Domestic water use accounts for 12 % of the global total (Hanasaki et al., 2008a, b; Flörke et al., 2013; Wada et al., 2014a, b). However, available global models and scenarios of domestic withdrawals are limited. Earlier attempts to model domestic water withdrawal are summarized in Table 3.

The WaterGAP model was the first global water model that included a sub-model to project future domestic water use globally at grid-scale resolution (Alcamo et al., 2003a, b). WaterGAP uses a multiple regression model with population and GDP per capita as independent variables. Historical change in domestic water use are explained by categorizing them as structural and technological changes. Structural change refers to the observation that water use intensity, or per capita water use, grows rapidly for countries with low but increasing income, and slows down in countries with high income. Technological change is the general trend that water use for each service becomes smaller over time due to improvement in the water use efficiency of newer devices. One of the key challenges of this approach is calibration of the parameters. Sufficient amounts of reliable data are essential for calibration, although published historical time series of water withdrawals are limited for many countries. Alcamo et al. (2003a, b) calibrated the key parameters regionally using the data compiled by Shiklomanov (2000a, b) and nationally where data were available. Flörke et al. (2013) updated the model and parameters by collecting country-level domestic water use data for 50 individual countries and 27 regions. Wada et al. (2014a, b) developed a similar model as Alcamo et al. (2003a, b) and Flörke et al. (2013) and projected national domestic water withdrawal for the whole 21st century.

Shen et al. (2008) proposed a model with different formulations from Alcamo et al. (2003a, b). They assumed that the future water use level of developing countries will converge with that of present developed countries as economic growth continues. They first plotted per capita GDP and water use at present by countries. Then they adopted a logarithmic model and regressed with the data that represent the present global relationship between per capita GDP and water use. Hayashi et al. (2013) adopted the same model as Shen et al. (2008), while they made regression separately from urban and rural areas since the accessibility to tap water is substantially different. Because their models do not require historical time-series data of regions and countries, it is easy to calibrate the model parameter. In contrast, the results are presented under a strong assumption that the path of growth in domestic water use is globally uniform.

The estimated model parameters mentioned above represent historical relationships between domestic water withdrawal and socio-economic factors. It remains uncertain whether maintaining these parameters throughout the 21st century is a valid approach, since future scenarios such as SSPs depict substantially different future conditions. Hanasaki et al. (2013a, b) developed a set of national projections on domestic water withdrawal globally for the 21st century based on the latest developed SSPs. They adopted a model similar to Alcamo et al. (2003a, b) and prepared parameter sets mainly based on literature review that are compatible with the five different views of a world in the future as depicted in the SSPs. Although arbitrariness is included in the parameter setting, this approach enables us to project water use for the world that is substantially different from that realized in the past.

In the current analysis, H08 uses the method described by Hanasaki et al. (2013a, b), PCR-GLOBWB uses Wada et al. (2014a, b), and WaterGAP uses the method described in Flörke et al. (2013) (see Table 3). In contrast to the industrial sector, the methods applied by the three water models to calculate domestic water use are similar, and are driven primarily by population numbers while based on per capita water use (or withdrawal) intensities. All three models calculate both water withdrawal and consumptive water use, the latter subtracting the return flow to the rivers and groundwater. National numbers of domestic water use are distributed to a 0.5∘ by 0.5∘ grid according to the gridded total population numbers for all three models. H08 primarily uses population numbers and per capita water use as input socio-economic variables. WaterGAP is driven by population numbers and GDP per capita, while PCR-GLOBWB is also driven by population numbers, but additionally considers GDP, total electricity production, and energy consumption for the calculation of per capita water use and associated future trend similar to the water use intensity calculation in the industrial sector (see Appendix A1). In addition, assumptions on technological change rates are considered by all three models whereas WaterGAP also takes into account structural changes.

As pressure grows on many of the world's river basins, it becomes increasingly critical to balance the competing needs among different water use sectors and ecosystems. Environmental flows refer to the amount of water that needs to be allocated for the maintenance of aquatic ecosystem services (Dyson et al., 2003; Pastor et al., 2014). Various factors contribute to the health of river ecosystems, including discharge (streamflow), the physical structure of the channel and riparian zone, water quality, channel management, level of exploitation, and the physical barriers to connectivity (Acreman and Dunbar, 2004; Smakhtin et al., 2004, 2006).

