Population Development of the Rhine-Neckar Metropolitan Area: A Stochastic Population Forecast on the Basis of Functional Data Analysis Philipp Deschermeier Abstract: Stochastic population forecasts are gaining popularity in these times of demographic change, as compared with the scenario technique frequently used for projections, they provide important additional information: the forecasted popula- tion lies within a prediction interval to which a probability of occurrence can be al- located. However, this approach requires long time-series and detailed information about the determinants of population development (fertility, mortality and net mi- gration), which are frequently not available in suffi cient depth at the regional level, but are generally subsumed into age groups. Stochastic population forecasts are therefore usually limited to the national level. Nonetheless, methods of functional data analysis enable us to disaggregate the required demographic variables into years of age and to use them as the data basis of a stochastic model, also at the regional level. This essay presents this approach and models based on it using the example of the population development of the Rhine-Neckar metropolitan area in Germany. Keywords: Rhine-Neckar metropolitan area · Stochastic population forecasts · Functional data analysis 1 Introduction Demographic change is dramatically transforming the German society: the ratio of young to old and of the gainfully employed to pensioners is shifting in favour of older persons. Both public debate and academic research in Europe are largely focused on the socio-political consequences of demographic change. In particular, the focus is on the problem of fi nancing the statutory pension insurance, which led to the negative sound of the term “aging population”. Frequently, the positive aspect of demographic change is forgotten: people have a higher life expectancy and a longer active lifespan (Schnabel et al. 2005: 3). Comparative Population Studies – Zeitschrift für Bevölkerungswissenschaft Vol. 36, 4 (2011): 769-806 (Date of release: 01.11.2012) © Federal Institute for Population Research 2012 URL: www.comparativepopulationstudies.de DOI: 10.4232/10.CPoS-2011-21en URN: urn:nbn:de:bib-cpos-2011-21en4 • Philipp Deschermeier770 Rising life expectancy, particularly the above-average drop in the mortality of older people, along with continuous low fertility, brings about far-reaching changes in the age structure. Although at the national level, only a slight decrease of the population is forecasted in the coming 20 years (from 81 million to approx. 77 to 79 million inhabitants), the number of people of working age will drop considerably (Statistisches Bundesamt 2009; Fuchs/Dörfl er 2005a/b; Börsch-Supan/Wilke 2009). Hence, in the coming decades, society will be faced with a substantial macro-eco- nomic structural transformation affecting all of the important markets: the labour market will lack young workers, the product markets will have to adjust to structur- ally changed consumer demands and on the capital market, savings behaviour and the demand for productive investments will be changing (Börsch-Supan 2007). At the regional level, numbers and structure of future population development diverge to a high degree due to different economic conditions, so that the consequences for demand of goods, the housing market and the supply of workers deviating from region to region will result in a growing need for regional population forecasts as quantitative decision-making bases for regional planning. In particular, stochastic models using methods from time series analysis are becoming increasingly popular. However, these approaches require very long time series for various demographic variables, such as mortality and information on age and gender-differentiated struc- ture, which are frequently not available at the regional level. Therefore, stochastic models can usually only be conducted at the national level. Furthermore, the data required for a population forecast are frequently processed differently by the differ- ent statistical offi ces of the federal states. For instance, the structure of age-specifi c variables of the federal states deviates, making population development diffi cult to model, especially for polycentric metropolitan areas which are located across federal state borders. This article provides a stochastic forecast of the population of the Rhine-Neckar metropolitan area broken down by years of age (up to 90+ years) and gender from 2010 until 2030. This region is situated at the intersection of the three federal states of Baden-Wuerttemberg, Hesse and Rhineland-Palatinate and is a growing eco- nomic region. On the one hand, its socio-cultural connections reach far back into the past, when the region formed a territorial unit called the “Electoral Palatinate”. On the other hand, since the 1970s economic ties have developed across federal state borders, which are associated with the initiative of the “Rhine-Neckar Triangle” (Lowack 2007: 132). The three centrally located regional metropolises of Mannheim, Heidelberg and Ludwigshafen shape the polycentric structure of the region, which, with about 2.4 million inhabitants, is Germany’s seventh-largest economic region. The area stretches across 15 rural and urban districts. Metropolitan areas are con- sidered “engines of economic development” (translated by CPoS, Bundesamt für Bauwesen, Raumordnung und Städtebau 1995), from which stimuli should emanate to structurally weaker regions. However, due to the future decline in population, they compete with one another, particularly for highly skilled workers (Klein 2008: 44). This “war for talent” (translated by CPoS, Chambers et al. 1998: 1) illustrates the signifi cance of regional population forecasts that help to identify future bottle- necks early and enable a suitable response, for example through educational and Population Development of the Rhine-Neckar Metropolitan Area • 771 advanced training programmes (Gans et al. 2009: 118). The Rhine-Neckar metropol- itan area is integrating the future challenges in the regional strategy “Demographic Change” via a network of regional decision makers, the “Lenkungskreis Demogra- phie” (translated by CPoS, Metropolregion Rhein-Neckar 2011: 26). The population forecast presented in the following paper uses functional data for an in-depth analysis of the demographic development of the Rhine-Neckar metro- politan area. This approach is based on the idea that the secondary statistical data basis of the demographic variables individually aggregated to age groups follows a measurable functional context. The objective of this study is to estimate these and utilise them for forecasts. In this way, the differently structured and broken down data sets of the detailed demographic variables of the statistical offi ces of the federal states of Baden-Wuerttemberg, Hesse and Rhineland-Palatinate can be smoothed to one study region and aggregated to a total region. Time series models are based on functional data analysis (Chapter 3), which reproduce the uncertainty of demographic development through confi dence intervals more precisely than the frequently used scenario technique. Chapter 4 illustrates the advantages of this ap- proach with the example of the Rhine-Neckar metropolitan area, and Chapter 5 con- cludes with an outlook. 2 Population forecasts It is diffi cult to foresee the demographic future of a region, as there are very many possible courses it could take. The literature differentiates two methodological ap- proaches for dealing with this challenge. The traditional approaches are determin- istic, and although they are frequently used by offi cial statisticians, they portray the development of the population without precise information about the uncertainty of future developments (Lipps/Betz 2004: 1). One possibility for this is the use of math- ematical extrapolation methods, which analyse the trends of the past and update them using growth functions. The use of growth rates might be suitable for large- scale projections such as that of the global population (O’Neill et al. 2001: 207). It is, however, less suitable for modelling at the regional level, where development is marked by a greater dynamic, particularly through migration movements. In order to take these aspects into consideration, conventional deterministic methods use the basic demographic equation as a basis (Bähr et al. 1992: 327): B t = B t-1 + G t-1,t - S t-1,t + M t-1,t whereby B t = population at point in time t; B t-1 = population at point in time t-1; G t-1,t = births, between t-1 and t; S t-1,t = deaths, between t-1 and t, as well as M t-1,t = net migration between t-1 and t. • Philipp Deschermeier772 A possible course of the future population is calculated from an initial popula- tion and assumptions about the demographic determinants (fertility, mortality and net migration) (Statistisches Bundesamt 2009: 9). The basic demographic equation differentiates various points in time, but views the