-working-age population (\( \mathrm{n}\mathrm{w} \)) and working-age population (\( w \)) add up to the beta coefficient of the overall model (\( g \)), and beta coefficients from the models for cohort turnover (\( \mathrm{c}\mathrm{t} \)), migration at working ages (\( \mathrm{mg} \)), and mortality at working ages (\( \mathrm{m}\mathrm{t} \)) effects on TSR growth add up to beta coefficient from the model for the working-age population dynamics’ effect. For the ease of notation, we will refer to the partial model using the above symbols in brackets.

To use further the additive feature of the models, we ran a separate regression for each partial change in TSR in each year, dividing the study period into four decades—for each of the decades, the initial TSR distribution is used as an explanatory variable. The temporal decomposition gives insight into how the convergence process evolves throughout the study period. Summing up, in this paper, we use two dimensions of the decomposition of convergence in aging: demographic factors of the change in the TSR and time.

### Software

The analysis and the necessary data preparation were conducted using *R*, a language and environment for statistical computing, version 3.3.2 (R Core Team 2016). The crucial additional packages include *dplyr* (Wickham and Francois 2015), *tidyr* (Wickham 2016b), *ggplot2* (Wickham 2016a), *viridis* (Garnier 2016), and *rgdal* (Bivand et al. 2015). All the scripts are in the attachment for reproducibility.