Early definitions of environmental flows were premised on the importance of maintaining a fixed minimum flow, but all aspects of a flow regime (including floods, medium, and low flows) are important, and changes to any part of the regimes may impact or influence the overall ecosystem and provision of ecosystem services (Pahl-Wostl et al., 2013; Acreman and Dunbar, 2004). Environmental flow requirements should therefore not only address the amount of water needed, but also issues of timing and duration of river flows (Smakhtin et al., 2006). In order to accommodate these seasonal and inter-annual variations, environmental flow requirements must vary over space and time in order to meet and supply the ecosystem services as outlined by various stakeholders (Pahl-Wostl et al., 2013). Action on environmental flow requirements have been offset and limited by (1) lack of understanding of environmental flow benefits, (2) uncoordinated management of water resources, (3) low priority given to environmental flows in allocation processes, (4) limiting environmental flows to low flow requirements, (5) not paying attention to the impacts of too much water, and (6) the difficulties of coordinating complex environmental flows (Richter, 2010).

Estimated calculations of environmental water requirements (EWRs), which are the sum of ecologically relevant low-flow and high-flow components to ensure a scenario of “fair” ecosystem service delivery, vary depend on hydrological regimes, but are generally in the range of 20–50 % of renewable water resources (Smakhtin et al., 2004). They are highest in the rivers of the equatorial belt (Amzaon and Congo), where there is stable rainfall, and for river systems that are lake-regulated (Canada, Finland), or those that are influenced by a high percentage of groundwater generated baseflow (northern and central Europe, or swamps (Siberia). However, estimates of EWRs are much lower for areas with highly variable monsoon-driven rivers, rivers of arid areas, and those with high snowmelt flows (Asia, Africa, and Arctics). Varying, simplistic approaches have been used to estimate EWRs. In IMPACT, for example, environmental flow is specified as a share of average annual runoff) (Rosegrant et al., 2012). When data are unavailable in a particular food producing unit, an iterative procedure is used. The initial value for environmental flows is assumed to be 10 % with additional increments of 20–30 % if navigation requirements are significant (for example in the Yangtze River basin); 10–15 % if environmental reservation is legally enshrined, as in most developed countries; and 5–10 % for arid and semi-arid regions where ecological requirements, such as salt leaching, are high (for example, Central Asia) (Rosegrant et al., 2012).

The H08 method uses an empirical model that estimates the amount of river discharge that should be kept in the channel to maintain the aquatic ecosystem, which is based on case studies of regional practices, while the river discharge should ideally be unchanged for the preservation of the natural environment (Hanasaki et al., 2008a, b). PCR-GLOBWB equates EFRs to Q90, i.e., the streamflow that is exceeded 90 % of the time, following the study of Smakhtin et al. (2004). WaterGAP also follows the method of Smakhtin et al. (2004), but also incorporates the concepts of hydrological variability and river ecosystem integrity. This paper focuses on domestic and industrial use, and therefore EWRs will not be analyzed with the results.

Within WFaS, qualitative scenarios of water availability and demand are being developed that are broadly consistent with scenarios being developed for other sectors and that incorporate feedback from stakeholders where possible (Fig. 1). In the first step (“fast-track”), the SSP storylines, already the result of a multi-year community effort across sectors, have been extended with relevant critical dimensions affecting water availability and use. The SSPs offer the possibility for experimentation by a wide range of researchers extending the “original” SSPs in various dimensions (O'Neill et al., 2015). However, SSPs were developed by the climate change community with a focus of the key elements for climate policy analysis, i.e., less or no information is given related to the water sector. Therefore WFaS has extended SSP storylines and has developed a classification system called hydro-economic (HE) classes to describe different conditions in terms of a country's or region's ability to cope with water-related risks and its exposure to complex hydrological conditions, which affect its development in the scenarios (Fischer et al., 2015). Critical water dimensions have been assessed qualitatively and quantitatively for each SSP and HE class (classified using GDP per capita and four indicators describing hydrologic complexity). Several climate and socio-economic pathways are being analyzed in a coordinated multi-model assessment process involving sector and integrated assessment models, water demand models and different global hydrological models. Integration and synthesis of results will produce a first set of quantified global water scenarios that include consistency in climate, socio-economic developments (e.g., population, economic, energy) and water resources, with this paper focusing on aspects of water demand.