population only in the aggrega- tion. In order to make the diverse changes in demographic change visible, the basic equation must be differentiated according to age (x = 0, 1, 2, …, k-1, k) and gender (M=men and F=women) of the population. The births and deaths of the individual age groups result from the product of the underlying demographic rates with the population of the previous year. The demographic development of a region is there- fore determined by fi ve components: the fertility rate, the male and female mortal- ity rate as well as the gender-differentiated net migration. The cohort component method formalises female population development (index “F”) into a matrix model (Lee/Tuljapurkar 1994: 1178): respectively (Pfl aumer 1988: 136): The (female) population of a particular year of age x of the ensuing period t+1 is calculated from the sum product of a line of matrix Ω (mortality rate m x,t or fertility rates f x,t ) with the column vector of the population at point in time t. Parameter v quantifi es the percentage of newborn females and αx,t the probability of survival, de- fi ned as (1-m x,t ). This product supplies a vector for natural population development. The number of newborns is calculated from the sum of women of childbearing age multiplied with the fertility rates. This value is distributed over the two genders and then multiplied by the probability of surviving the fi rst six months of life (α *,t ). The population at time t+1 is the result of adding the vector for natural population devel- opment to the vector of net migration. Since the cohort component method examines the individual components of the basic demographic equation separately and differentiates cohorts (usually years of age), it can precisely indicate those infl uences that affect only certain age ranges. This approach was used as early as the 1920s by Whelpton (1928) to forecast the population of the United States from 1925 until 1975 (Bähr et al. 1992: 500). Today, this is the standard method for models of population development and is used by the statistical offi ces (for example, Statistisches Bundesamt 2009). In order to take . , Population Development of the Rhine-Neckar Metropolitan Area • 773 the uncertainty of the demographic future into account, different progressions are calculated using different assumptions about the demographic rates. These scenar- ios (Lutz et al. 1998a: 140) usually include an optimistic, a neutral and a pessimistic trend. When interpreting these scenarios, a frequent error is to interpret the range of variation between the pessimistic and optimistic progression as a measure of un- certainty (Keyfi tz 1972: 353). Combining different assumptions for regional forecasts seems unsuitable, since future migration movements in particular have a major im- pact on the population structure, yet are affl icted with very great uncertainty.1 Although the offi cial bodies have been using the deterministic approach for years, it is not unproblematic (Lee 1999; Lipps/Betz 2003; Keilman et al. 2002). With the exception of the range of possible developments, the deterministic approach cannot say anything about the occurrence of the different scenarios. Stochastic approaches, by contrast, enable forecasts that cite the probability with which the result lies within a certain range of variation. This is valuable additional informa- tion, for example for policy consulting. Regional planners would like to know, for instance, which of the scenarios is most likely to occur, in order to avoid “unpleas- ant surprises” and to draw up more concrete plans. Furthermore, the assumption of fi xed demographic components contradicts the use of scenarios. For example, the number of live births of a woman is frequently set at 1.4 in Germany, whereby the future values are completely determined by the previous periods. Lipps and Betz (2003: 4-5) use a simple arithmetic example to show that the development of a de- mographic component assumed as fi xed (perfect autocorrelation) with a time frame of 20 years deviates considerably from a forecast based on time series models, in spite of marginal differences in the individual periods. Furthermore, perfect auto- correlation between the demographic variables also results from the combination of the assumptions made. A possible scenario of “high population growth” arises from the combination of high fertility and low mortality or high life expectancy. In this scenario, a high birth rate implies high life expectancy (Keilman et al. 2002: 410). This autocorrelation between the demographic variables produces inconsistencies, as extreme assumptions about one variable need not necessarily result in extreme assumptions about another variable. A population forecast should therefore always demonstrate two components (Keilman et al. 2002: 410): 1. a range of possible development and 2. an a probability of occurrence for this range. In addition, a forecast should be able to take four types of correlation into ac- count: over time, between the different age groups, between the genders and be- tween the demographic components.2 However, a possible correlation between 1 In addition, interested readers can fi nd a detailed portrayal and discussion of various approach- es for regional population forecasts in Rogers (1985). 2 Keilman et al. (2002: 412-414) contains a detailed discussion of the different types of correla- tion. • Philipp Deschermeier774 the different demographic rates plays merely a minor role in developed countries (Keilman et al. 2002: 412). Only the stochastic approach can fulfi l these demands on a forecast, because the future values lie within a confi dence interval that serves as the measure of the anticipated precision. Three approaches can be distinguished (Lipps/Betz 2003: 5): analysis of historical forecasting errors• assumptions by expert groups• time series models• It was Keyfi tz (1981) and Stoto (1983) who began analysing historical forecasting errors. By comparing earlier forecasts with actual developments, forecasting errors can be derived, from which a forecast interval can be determined for the future (Keilman et al. 2002: 415). However, this approach only supplies viable results, if the future deviations of demographic rates are similar to previous ones and if misinter- pretations also apply to the future (Lipps/Betz 2003: 5). In addition, frequently only a few periods are available from which the error and the construction of the forecast intervals can be derived, which is why the quality of the approaches appears ques- tionable. Furthermore, early forecasts are based on methods that frequently are no longer state of the art and therefore must be classifi ed as less precise than forecasts made using more recent methods. At the regional level, this approach is usually not implementable or diffi cult to implement due to the lack of forecasts. Lutz et al. (1996, 1998a/b) use the assumptions of expert groups to project the demographic rates and their uncertainty (Lutz et al. 1996). An expert group agrees both on a point estimate and on a confi dence interval. In the next step, a projection of population development with a range of variation is calculated using a simulation under a distribution assumption for the demographic rates and assumptions about their correlation. Keilman et al. (2002: 415) speak against the use of this approach citing that even experts cannot reliably differentiate between a 95 % and a 99 % confi dence interval. In addition, the risk of a serial correlation is high, which leads to inconsistent confi dence intervals. Lee (1999: 172) discusses that there is no objec- tive way to derive the uncertainty about future demographic developments from expert opinions. Furthermore, suitable expert groups may not be available at the regional level, which may lead to this approach being impossible. Time series models are based on the assumption that the demographic develop- ments of the past can be explained by using a statistical model and that this context will also be valid in future. Probably, the most well known time series model is that of Lee and Carter (1992) designed for modelling mortality. The fi rst step of this two-step approach for estimating future trends of age and gender-specifi c mor- tality rates matches the model to the time series of mortality rates and then, in a second step, transfers their development into the future. Since the approach is easy to implement and has proven fl exible, it has been frequently adapted and devel- oped further (Booth 2006). Hyndman and Booth (2008: 324) provide an overview of other approaches for modelling demographic rates. Time series models are suitable only for short and medium-term forecasts (Keilman et al. 2002: 414). A time span Population Development of the Rhine-Neckar Metropolitan Area • 775 that is too long, results in unrealistic forecasts and very broad confi dence intervals (Sanderson 1995: 274). The main advantages of time series models over the two stochastic alternatives for a regional stochastic population forecast are that the un- certainty can be illustrated consistently and requires neither earlier forecasts nor expert groups. Since a lot of data are frequently only available for aggregated age groups at the regional level, the paradigm of functional data and models based on them appear helpful for a regional population forecast.3 This approach enables an effective disaggregation of secondary data. Based on this, the models by Hyndman and Ullah (2007) as well as Hyndman et al. (2011) demonstrate smaller forecast intervals than alternative approaches (Hyndman/Ul- lah 2007: 4953; Hyndman et al. 2011: 25). Therefore, these models are the basis for the stochastic population forecast of the Rhine-Neckar metropolitan area. 3 The paradigm of functional data 3.1 Basic idea An in-depth analysis of the impacts of demographic change on the population struc- ture requires demographic variables broken down by years of age. When the data situation is not optimal, the use of functional data proves quite helpful, as particu- larly at the regional level, secondary data are frequently only available differen- tiated by age groups. The analysis of functional data (“functional data analysis,” subsequently referred to as FDA) is an approach for using statistical series, which joins a curve of single data points that represent a connected series (Ramsay/Silver- man 2001: 5822). The objective of the FDA is to estimate this function and use it as a basis for forecasts. In doing so, each secondary statistical observation Y t (x) of a demographic component serves as a knot for a smooth function s t (x). Hence, for a horizon of N points in time, there are N curves. Each observation is subject to a measurement error ε t (x) resulting in the following model (Ramsay 2008: 5): Y t (x) = s t (x)+ε t (x) In order to acquire the necessary information about the smooth functions s t (x) for the FDA, the individual demographic parameters must be disaggregated with their information for age groups to years of age. Statistical smoothing is an ap- proach that attempts to estimate considerable but unavailable patterns of a data set and thereby eliminate any possible noise. According to Wood (1994: 27), cubic spline functions deliver the best results for demographic data. The medians of the individual age groups form the supporting points or nodes, of which the fi rst is lo- cated at the beginning of the fi rst age group (age=0), the next node at the median 3 Interested readers can fi nd a detailed discussion of the various methodological approaches in O’Neill et al. (2001: 210-222). • Philipp Deschermeier776 of the second group and so forth. The fi nal node is located at the end of the upper age group. The years of age comprising an age group form a section. The smooth- ing method joins the nodes through different cubic polynomials for the individual sections to a curve, which is always differentiable for all points. The course of the spline functions allows us to read a value for the modelled demographic component for each year of age. Figure 1 illustrates the procedure using the example of a fertil- ity rate: the median values of the individual age groups form the nodes (squares), between which cubic splines run, which compose the smoothed curve. Functional data, or the individual smooth functions for the N points in time, un- derlie two variations: amplitude variation and phase variation (Ramsay 2008: 3). Amplitude variation is a vertical variation of the functions over time. Hence, the “amount” of the function values is not constant. Phase variation indicates the hori- zontal variation that, for example, the age-specifi c characteristics of the demo- graphic parameters show a specifi c trend over N points in time. For instance, in the past decades the age of women who reached the maximum of the age-specifi c birth rate rose continuously. These variations in the curves complicate modelling with time series models, as by smoothing data to years of age, models with very Fig. 1: Smoothing secondary data using the example of the age-specifi c fertility rates of one year 0 10 20 30 40 50 60 70 80 90 100 10 15 20 25 30 35 40 45 50 Age Smoothed curve Secondary data Fertility rate per 1,000 women in the respective age group Source: own design Population Development of the Rhine-Neckar Metropolitan Area • 777 many parameters would be needed to portray all of the developments in age and time adequately. For this reason, the FDA uses the so-called “time-warping” ap- proach to illustrate the characteristic variables of a function with a low number of parameters and to minimise their variation through amplitude variation and phase variation (Ramsay/Silverman 2001: 5823): A linear combination of basis functions φ k (k=1, …, K) corrects the biases: The basis functions contain information about the age and time-specifi c develop- ments of the smooth function. The more basis functions are used, the more fl exible a time series model becomes. The big challenge is keeping the function simple and the number of parameters used low. For the more basis functions are used in calcu- lating the smooth function, the more sensitively the function reacts to measurement errors. This approach is widespread, since the basis functions can be simply esti- mated by using a principal component analysis, which describes the development of all of the main variables of the curves over the course of time. This alignment of functions over the time frame, also called registration, facilitates further assess- ments and improves comparability between different points in time. The remaining variation of the curves is merely a result of pure amplitude variation. 3.2 Models Hyndman and Ullah (2007) use functional data to develop a model that can reliably indicate different demographic parameters. For instance, Y t (x) describes the value of a demographic rate for age x at time t. Based on the basic idea of FDA, the model is based on a smooth function s t (x), which is observed with one error to a discrete point in time depending on the age. For the (secondary statistical) data points {x i , y t (x i )}, whereby t = 1, …, n and i = 1, …, “highest age group”, therefore: Y t (x) = s t (x)+ σ t (x)ε t (x). The error terms ε t,x are independently and identically distributed and weighted by the term σ t depending on the age. This is advisable, because for small function values in certain age ranges, for example the fertility rate for women under 15 and over 45 years of age, the measurement errors are also correspondingly small. Not weighting them could lead to the impression that the model in these sections indi- cates the data better than in other age ranges. The modelling aims to forecast Y t (x) for the range of h points in time (t = n+1, …, n+h) and all defi ned age groups. This approach is based on functional data and attempts to identify and record the ampli- tude variation and the phase variation as sources of measurement errors: 1. For a population forecast, data are desirable that are broken down by years of age, for this is the only way, for example, to make the diverse effects of demo- graphic change on the age structure of the population visible. However, sec- ondary statistical data are frequently aggregated to age groups. In this case, . • Philipp Deschermeier778 the age structure is smoothed using non-parametric smoothing methods for each available point in time, in order to be able to estimate the function that Y t is based upon. A separate function is estimated for each point in time. 2. Modelling all main characteristic attributes of a demographic variable requires a number of parameters. The lengthy time series data needed are rarely avail- able, particularly at the regional level, due to regional reforms and other cir- cumstances. Instead, basis functions break down the functions estimated in step one into individual elements containing all of the main variables, which, however, have to do with a lesser number of parameters: whereby μ(x) is the mean of s t (x) over all observed years, {φ k (x)} is a set of basis functions and e t (x) is a normally distributed error term with mean 0 and variance var(x). The basis functions are the result of a principle compo- nent analysis. The mean shows the essential structure of the modelled demo- graphic variable, while the developments deviating in age and time from the mean through the amplitude and phase variation are contained in the basis functions. 3. A univariate time series model is estimated for every coeffi cient β t,k with k = 1, …, K. Hyndman and Booth (2008: 327) show that the method reacts insensitively to the choice of K, if based on a suffi ciently high value. For this reason, it is advisable to form as many basis functions as possible. However, this increases the duration of the model calculation, and the basis functions with a low explanatory content are in some cases diffi cult or impossible to interpret. 4. The variation caused by the amplitude and phase variation is subject to chron- ological deviations. The parameter β t,k indicates these developments and is forecasted for a time frame of h points in time: whereby marks the estimator at time n+h for . 