The focus of this chapter is to describe the water demand modeling, i.e., the underlying drivers and assumptions as well as the model results. The WFaS assessment has initially employed a “fast-track” analysis to produce well-founded yet preliminary scenario estimates following the SSP storylines and to apply available quantifications of socio-economic variables and climate model projections of the RCPs from the Inter-Sectoral Impact Model Intercomparison Project (ISI-MIP; Warszawski et al., 2014).

In WFaS the SSP narratives were enriched with relevant critical dimensions of the main water use sectors agriculture, industry, and domestic for the development of a first set of assumptions applied in global water models. This is achieved for various conditions in terms of a country or region's ability to cope with water-related risks and its exposure to complex hydrological conditions. For this purpose a hydro-economic (HE) classification has been developed, assigning each country in a two-dimensional space of coping capacity and hydrologic complexity (see Appendix A2). Critical water dimensions were evaluated qualitatively and quantitatively for each SSP and HE class classified with GDP and available renewable water resources (Fischer et al., 2015). In the WFaS “fast-track” analysis we have selected three SSP-based scenarios for the quantification of spatially explicit global water use until 2050 using the state-of-the-art global water models H08 (Hanasaki et al, 2008a, b), PCR-GLOBWB (Van Beek et al., 2011; Wada et al., 2014b), and WaterGAP2.2 (Flörke et al., 2013; Müller Schmied et al., 2014). These SSPs were chosen to envelop an upper (SSP3-RCP6.0), a middle (SSP2-RCP6.0), and a lower (SSP1-RCP4.5) range of plausible changes in future socio-economics and associated greenhouse gas emissions based on data availability of SSP scenarios when the WFaS “fast-track” analysis was conducted. Tables 4 and 5 summarize quantitative scenario assumptions applied in the water model calculations. The Appendix A3 summarizes how we generate scenario assumptions based on SSP and HE classification.

Note that future land use changes including irrigated areas and livestock numbers according to the new SSP scenarios are still under development, therefore, we were not able to include irrigation and livestock sector in this “fast-track” analysis. For a comprehensive assessment of future irrigation under the latest RCP scenarios, we refer to Wada et al. (2013b) who used a set of seven global water models to quantify the impact of projected global climate change on irrigation water demand by the end of this century, and to assess the resulting uncertainties arising from both the global water models and climate projections. In addition, due to limited data available for future ecosystem service, we did not include the assessment of environmental flow requirements. We refer to Pastor et al. (2014) for a comprehensive assessment of global environmental flow requirements. Thus, here we primarily focus on the industrial (electricity and manufacturing) and domestic sectors.

Ensemble results of global industrial water withdrawals highlight a steep increase in almost all SSP scenarios (Fig. 2). It should be noted that WaterGAP makes a distinction between thermoelectric and manufacturing water use, while the other two global water models, PCR-GLOBWB and H08, calculate aggregated industrial water demands only.

Global withdrawals are projected to reach nearly 2000 km3yr-1 by 2050, more than double the present industrial water use intensity in 2010 (850 km3yr-1). A different trend can be seen in a reduction of water use (40 %) projected by H08 for SSP1 compared to PCR-GLOBWB and WaterGAP, which project about 50 and 100 % increases, respectively. Under the SSP2 and SSP3 scenarios, the results are more consistent. Global industrial water withdrawal is projected to increase by 70–120 % under the “business-as-usual” SSP2 scenario and by 45–120 % under the “divided world” SSP3 scenario. H08 results show the largest range among the SSP projections, falling between a -40 % decrease (SSP1) and an 80 % increase (SSP3). PCR-GLOBWB has a relatively a narrow range between an increase of 50 % (SSP1) to 70 % (SSP3). The range is even narrower for WaterGAP with an increase of 105 % for SSP1 and 119 % for SSP2. By 2050 WaterGAP projects the largest net increase under SSP2, while the other models project that under SSP3.

In order to investigate reasons for the major differences among the three global water models we now scrutinize regional trends in industrial water withdrawals projections under the same sets of SSP scenarios. Figure 3 shows regional trends in projected industrial water withdrawals among the three models to highlight the uncertainty in water use projections. We selected regional major water users with significantly different projections across the three models. Each country has been assigned to a HE classification (Appendix A2), for which a consistent set of socio-economic scenarios and assumptions for technological and structural change has been developed under each SSP (see Tables 4 and 5). In the mature, industrialized economy of the USA and Germany, the projected industrial water withdrawals exhibit a steadily decreasing trend toward the year 2050 for almost all projections. However, H08 features an increasing trend (after a sharp drop in 2020) for both countries under the SSP3 scenario.