5. The estimated and forecasted values are the basis for the calculation of future values of s t (x) from step two, as the forecasts for all age groups are calculated from the point estimators multiplied by the estimated basis functions. 6. The forecast intervals are calculated from the estimated and forecasted vari- ances of the error terms (step two). These confi dence intervals are valuable additional information compared with deterministic approaches, as rath- er than indicating the range of variation of possible developments simply through a variation of scenarios with different assumptions, this approach is based upon statistical methods and distributions. The advantage of this approach is that it allows for smooth functions, is robust against structural breaks, such as crises, wars or disasters, and supplies a model , , Population Development of the Rhine-Neckar Metropolitan Area • 779 framework that enables restrictions,4 yet is simultaneously useable for different (demographic) models. For instance, Erbas et al. (2007) forecast the risk of breast cancer in Australian women. The model is a generalisation of the groundbreaking Lee-Carter model (Lee/Carter 1992) for estimating and forecasting mortality rates, which in many enhancements can also be used for other demographic variables, for example fertility data (Lee 1993). The basic idea of the Lee-Carter model is also refl ected in the FDA models, however it differs in some points, for example in that Lee and Carter did not require any smooth functions (Hyndman/Ullah 2007: 326). The approach by Hyndman and Ullah (2007) estimates the subpopulations of the modelled demographic variable (frequently broken down by male and female) independently of one another. In this case, the forecasted development of both fi g- ures can diverge in a lengthy time frame. However, this effect can usually not be explained or founded, or only with diffi culty. Wilson (2001: 167) observes a global convergence for the mortality rates both between different countries and between the sexes. Approaches that take up and explicitly model this aspect are called “co- herent models” (Li/Lee 2005: 581) in the literature. Forecasts based on this way of thinking should demonstrate a certain structural correlation. For mortality rates, this was implemented through methodological enhancements of the classical Lee- Carter approach, for example by Lee and Nault (1993), Lee (2000) as well as Li and Lee (2005). Hyndman et al. (2011) used functional data and enhance the approach of Hyndman and Ullah (2007) in order to avoid a diverged development as the result of imprecise estimations. Since differentiation is made in the stochastic population forecast only according to gender, the following remarks are limited to the case of two subpopulations. The approach also proves very fl exible and can be expanded to a larger number of subpopulations. The secondary statistical values Y t,F (x) of a demographic component are based on a smooth function s t,F (x), which is observed with an error. Among two subpopu- lations, the following applies to the women (subscript F): whereby x i is the mean of the age group i (i = 1, …, p), ε t (x i ) is an independent and identically distributed random variable and σ t,F is a term through which the measure- ment errors vary in age. With the exception of the logarithmic formulation, this ap- proach corresponds to the described approach by Hyndman and Ullah (2007). The smooth function is observed for both men and women. Modelling a coherent de- velopment requires the following two terms: the geometric average (P t for product 4 The approach allows for qualitative restrictions; for example the assumption that the probabil- ity of death rises monotonically from a certain age can be integrated. The restrictions employed for this study are contained in Chapter 4.2. , • Philipp Deschermeier780 model) and the square root from the ratio (R t for ratio model), each for the smoothed fi gures of a demographic component s t,F / M (x): The product model describes the age-specifi c progression of a demographic variable, while the ratio model adapts the gender-specifi c variation. There are hard- ly any corrections at values close to one of R t , while values deviating from one indicate gender-specifi c deviations in a specifi c age range. The advantage of this approach is that through the logarithmic formulation, the product in P t becomes a sum or a difference from the ratio in R t . Tukey (1977) showed that both are practi- cally uncorrelated. The product model and the ratio model also underlie variations through the am- plitude variation and the phase variation. To prevent a diverged development of the demographic rate and also indicate its age and gender-specifi c change, P t and R t are portrayed by the basis functions model for functional data by Hyndman and Ullah (2007): The basis functions φ k (x) and ψ l (x) contain the variables of P t or R t deviating from the mean (μ p and μ r ). The change of the basis functions over time is shown by the coeffi cients β t,k and γ t,l . The model errors measure the error terms e t (x) and w t (x). To ensure that the estimators for the forecast are coherent, or do not diverge, the coeffi cients of the basis functions {β t,k } and {γ t,l } for the time series models used must be stationary processes. The forecasted estimators are multiplied by the basis functions. The results are forecasts until time n+h for future P t (x) and R t (x) from which the future gender-specifi c values for the modelled demographic component result: The forecast of βn,k,h and γn,l,h for point in time n+h for the individual genders (sub- script j marks “male” or “female”) is determined by:5 . . . 5 The exact procedure for forecasting the individual coeffi cients is found in Hyndman et al. (2011: 6-7). . Population Development of the Rhine-Neckar Metropolitan Area • 781 The presented models based on functional data are suitable for short- and medi- um-term forecasts of approx. 20 years (Hyndman/Booth 2008: 339). For a longer time frame, the accuracy decreases and the width of the confi dence intervals increases. The forecasts of the demographic variables are based on information about the development in the past, which is why they cannot indicate a future trend reversal. However, both are basically true for the use of time series models with no infl uence on the choice of model for the forecast of the population. In a direct comparison to alternative specifi cations, both approaches show greater accuracy (cf. Hyndman/ Ullah 2007: 4953 and Hyndman et al. 2011: 17). The gender-differentiated demo- graphic variables (net migration and mortality) are modelled using the approach by Hyndman et al. (2011), while the model by Hyndman and Ullah (2007) is assumed for the fertility rate. Both approaches are particularly suitable for regional population forecasts, as through the use of basis functions the results are very robust against outliers and structural breaks. This is an advantage since at the regional level, for example because of regional reforms, long time series are frequently not available. In addition, social developments such as the possible trend reversal from subur- banisation to reurbanisation can be indicated at a correspondingly lower aggrega- tion level through the basis functions. Furthermore, many variables offered by the statistical offi ces of the federal states are only broken down by age groups. Using the ideas of the paradigm of functional data, this information can be disaggregated to single years of age. The basis functions also allow us to model demographic trends and structural breaks, which are manifested in the form of phase variation or amplitude variation in the time series. Whether their content can be interpreted or serve merely to correct statistical effects depends on their explanatory content. 4 The population development of the Rhine-Neckar metropolitan area 4.1 Algorithm Five demographic components infl uence population development: the fertility rate, the gender-specifi c mortality rates as well as male and female net migration. The cohort component method formalises this correlation to a system of equations, however, without precisely specifying the processes within one year. This popula- tion forecast is based on the algorithm by Hyndman and Booth (2008: 340), which is diagrammatically portrayed below (Fig. 2). It begins with the population on 31 December of the previous year. The migration movements do not occur completely at one fi xed point in time, but distributed throughout the year. During this time, im- migrants can theoretically have children or die. The algorithm takes this aspect into consideration by distributing the number of simulated age-specifi c net migration changes, half at 1 January and half at 31 December of one year. The population on 30 June of a year corresponds to the population on 31 December of the previ- ous year plus the fi rst half of the net migration. The deaths are calculated from the population on 30 June multiplied by the mortality rates. The number of live births is calculated along these lines, corrected by the infant mortality rate. The newborns • Philipp Deschermeier782 are distributed to the genders on the basis of the moving average of the past fi ve years. The female population at time t+1 is calculated: The population at the end of the year (31 December) is determined through the sum of the population on 30 June of a year, the deaths, the adjusted live births and the second half of the net migration. This population is then used as the starting point for calculating the subsequent year, for which the algorithm is passed through from the beginning. Unlike deterministic models, which project a single popula- tion development per scenario each, due to the uncertainty about future develop- ment, stochastic approaches require more complicated simulations, since each of the forecasted values are dispersed about one mean each. On the basis of the as- sumed distributions for the individual demographic rates, a predetermined number of possible trajectories of the future population are simulated and stored in a data- base. The confi dence interval, in which the future population lies, ranges here from (100-α) / 2nd percentile to (100+α) / 2nd percentile (Keilman et al. 2002: 416). Separate simulations of the models from Section 3.2 for the fertility rate, the male and female mortality rates as well as the gender-specifi c net migration for each year from 2010 until 2030 form the basis for the population forecast. The fer- tility and mortality rates refer to the population on 30 June (cf. Fig. 2). The births and deaths follow the assumption of the Poisson distribution and are calculated by random drawing from the distribution and summation of these values over the relevant age range. Hyndman and Booth (2008: 328) provide a detailed overview of the generation of the respective trajectories and their integration into the population forecast. . Fig. 2: Algorithm for the cohort component method Source: own design according to Hyndman and Booth (2008) Population Development of the Rhine-Neckar Metropolitan Area • 783 All of the following calculations were carried out using the statistical program “R” (Version 2.12.0). This free statistical software is available on all prevalent platforms, and the add-on package “Demography” also contains the described approaches for functional data as well as the presented algorithm for the cohort component meth- od. The forecasts require the “Forecast” package. A total of 1,000 progressions of population development were simulated and then exported into an Excel database to calculate the forecast intervals from “R.” 4.2 Data: Sources and preparation The data used for the population forecast were provided by the statistical offi ces of the federal states Baden-Wuerttemberg, Hesse and Rhineland-Palatinate. All of the aggregated values for the metropolitan area arise from the sum of the individual values from the 15 rural and urban districts. Calculation of the fertility rate requires the number of live births according to age groups of the mothers {10-14, 15-19, …, 40-44, 45-49}. Since the data do not differentiate between the genders of the live births, we assume that per 100 females born there are 105 males born. The mortal- ity rate is calculated from the number of gender-specifi c deaths per age group {<1, 1-4, 5-9, …, 70-74, 75 and older} to the number of men or women in the correspond- ing cohort. Both variables are available for the years 1981 to 2009. These variables are referenced to the population data of the corresponding time period, broken down by age groups {<3, 3-5, 6-9, 10-14, 15-17, 18-19, 20-24, 25-29, …, 70-74, 75 and older} and gender in order to calculate the rates. Calculating mortality rates for boys and girls under 10 years of age proved problematic, since the age groups of the deaths and risk population diverged. For this reason, the population data of a group were distributed evenly over the individual years of age and aggregated to the age groups of the deaths. The data on net migration for the years 2002 to 2009 are dif- ferentiated according to gender and divided into the age groups {<18, 18-24, 25-29, 30-49, 50-64, 65 and older}. The author’s analyses based on the residual method presented above supplement these series by the years 1995 until 2001. However, these data sets do not fulfi l the demands of the models presented here. An analysis of functional data requires smooth (age) functions to base the second- ary data upon. Therefore, the median is formed for each age group of the individual demographic rates and used as a node between the smoothed demographic rates. The mortality rates should rise monotonically from a certain threshold (for exam- ple years of age greater than 50) (Hyndman/Ullah 2007: 4945), because the older a person is, the higher their probability of death. This qualitative restriction can be in- tegrated into the smoothing of the data with weighted penalised regression splines according to Wood (1994). This procedure reduces biases in the estimated curves of the upper age cohorts. For the same reason, the data sets of the fertility rates are assumed as concave and prepared with weighted B regression splines according to the method by He and Ng (1999). Net migration is not a relative value and can be both positive and negative. A weighted locally quadratic regression is carried out (Moore et al. 1997) as the smoothing method. These three smoothing methods are contained in the “Demography” package for “R.” • Philipp Deschermeier784 Figure 3 shows the chronological development of the age-specifi c fertility rate for the time range between 1981 and 2009 based on the curves for 1981, 1990, 2000 and 2009. The shift of the maximum of the age-specifi c rates refl ects the rise in the median age of mothers at births of 26.9 years (1981) to 29.6 years (2009). This development reveals the advantage of modelling based on functional data, as the empirical curves for a specifi c demographic parameter deviate over the range of N points in time from the mean of this parameter. The development corresponds to a phase variation taken into consideration while modelling the time series through the basis functions of the FDA model. The trend observed in many developed countries and regions with a constantly rising life expectancy is also observed in the mortality rates of the men and women in the Rhine-Neckar metropolitan area (Fig. 4). Women across all years of age have a higher life expectancy and lower mortality rates than men. Between 1981 and 2009, the fi gures drop equally for both. The main difference is found in the male mortality rates between 35 and 65 years, which run almost linear, while the curves for women show a rather concave progression. The mortality rates between the genders differ the most in this interval. In the higher age groups, the gender-specifi c curves converge. For all people over 75 years, the (logarithmised) fi gures could not be smoothed due to the data situation and therefore had to be linearly interpolated. The target values for the age group “90+ years” of the individual years were tak- Fig. 3: Smooth age-specifi c fertility rate per 1,000 women for 1981, 1990, 2000 and 2009 0 20 40 60 80 100 120 10 15 20 25 30 35 40 45 50 Age 1981 1990 2000 2009 Fertility rate per 1,000 women Source: Author’s calculation of the functional data using B regression splines according to the method by He and Ng (1999) and on the basis of the data from the sta- tistical offi ces of the federal states Baden-Wuerttemberg, Hesse and Rhineland- Palatinate. Population Development of the Rhine-Neckar Metropolitan Area • 785 en from the mortality tables of the German Federal Statistical Offi ce (Statistisches Bundesamt 2010: 59). The population in the age from 90 years is contained in the upper age group “90 and older.” Net migration underlies a variety of infl uences, which are effective both in the respective regions of origin and destination and which lead to considerably greater chronological fl uctuations than in the fertility and mortality rates (Fig. 5). After the infl uxes and outfl uxes connected to German reunifi cation in the early 1990s ebbed, the series stabilised noticeably. And since about the middle of the 1990s, the mi- gration surpluses of women are higher than that of men in the metropolitan area (Fig. 5). The data show that particularly young men and women of the ages between 18 and 25 years migrate into the metropolitan area (Fig. 6), which underscores its signifi cance as an educational and university location. Slight migration losses occur only in the upper age groups. Source: Author’s calculation of the functional data using weighted penalised regression splines according to Wood (1994) and on the basis of the data from the statistical offi ces of the federal states Baden-Wuerttemberg, Hesse and Rhineland-Palati- nate. -4.5 -4.0 -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0 