For the emerging economies (China, Brazil, and Russia), the ensemble projections show large differences among the three global water models. WaterGAP projects a much larger net increase in industrial water withdrawals for China and Brazil by 2050 under all SSPs, while H08 results show a net decrease under SSP1 (China, Brazil, Egypt and Russia) and SSP2 (Brazil and Russia). PCR-GLOBWB follows a similar trend with WaterGAP for China and Russia, but shows a much lower net increase for Brazil compared to WaterGAP. For PCR-GLOBWB and WaterGAP, the relative increase is similar for China and Russia. However, the different quantities of industrial water withdrawals at the starting year of the simulations lead to large differences in the absolute amounts by 2050 among the water models (due to the use of different data sets at the reference year of 2005). This is particularly obvious for Russia, where industrial water withdrawals differ by a factor of 4 at the reference year between PCR-GLOBWB and WaterGAP. H08 results show a decreasing trend for SSP1 in these countries as shown in the global trend. The higher industrial water withdrawal estimated by WaterGAP in emerging economies is often due to an increase in manufacturing water use. H08 and PCR-GLOBWB do not disaggregate the industrial sector into manufacturing and thermal electricity, which results in a homogeneous response in projected trends among these sub-sectors. In India, Brazil, and China, where economies are projected to grow rapidly in the coming decades, industrial water withdrawals are projected to increase by a factor of more than 2 by 2050. Here H08 again shows a decreasing trend for India and Egypt under SSP1, while PCR-GLOBWB and WaterGAP project a steep increase. For WaterGAP, the large increase in industrial water withdrawals is partly explained by a sharp increase in manufacturing water use. In Saudi Arabia, the use of different data sets for the reference year causes a large spread in the ensemble projections. The net decrease in projected industrial water withdrawals is estimated by PCR-GLOBWB and WaterGAP, while H08 alone shows an increasing trend under all SSP scenarios considered.

Figure 4 shows ensembles of global domestic water withdrawal projections from the three global water models. Due to the rapid increase in world population, ensemble results among the three models show a sharp increase in domestic water withdrawals under all SSP scenarios. Depending on the scenario, global volume is projected to reach 700–1500 km3yr-1 by 2050, which is an increase of 50 to 250 % compared to the present water use intensity (400–450 km3yr-1 in 2010). In contrast to the industrial sector, the models agree in projecting a consistently increasing trend for future domestic water use by 2050, with a minor exception for WaterGAP, which projects a slight decrease in domestic water use after 2040 under the SSP1 scenario. However, compared to the present water use, WaterGAP still projects a 70 % increase by 2050 under SSP1. However, PCR-GLOBWB projects a much higher increase in domestic water use by 2050 compared to H08 and WaterGAP. The increase by 2050 ranges between 40 and 70 % (SSP1), 70 and 140 % (SSP2), and 90 and 150 % (SSP3) for H08 and WaterGAP, respectively. For PCR-GLOBWB, the increase is projected to be much higher and reaches 170 % (SSP1), 230 % (SSP2), and 250 % (SSP3).

Model results are shown in Fig. 5 for domestic water withdrawals for the same set of countries as shown in the industrial sector (Fig. 3). Although the agreement among modeled trends is high for the global sums, trends are not clear on the country scale. For example, for the USA and Germany, the projected trends in domestic water withdrawals show different signals by 2050 across the models. H08 projects an steadily increasing trend for both countries under all SSPs. For WaterGAP, the domestic water withdrawals are projected to increase up to 2020 or 2030 but decrease thereafter under all scenarios as a result of structural change and population development. The decrease is much larger under SSP1, where the domestic water withdrawals are projected to decrease by 10–20 % compared to the present water withdrawal. PCR-GLOBWB projects for the USA a rapid increase in domestic water withdrawals by 2050 under all scenarios, but for Germany, only a moderate or negligible increase under SSP1 and SSP2 and a large increase under SSP3.