10 20 30 40 50 60 70 Age Men 1981 Men 2009 Women 1981 Women 2009 Logarithmized mortality rates Fig. 4: Smoothed and logarithmised mortality rates for men and women for 1981 and 2009 • Philipp Deschermeier786 Fig. 6: Gender differentiated net migration according to years of age for 2009 -800 -400 0 400 800 1,200 1,600 0 10 20 30 40 50 60 70 80 90 Age Women 2009 Men 2009 Net migration Source: Author’s calculation on the basis of the data from the statistical offi ces of the federal states Baden-Wuerttemberg, Hesse and Rhineland-Palatinate. Fig. 5: Development of net migration (1975-2009) Source: Author’s calculation of the functional data using weighted “locally quadratic re- gression” according to Moore et al. (1997) and on the basis of the data from the statistical offi ces of the federal states Baden-Wuerttemberg, Hesse and Rhine- land-Palatinate. -10,000 -5,000 0 5,000 10,000 15,000 20,000 25,000 1975 1985 1995 2005 Women Men Net migration Time Population Development of the Rhine-Neckar Metropolitan Area • 787 4.3 The demographic components: model and forecast The smoothed data of the fertility rate consist of 29 curves for the years 1981 to 2009. A model with K=6 basis functions was estimated for this period. Hyndman and Booth (2008: 327) showed that the results of the method are independent of the number of basis functions and K=6 is suffi ciently large. The basis functions encompass 82.7 %, 11.3 %, 3.3 %, 1.9 %, 0.6 % and 0.1 % or 99.9 % of the vari- ation deviating from the mean. The coeffi cient of the fi rst basis function (Fig. 7) shows the drop in the fertility rate (β t.1 falls within the period of observation), which is strongest among women between 20 and 25 years (highest positive function values of the basis function φ 1 ). The negative values of the basis function for ages between 35 and 40 years show that births increase in this range: The fi rst basis function thus describes the social trend of shifting births to a later point in time. The remaining basis functions as a whole comprise an insuffi cient share of the variance of the smoothed function to portray the age-specifi c fertility rates for the period from 1981 until 2009 and are therefore not suitable for contextual interpreta- tion.6 They counteract systematic statistical biases, which are caused by amplitude variation and phase variation. The forecast of the fertility rate was conducted for the years 2010 until 2030 (Fig. 8). The maximums of the age-specifi c fertility rates rise over the entire period, and the individual curves shift over the course of time into higher age groups. The average age of mothers rises from 29.85 (2009) to 31.36 years (2030) and lies within the 80 % forecast interval [31.06; 31.65]. The total fertility rate (TFR) increases con- 6 All basis functions of the models for fertility, mortality and net migration are illustrated in the Appendix. Fig. 7: The fi rst basis function of the fertility model including the forecast of the coeffi cients β t.1 for 2010 until 2030 and the 80 % forecast interval (dotted lines) Source: Author’s calculation on the basis of the data from the statistical offi ces of the fed- eral states Baden-Wuerttemberg, Hesse and Rhineland-Palatinate. -0.30 -0.20 -0.10 0.00 0.10 0.20 0.30 0 5 10 15 20 25 30 35 40 45 50 Age Basis function (x) -20 -15 -10 -5 0 5 10 1981 1988 1995 2002 2009 2016 2023 2030 Coefficient t, Time • Philipp Deschermeier788 sistently over the forecast range. Figure 9 contains the point estimators and the 80 % forecast intervals. The mean TFR therefore corresponds with expectations of the development of the national TFR with 1.4 births per woman (Statistisches Bundesamt, 2009: 27). Heckman and Walker (1990) cite a high correlation between the childbearing behaviour and the income of women as the reason for the shift of births to later in life. This economic interpretation is, however not comprehensive enough. Ott et al. (2006) sketch out a three-phase model that takes children into account in life plan- ning only after reaching certain vocational objectives. Gustaffson (2001) bases the shift on disadvantages of motherhood on the career of potential mothers. Börsch- Supan and Wilke (2009) assume that the emerging decrease of people of working age will lead to a growth in female employment, and they will thus approximate the employment behaviour of men. Since the postponement of births is, however, biologically limited, births must be increasingly made up for in later years of age. These effects are refl ected in the forecast: the births shift chronologically, the aver- age age of mothers at childbirth rises and the fertility rates of the higher age groups Fig. 8: Forecast of the fertility rate per 1,000 women for 2010 (black) and 2030 (grey) with the 80 % forecast intervals (dotted lines) Source: Author’s calculation on the basis of the data from the statistical offi ces of the fed- eral states Baden-Wuerttemberg, Hesse and Rhineland-Palatinate. 0 20 40 60 80 100 120 140 10 15 20 25 30 35 40 45 50 Age 2030 80 % forecast interval 2030 2010 80 % forecast interval 2010 Fertility rate per 1,000 women Population Development of the Rhine-Neckar Metropolitan Area • 789 rise considerably. In future, women in the high age groups will contribute most to the modelled TFR (Fig. 9). To prevent the mortality rates from diverging in the forecast, the model is based on the coherent model of functional data, which contains six basis functions both for the models of the product and for the ratio. The basis functions of the geometric mean of both genders comprise 96.6 %, 1.5 %, 1.0 %, 0.5 %, 0.2 % and 0.1 % of the variation of the smooth functions on which the mortality rates are based. The coeffi cient β t.1 of the dominant fi rst basis function (Fig. 10) shows a drop in mortal- ity rates, which is refl ected the most in infant mortality (< 1 year) and lessens with age. The basis functions of the ratio model comprise 60.2 %, 17.6 %, 10.2 %, 6.8 %, 3.0 % and 1.8 % of the variation. The fi rst basis function shows that the greatest de- viations in the mortality rates between men and women are at about 20 years. This variation is, however, subject to strong fl uctuations over time. The mortality forecast indicates the trend of consistently declining fi gures, which slows down, however, over the course of time (Fig. 11). The life expectancy (Fig. 12) of women rises until 2030 to 80.55 years and lies within the 80 % forecast interval [80.26; 80.82], while that of the men rises to 78.75 years with the 80 % forecast in- terval [78.47; 79.06]. Carnes and Olshansky (2007: 377) cite the obesity of children and young people as well as new infectious diseases as sources that could possibly weaken or even reverse this trend. These possible future infl uences cannot, how- ever, be included in a forecast based on time series models. Fig. 9: Development of TFR per 1,000 women for 1981-2030 with the 80 % forecast interval 1,000 1,100 1,200 1,300 1,400 1,500 1,600 1981 1988 1995 2002 2009 2016 2023 2030 TFR 80 % forecast interval TFR per 1,000 women Time Source: Author’s calculation on the basis of the data from the statistical offi ces of the fed- eral states Baden-Wuerttemberg, Hesse and Rhineland-Palatinate. • Philipp Deschermeier790 The aggregated net migration (Fig. 5) shows that the values for the two genders run similarly, which justifi es modelling using the coherent model. The time series prior to 1995 is subject to great dynamics, which had a negative effect on the model and the subsequent forecast. For this reason, only the values between 1995 and 2009 are included in the model. The high male and female net migration between 18 and 25 years is the dominant effect in the model. The fi rst of the six basis func- tions of the product model indicate the change over the course of time (Fig. 13) and records 81.6 % of the variation from the mean, the remainder by contrast only 10.5 %, 5.2 %, 1.4 %, 0.7 % and 0.4 %. The fi rst basis function of the ratio model il- lustrates the gender-specifi c differences of educational migrants (variance percent- age 53.9 %). The coeffi cient shows that male net migration between 2000 and 2003 was higher but that the trend weakened and even reversed in ensuing years. This Fig. 10: The fi rst basis functions of the mortality model including the forecast of the coeffi cients β t.1 and γ t.1 for 2010 until 2030 and the 80 % forecast interval (dotted lines) -0.05 0.00 0.05 0.10 0.15 0.20 0.25 0 10 20 30 40 50 60 70 80 90 Basis function (x) Age -12 -8 -4 0 4 1981 1988 1995 2002 2009 2016 2023 2030 Coefficient t, Time -0.1 0.0 0.1 0.2 0.3 0 10 20 30 40 50 60 70 80 90 Basis function (x) Age -1.0 -0.5 0.0 0.5 1.0 1981 1988 1995 2002 2009 2016 2023 2030 Coefficient t, Time Source: Author’s calculation on the basis of the data from the statistical offi ces of the fed- eral states Baden-Wuerttemberg, Hesse and Rhineland-Palatinate. Population Development of the Rhine-Neckar Metropolitan Area • 791 Fig. 11: Age and gender-specifi c mortality rates for 2009 (black) and the forecast for 2030 (grey) with