For China, Brazil, India, and Egypt, ensemble projections show rather a consistent pattern across the models. For those countries, present domestic water withdrawals share altogether one-third of the global total, and population is projected to grow more rapidly than in other countries. H08 projects an increasing trend by 2050 under all scenarios, but the increase is much larger for SSP2 and SSP3 than SSP1. For PCR-GLOBWB, the projections show a steep increase under all scenarios. There is a pronounced increase in countries with large population growth (China, India, Egypt, Brazil), where the domestic water withdrawals are projected to quadruple in almost all scenarios and models. In Brazil WaterGAP shows a similar increasing trend with PCR-GLOBWB. However, the increase in domestic water withdrawals is much milder for the other countries in WaterGAP, particularly after the 2030s, where the domestic water withdrawals start decreasing for China, India, and Egypt under the SSP1 scenario due to a stabilization or decreasing trend in population. For Russia, PCR-GLOBWB projects a pronounced increase that is similar in China, Brazil, India, and Egypt under all scenarios, while H08 and WaterGAP show rather a constant or decreasing trend towards 2050 under almost all scenarios, except for a slight increase under the SSP3 scenario for H08. Similar to the industrial sector, the initial value at the reference year (2005) has a large difference between PCR-GLOBWB and the other two models, leading to a large spread in absolute values by 2050. This is also the case for Germany, but between WaterGAP and the other two models. The ensemble projections show a consistent pattern for Saudi Arabia among the three models under all scenarios, where domestic water withdrawals are projected to increase by 100–200 % until 2050 due to a growing population.

A major difference among the employed water models relates to the sector specific details and the number of input socio-economic variables employed in the calculation procedures. As discussed in the method section (Sect. 2), existing global water models use different methodological approaches to estimate sectoral water use. This is also true for the three water models applied in this study. As previously noted, H08 and PCR-GLOBWB determine water use for an aggregated industry sector. However, H08 uses primarily total electricity production, while PCR-GLOBWB uses GDP and total energy consumption in addition to total electricity production. For H08 and PCR-GLOBWB, these variables are used to estimate the future change in water use intensity by constructing the future trend, rather than actually calculating the absolute amount of industrial water use. In contrast, WaterGAP separates water use for thermal electricity production (e.g., technologies and cooling system types) and manufacturing, and uses those for the calculation of absolute amounts of these industrial sub-sectoral water uses for each year. This results in more complex functions where either electricity water use or manufacturing water use can dominate the future change in industrial water use. For example, projected industrial water use is dominated by the manufacturing sector in Brazil, Pakistan, Indonesia, and Mexico, and by the thermal electricity sector in China, the USA, and Canada. In the H08 and PCR-GLOBWB models detailed changes in manufacturing or thermal electricity water use cannot be captured. A simple approach may neglect future dynamic changes in sub-sectoral water use within the industrial sector. For example, SSP scenario narratives correspond to different sources of energy and changes in the economy including the structure of GDP. This may result in large variations of sub-sectoral water use intensity across countries, which can be important in capturing regional water use characteristics.

In addition to the different methodological approaches, we found that the use of different data sets for the reference year (2005) causes a remarkable difference in future amounts of industrial water use. In H08, industrial water use at the reference year (2005) is globally 10 % lower compared to PCR-GLOBWB and 20 % lower than WaterGAP, i.e., meaning that the models start their simulations from a different starting point. The difference among the models is less obvious for the domestic sector (±5 %). H08 and PCR-GLOBWB project the same future trend in industrial water use, however, the use of different data sets for the reference year (i.e., the starting point) immediately impacts the results and subsequent amounts of future water use. This was clearly demonstrated in some countries such as Russia and India. Although we harmonized the model drivers of socio-economics (GDP, population, energy) and assumptions on technological and structural change, the use of the same reference data set was not considered in the WFaS “fast-track” assessment. This is partly due to a lack of available data for many countries of the world on water withdrawals and consumptive use, particularly in industry. Locations of water users, water efficiency technological changes over time, and quantities of water withdrawals are largely unknown, and although the general factors that influence water demand are known, we often do not have enough information to show statistical significance.

H08 and PCR-GLOBWB estimate their initial water withdrawal based on the widely used AQUASTAT data from the FAO. AQUASTAT compiles country reported statistics of sectoral water use including a quality check. In WaterGAP the initial water use for the year 2005 is based on a separate compilation of statistical sources from individual countries. Reasons for apparent differences between these two approaches, both using statistical data reported by countries, were not investigated and are therefore unknown. Improvements in available data could be achieved by bottom-up assessments such as investigation of individual water uses within the sectors and their influence on the total water demand for that sector. For example, household water uses for toilets, showers, washing machines, and dishwashers can be assessed along with technological changes in the appliances leading to improved water use efficiency over time, methods that are being investigated in the WaterGAP modeling framework. For industry the information sources used for water footprinting can be applied to better estimate water uses for different types of industry. Environmental economic accounting systems and water extended input–output modeling can provide data sources of water use intensities across sectors and can be used to assess changes over time in these industries. Applying this at the global scale may be challenging and involve significant data compilation work. Nevertheless, the use of the same reference data set for the start year could be considered in the next water use model intercomparison. Improved information can lead to the use of global water models for policy guidance and assessment of water management.

Using different sets of socio-economic driver variables also results in significant differences. Future trends in industrial water use projections are similar among the three models for developed countries that correspond to the HE-2 classification (e.g., USA and Germany). H08 projects a decreasing trend under SSP1 for those emerging economies that correspond to HE-1 and HE-4. Apparently, projected increases in total electricity production are counterbalanced by assumed improvements in water use intensity due to technological changes. In contrast, PCR-GLOBWB and WaterGAP project a consistently increasing trend under the same scenario due to increasing GDP. However, it should be noted that the composition (sub-sectors) of GDP in the “Sustainability” scenario SSP1 is not known. There are some differences in projected trends between PCR-GLOBWB and WaterGAP, but these are mainly attributable to the difference in sub-sectoral water use calculation (aggregated vs. disaggregated). The use of different socio-economic variables such as GDP and energy consumption creates a different trend in PCR-GLOBWB and WaterGAP compared to that in H08. This was also the case for the domestic sector in which PCR-GLOBWB projects a much higher increase in water use intensity by 2050. GDP projections in the SSP scenarios increase significantly for almost all countries, particularly in emerging economies. The increase in total electricity production is much milder due to improvement in energy use intensity (i.e., higher electricity production per unit energy use), and technological and structural improvement. The calculation of (sub-)sectoral water use intensity using different sets of socio-economic variables should be further investigated.

While the discussion above has focused on the difference in water use projections, there are also many regions where the estimated signals or trends are in agreement across the water models. Figure 6 shows global maps of projected domestic water withdrawals calculated by the three models. Since the projected trends and variability among the models are rather similar under the three SSP scenarios, here we show only the projections under the SSP2 scenario and we refer to Appendix A8 for the results of the SSP1 and the SSP3 scenario. For the domestic sector, the model agreement is rather high for almost all countries under the present condition (CV < 0.3). However, by 2050, the ensemble projections diverge and the model agreement becomes much lower for some countries such as Russia, China, Australia, and some countries in Central Asia (e.g., Afghanistan) and Africa (e.g., Ethiopia).

The model agreement for the industry sector is low (CV > 0.5) for the current conditions in many countries (Fig. 7). By 2050, the spread across the models becomes even wider for many countries in Asia, Africa, and South America by 2050 (CV > 0.75). For both the industrial and domestic sector, the model agreement is particularly high for countries in North America (e.g., the USA), western Europe (e.g., Germany), and Japan both for present condition as well as the future projections (CV < 0.3). These are countries where long time series of measured data do exist. Despite the differences in methodology and input data, the water models produce a smaller range in industrial and domestic water use projections for these countries compared to countries in the developing world and emerging economies. Thus future changes in water use projections of industrialized countries are apparently more robust. We consider the following reasons for attributing a higher confidence in future water use calculations of developed countries: (i) the scenario assumptions (i.e., technological changes according to SSPs narratives) and associated input data sources (e.g., GDP, electricity production, energy consumption) are more consistent with one another; (ii) the future change in socio-economic development is relatively stable so that the change is rather insensitive to the different methodological approaches of the models, and (iii) the input variable of total electricity production (which does not increase as strongly as in the developing world) dominates the calculation of (sub-)sectoral water use intensity for the three models. In addition, another important reason is that data availability is also higher in industrialized countries, where global water models produce their regression equations calculating water use intensity based on data in these areas. Therefore, the regressions are better fits in these areas, and extrapolations to other areas, particularly with extreme growth changes, will result in large extrapolation error.