the 80 % forecast intervals (dotted lines) -4.5 -4.0 -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0 10 20 30 40 50 60 70 80 90 Age Logarithmized mortality rate Women 2009 Forecast for 2030 80 % forecast interval -4.5 -4.0 -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0 10 20 30 40 50 60 70 80 90 Age 2009 Forecast for 2030 80 % forecast interval Logarithmized mortality rate Men Source: Author’s calculation on the basis of the data from the statistical offi ces of the fed- eral states Baden-Wuerttemberg, Hesse and Rhineland-Palatinate. Fig. 12: Development of the life expectancy of men (black) and women (grey) for 1981-2030 and the 80 % forecast intervals (dotted lines) 70 72 74 76 78 80 82 1981 1988 1995 2002 2009 2016 2023 2030 Time Women 80 % forecast interval Men 80 % forecast interval Life expectancy Source: Author’s calculation on the basis of the data from the statistical offi ces of the fed- eral states Baden-Wuerttemberg, Hesse and Rhineland-Palatinate. • Philipp Deschermeier792 development continues in the forecast, as in 2030 (Fig. 14) compared to 2009 (Fig. 6) the values of the women in the age range between 18 and 25 years is considerably higher than that of the men (Fig. 14). In the model, the Rhine-Neckar metropolitan area remains an attractive education or study location for young people. Nonethe- less, due to the wide forecast intervals for people under 50 years of age, the fore- casts of net migration are the greatest source of uncertainty about future population development. Fig. 13: The fi rst basis functions of the migration model including the forecast of the coeffi cients β t.1 and γ t.1 until 2030 and the 80 % forecast interval (dotted lines) 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0 10 20 30 40 50 60 70 80 90 Basis function (x) Age -400 -200 0 200 400 600 1995 2002 2009 2016 2023 2030 Coefficient t, Time -0.1 0.0 0.1 0.2 0.3 0.4 0 10 20 30 40 50 60 70 80 90 Basis function (x) Age -400 -200 0 200 400 600 1995 2002 2009 2016 2023 2030 Coefficient 1(x) Time Source: Author’s calculation on the basis of the data from the statistical offi ces of the fed- eral states Baden-Wuerttemberg, Hesse and Rhineland-Palatinate. Population Development of the Rhine-Neckar Metropolitan Area • 793 4.4 The population development of the Rhine-Neckar metropolitan area Since the 1990s, the male and female population of the Rhine-Neckar metropolitan area has been growing steadily due to migration gains caused by the political de- velopments in Germany (reunifi cation) and Europe (immigration of repatriates of German origin) (Gans/Schmitz-Veltin 2006: 316). Population development for the years 2010 until 2030 is calculated using the algorithm described in Section 4.1 on the basis of the cohort component method and the recorded births, deaths and migrants from the demographic components. The male population will rise slightly until 2022 (Fig. 15) and then drop measured on the median of the simulation by Fig. 14: Forecast of age distribution of net migration of men and women for 2030 with the 80% forecast intervals (dotted lines) -300 -100 100 300 500 700 0 10 20 30 40 50 60 70 80 90 Net migration 80 % forecast interval Net migration Women Age -300 -100 100 300 500 700 0 10 20 30 40 50 60 70 80 90 Net migration 80 % forecast interval Net migration Men Age Source: Author’s calculation on the basis of the data from the statistical offi ces of the fed- eral states Baden-Wuerttemberg, Hesse and Rhineland-Palatinate. Fig. 15: Development of the male and female populations between 2010 and 2030 with the 80 % forecast intervals (dotted lines) 960,000 980,000 1,000,000 1,020,000 1,040,000 1,060,000 1,080,000 1,100,000 1990 2000 2010 2020 2030 Population development 80 % forecast interval Population Women Time Source: Author’s calculation on the basis of the data from the statistical offi ces of the fed- eral states Baden-Wuerttemberg, Hesse and Rhineland-Palatinate. 960,000 980,000 1,000,000 1,020,000 1,040,000 1,060,000 1,080,000 1990 2000 2010 2020 2030 Population development 80 % forecast interval Population Men Time • Philipp Deschermeier794 2030 to 1.0349 million inhabitants, which corresponds to a decrease of 2.31 % for the forecast range, and lies within the 80 % forecast interval [1.0292; 1.0403]. The number of women will drop by 2030 by 4.45 percent to 1.0280 million inhabitants (80 % forecast interval [1.0234; 1.0327]). According to this, the female population will decrease somewhat more strongly than the male. The population pyramid in Figure 16 shows the anticipated age structure for the year 2009 as the last year of the time series data and the forecast for 2030 with the 80 % forecast intervals. The largest percentage of the total population in 2009 was the age groups between 20 Fig. 16: Male and female population according to years of age 2009 (grey) and the 0.5 quantile of the simulation for 2030 (black) with the 80 % forecast intervals (dotted lines) Source: Author’s calculation on the basis of the data from the statistical offi ces of the fed- eral states Baden-Wuerttemberg, Hesse and Rhineland-Palatinate. 30,000 20,000 10,000 0 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 0 10,000 20,000 30,000 Age Population Population 2009 Forecast for 2030 80 % forecast interval Men Women Population Development of the Rhine-Neckar Metropolitan Area • 795 and 30 years. The fertility rates lie considerably below the level for maintaining the population and even the positive net migration is not suffi cient to balance out the mortality surpluses. This makes a distinct decrease in the number of people under 30 years of age apparent. The number of people of working age under 30 years decreases enormously, while the group of the 30 to under 65 year olds will remain at about the level of 2009. This effect arises mainly from the shift of the largest age groups from 2009 to the older age groups in 2030. The forecast interval is greatest in the age range of under 20 years, since the number of these people come from the fertility forecast; while in the year 2030 the people in the age range of over 20 years had already been born by the fi rst forecast year (2010) and therefore are no longer a source of additional uncertainty. The width of the confi dence intervals additionally is determined by the variation of net migration, which will distinctly stabilise from 50 years (cf. Fig. 14), whereby the population from 50 years of age can be forecasted with greater accuracy. 5 Conclusions The demographic future of a region is uncertain. The deterministic models fre- quently used by the offi cial statistical offi ces for forecasts are of the “if ..., then ...” character, whereby the demographic determinants develop in different scenarios across predetermined assumptions. These should indicate the breadth of possible developments, however without making statements about their incidence probabili- ties. The stochastic approach, which uses time series models to calculate both the future population as well as a forecast interval for the demographic parameters, proves more problem-oriented. This approach, however, requires long time series and data broken down by years of age over all of the components of the basic de- mographic equation. At the regional level, though, such data are frequently not at all available, or only in rough age groups or with insuffi cient structure. Furthermore, the various statistical offi ces of the federal states compile the variables in different age groups; aggregation to a cross-regional area such as the Rhine-Neckar metro- politan area, which consists of urban and rural districts in Baden-Wuerttemberg, Hesse and Rhineland-Palatinate, on such a basis is in actuality impossible. The use of functional data proves exceedingly helpful for these problems, as per assumption age-grouped secondary data follow a functional context that can be estimated with the presented FDA models. In this way, data sets are created broken down by years of age that can therefore be aggregated to cross-border functional regions. Hence the paradigm of functional data and the models based on it close data gaps in sec- ondary statistics, particularly at the regional level, and allow for more reliable state- ments about demographic development than the use of deterministic approaches. Without the approaches from Chapter 3, stochastic models are not implementable at the regional level. Moreover, the use of forecast intervals on the basis of the time series models provide more valuable additional information for regional planners than deterministic forecasts. At the regional level in particular, there is a great need for information about population development to serve as the basis for estimat- • Philipp Deschermeier796 ing the consequences of demographic change, for example on the labour supply and particularly on the number of the working population. Furthermore, regional population forecasts serve as the basis for estimating the future demand for living space. The application of the models for functional data on the Rhine-Neckar metro- politan area illustrates the consequences of demographic change for the seventh- largest economic area in Germany and is a possibility for realizing stochastic popu- lation forecasts at the regional level. Although the population will only decrease slightly by 2030, a considerable drop in the number of young people of working age under 30 years will be contrasted by a notable rise in people in the upper age groups and particularly in retirement age. This shortage can be alleviated by im- migration. The basic prerequisite for this is to recognise the strengths of the region and to create incentives for young people to move to the region and also to remain there following their studies or training. The impacts of demographic change will be particularly noticeable in the labour market. Regions will be competing for young and well-educated employees. 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Tukey, John Wilder 1977: Exploratory Data Analysis. London: Addison-Wesley. Whelpton, Pascal 1928: Population of the United States, 1925 to 1975. In: American Jour- nal of Sociology 34: 253-270. Wilson, Chris 2001: On the scale of Global Demographic Convergence 1950-2000. In: Popu- lation and Development Review 27,1: 155-171 [doi: 10.1111/j.1728-4457.2001.00155.x]. Wood, Simon 1994: Obtaining Birth and Mortality Patterns From Structured Population Trajectories. In: Ecological Monographs 64,1: 23-44 [doi: 10.2307/2937054]. • Philipp Deschermeier800 Translated from the original text by the Federal Institute for Population Research, for information only. The reviewed and author’s authorised original article in German is available under the title “Die Bevölkerungsentwicklung der Metropolregion Rhein-Neckar. Eine stochastische Bevölkerung- sprognose auf Basis des Paradigmas funktionaler Daten”, DOI 10.4232/10.CPoS-2011-21de or URN urn:nbn:de:bib-cpos-2011-21de5, at http://www.comparativepopulationstudies.de. Date of submission: 29.08.2011 Date of Acceptance: 25.04.2012 Philipp Deschermeier ( ). Universität Mannheim, Lehrstuhl für Wirtschaftsgeographie, Fakultät für Rechtswissenschaft und Volkswirtschaftslehre. Mannheim, Germany. E-Mail: philipp.deschermeier@uni-mannheim.de URL: http://gans.vwl.uni-mannheim.de/1465.0.html Population Development of the Rhine-Neckar Metropolitan Area • 801 F ig . A 1: T h e s ix b a si s fu n c ti o n s o f th e f e rt ili ty m o d e ls 0 10 30 50 0510 M ai n ef fe ct s A ge Mean 0 10 30 50 -0.20.00.10.2 A ge Basis function 1 In te ra ct io n Ti m e Coefficient 1 19 80 19 90 20 00 20 10 -6-20246 0 10 30 50 -0.050.050.150.25 A ge Basis function 2 Ti m e Coefficient 2 19 80 19 90 20 00 20 10 -2024 0 10 30 50 -0.10.00.10.2 A ge Basis function 3 Ti m e Coefficient 3 19 80 19 90 20 00 20 10 -2-101 0 10 30 50 -0.10.00.10.20.3 A ge Basis function 4 Ti m e Coefficient 4 19 80 19 90 20 00 20 10 -1012 0 10 30 50 -0.20.00.20.4 A ge Basis function 5 Ti m e Coefficient 5 19 80 19 90 20 00 20 10 -1.00.00.51.0 0 10 30 50 -0.20.00.2 A ge Basis function 6 Ti m e Coefficient 6 19 80 19 90 20 00 20 10 -0.20.20.40.6 S o u rc e : A u th o r’ s ca lc u la ti o n o n t h e b as is o f th e d at a fr o m t h e s ta ti st ic al o ffi c e s o f th e f e d e ra l s ta te s B ad e n -W u e rt te m b e rg , H e ss e a n d R h in e la n d -P al at in at e . A p p e n d ix • Philipp Deschermeier802 F ig . A 2 : T h e s ix b a si s fu n c ti o n s o f th e p ro d u c t m o d e l o n m o rt a lit y 0 20 40 60 80 -8-6-4-20 M ai n ef fe ct s A ge Mean 0 20 40 60 80 0.000.100.20 A ge Basis function 1 In te ra ct io n Ti m e Coefficient 1 19 80 19 90 20 00 20 10 -202 0 20 40 60 80 -0.3-0.2-0.10.00.1 A ge Basis function 2 Ti m e Coefficient 2 19 80 19 90 20 00 20 10 -0.40.00.4 0 20 40 60 80 -0.10.00.10.2 A ge Basis function 3 Ti m e Coefficient 3 19 80 19 90 20 00 20 10 -0.6-0.20.20.4 0 20 40 60 80 -0.5-0.3-0.10.1 A ge Basis function 4 Ti m e Coefficient 4 19 80 19 90 20 00 20 10 -0.20.00.20.4 0 20 40 60 80 -0.20-0.100.000.10 A ge Basis function 5 Ti m e Coefficient 5 19 80 19 90 20 00 20 10 -0.10.00.10.2 0 20 40 60 80 -0.10.10.3 A ge Basis function 6 Ti m e Coefficient 6 19 80 19 90 20 00 20 10 -0.15-0.050.05 S o u rc e : A u th o r’ s ca lc u la ti o n o n t h e b as is o f th e d at a fr o m t h e s ta ti st ic al o ffi c e s o f th e f e d e ra l s ta te s B ad e n -W u e rt te m b e rg , H e ss e a n d R h in e la n d -P al at in at e . Population Development of the Rhine-Neckar Metropolitan Area • 803 F ig . A 3: T h e s ix b a si s fu n c ti o n s o f th e r a ti o m o d e l o n m o rt a lit y 0 20 40 60 80 0.00.20.4 M ai n ef fe ct s A ge Mean 0 20 40 60 80 -0.050.050.15 A ge Basis function 1 In te ra ct io n Ti m e Coefficient 1 19 80 19 90 20 00 20 10 -0.6-0.20.20.6 0 20 40 60 80 0.050.150.25 A ge Basis function 2 Ti m e Coefficient 2 19 80 19 90 20 00 20 10 -0.4-0.20.00.2 0 20 40 60 80 -0.3-0.10.1 A ge Basis function 3 Ti m e Coefficient 3 19 80 19 90 20 00 20 10 -0.3-0.10.10.3 0 20 40 60 80 -0.20.00.1 A ge Basis function 4 Ti m e Coefficient 4 19 80 19 90 20 00 20 10 -0.20.00.10.2 0 20 40 60 80 -0.10.00.10.2 A ge Basis function 5 Ti m e Coefficient 5 19 80 19 90 20 00 20 10 -0.15-0.050.050.15 0 20 40 60 80 -0.8-0.40.0 A ge Basis function 6 Ti m e Coefficient 6 19 80 19 90 20 00 20 10 -0.100.000.10 S o u rc e : A u th o r’ s ca lc u la ti o n o n t h e b as is o f th e d at a fr o m t h e s ta ti st ic al o ffi c e s o f th e f e d e ra l s ta te s B ad e n -W u e rt te m b e rg , H e ss e a n d R h in e la n d -P al at in at e . • Philipp Deschermeier804 F ig . A 4: T h e s ix b a si s fu n c ti o n s o f th e p ro d u c t m o d e l o n m ig ra ti o n 0 20 40 60 80 050150250 M ai n ef fe ct s A ge Mean 0 20 40 60 80 0.000.100.20 A ge Basis function 1 In te ra ct io n Ti m e Coefficient 1 19 96 20 02 20 08 -2000200400 0 20 40 60 80 -0.20.00.10.2 A ge Basis function 2 Ti m e Coefficient 2 19 96 20 02 20 08 -1000100200 0 20 40 60 80 -0.3-0.10.00.1 A ge Basis function 3 Ti m e Coefficient 3 19 96 20 02 20 08 -100050100 0 20 40 60 80 -0.4-0.20.0 A ge Basis function 4 Ti m e Coefficient 4 19 96 20 02 20 08 -40-20020 0 20 40 60 80 -0.10.00.10.2 A ge Basis function 5 Ti m e Coefficient 5 19 96 20 02 20 08 -20-1001020 0 20 40 60 80 -0.10.00.10.20.3 A ge Basis function 6 Ti m e Coefficient 6 19 96 20 02 20 08 -2001020 S o u rc e : A u th o r’ s ca lc u la ti o n o n t h e b as is o f th e d at a fr o m t h e s ta ti st ic al o ffi c e s o f th e f e d e ra l s ta te s B ad e n -W u e rt te m b e rg , H e ss e a n d R h in e la n d -P al at in at e . Population Development of the Rhine-Neckar Metropolitan Area • 805 F ig . A 5: T h e s ix b a si s fu n c ti o n s o f th e r a ti o m o d e l o n m ig ra ti o n 0 20 40 60 80 -80-400 M ai n ef fe ct s A ge Mean 0 20 40 60 80 0.000.100.200.30 A ge Basis function 1 In te ra ct io n Ti m e Coefficient 1 19 96 20 02 20 08 -100050 0 20 40 60 80 -0.050.050.15 A ge Basis function 2 Ti m e Coefficient 2 19 96 20 02 20 08 -100-50050 0 20 40 60 80 -0.4-0.20.00.2 A ge Basis function 3 Ti m e Coefficient 3 19 96 20 02 20 08 -80-40020 0 20 40 60 80 0.000.100.20 A ge Basis function 4 Ti m e Coefficient 4 19 96 20 02 20 08 -101030 0 20 40 60 80 -0.25-0.100.05 A ge Basis function 5 Ti m e Coefficient 5 19 96 20 02 20 08 -15-5515 0 20 40 60 80 -0.3-0.10.1 A ge Basis function 6 Ti m e Coefficient 6 19 96 20 02 20 08 -6-20246 S o u rc e : A u th o r’ s ca lc u la ti o n o n t h e b as is o f th e d at a fr o m t h e s ta ti st ic al o ffi c e s o f th e f e d e ra l s ta te s B ad e n -W u e rt te m b e rg , H e ss e a n d R h in e la n d -P al at in at e . © Federal Institute for Population Research 2012 – All rights reserved Published by / Herausgegeben von Prof. Dr. Norbert F. Schneider Federal Institute for Population Research D-65180 Wiesbaden / Germany Managing Editor / Verantwortlicher Redakteur Frank Swiaczny Editorial Assistant / Redaktionsassistenz Katrin Schiefer Language & Copy Editor (English) / Lektorat & Übersetzungen (englisch) Amelie Franke Copy Editor (German) / Lektorat (deutsch) Dr. Evelyn Grünheid Layout / Satz Beatriz Feiler-Fuchs E-mail: cpos@destatis.de Scientifi c Advisory Board / Wissenschaftlicher Beirat Jürgen Dorbritz (Wiesbaden) Paul Gans (Mannheim) Johannes Huinink (Bremen) Marc Luy (Wien) Clara H. Mulder (Groningen) Notburga Ott (Bochum) Peter Preisendörfer (Mainz) Board of Reviewers / Gutachterbeirat Martin Abraham (Erlangen) Laura Bernardi (Lausanne) Hansjörg Bucher (Bonn) Claudia Diehl (Göttingen) Andreas Diekmann (Zürich) Gabriele Doblhammer-Reiter (Rostock) Henriette Engelhardt-Wölfl er (Bamberg) E.-Jürgen Flöthmann (Bielefeld) Alexia Fürnkranz-Prskawetz (Wien) Beat Fux (Zürich) Joshua Goldstein (Rostock) Karsten Hank (Köln) Sonja Haug (Regensburg) Franz-Josef Kemper (Berlin) Michaela Kreyenfeld (Rostock) Aart C. Liefbroer (Den Haag) Kurt Lüscher (Konstanz) Dimiter Philipov (Wien) Tomáš Sobotka (Wien) Heike Trappe (Rostock) Comparative Population Studies – Zeitschrift für Bevölkerungswissenschaft www.comparativepopulationstudies.de ISSN: 1869-8980 (Print) – 1869-8999 (Internet)