The Spatiotemporal Elasticity of Age Structure in China’s Interprovincial Migration System

: The supply and demand of labour in the market can often experience profound transformations as a result of an ageing population. This can substantially impact the sustainable development of human society. Since the 1970s, China’s internal migration has continued to increase, but there has been a shift toward an ageing trend since the year 2000. How does the change of age structure interact with socioeconomic development to produce changes in the supply and demand of labour over space and time? This study constructs a spatial dynamic panel data model of interprovincial migration ﬂows in China from 1985 to 2015 in order to quantify the spatiotemporal impacts of age structure on migration. The preliminary results indicate that age structure plays the most important role among regional socioeconomic characteristics of migration, dominated by the large supply, demand, and cross elasticities of labour population. Labour demand and the cross elasticities of total dependency ratio rank second. Comparatively, the total elasticities of regional GDP and wage levels on migration ﬂows are not as signiﬁcant as expected. This study lays the groundwork for identifying the interaction mechanisms of migration systems and provides important insights on regional sustainable development from the perspective of ageing.


Introduction
Population migration is one of the most important geographic processes with respect to its scale and influence over space and time [1,2].It affects the sustainable and circular development of society, culture, and economy through network systems.As the most influential driving factor in the migration system, the supply and demand of labour in the market transforms through changes in the population age structure, which itself is very closely related to migration systems [3][4][5].Over the past several decades, the impact of migration on age structure, education level, the labour market, and gross domestic product (GDP) has been studied qualitatively and quantitatively [6][7][8][9][10][11][12][13][14][15].In developed regions, immigration is expected to slow the decline in the labour force and to reduce the old-age dependency ratio, thereby providing the necessary impetus for socioeconomic sustainability [9].
Since the reform and opening-up policy in the late 1970s, migration within China has been continuously accelerating; this has contributed to rapid economic development and urbanization in the country [16,17].China's population in the 21st century has entered the demographic 'new normal' following the strict family planning measures of the past: the working-age population has decreased but the ageing population has increased [18].Rural-to-urban migration can temporarily slow down the ageing trend in urban areas, but it invariably accelerates the ageing trend in rural areas [19][20][21][22].There is evidence that migration has aggravated the ageing trend in the central and western regions of China while alleviating it in the eastern regions [23][24][25].In the meantime, the change in the age structure has led to shifts in migration patterns, resulting in the appearance of destination clusters in China [26].
Among other factors, labour supply and demand, as significant factors affecting economic growth, are closely related to the interaction mechanisms of migration systems and age structure.Recently, the Central Committee and State Council of the Communist Party of China (CPC) [27] have clearly stated that it is necessary to deepen reform and opening-up and to remove the institutional barriers that hinder the free flow of production factors in order to lay a solid foundation for high-quality development.Most social studies on the age structure in China have focused on population ageing and its consequences, including changes in the old-age dependency ratio, pension problems, household consumption [28][29][30][31], and the impact of population ageing on the labour market [32][33][34][35][36][37].However, little attention has been paid to the temporal and spatial changes in labour supply and demand in the migration system under the influence of regional socioeconomic characteristics.
Owing to the interaction between the supply and demand of labour and regional socioeconomic factors, as well as their dependence over time and space, we employed a spatial dynamic panel data model to study the impact of age structure on China's interprovincial migration flows from 1985 to 2015.We selected labour population ratio and total dependency ratio as exploratory variables, as well as regional GDP, population size, real wages, and railway travel time between provincial capitals as control variables.In addition, we incorporated temporal autocorrelation, spatial autocorrelation, and spacetime diffusion of migration flows as endogenous variables to investigate path-dependent migration processes.The Bayesian Markov chain Monte Carlo (MCMC) approach was used to estimate model parameters.In order to interpret the impacts of regional socioeconomic factors on migration flows, we adopted the concept of elasticity in economics.Supply or demand elasticity can be defined as the percentage of change in migration flows arising from a 1% change in a certain explanatory variable at an origin or a destination area.Similarly, cross elasticity is the percentage of change in migration flows in the surrounding areas arising from a 1% change in a certain explanatory variable in an area.At the same time, we can derive their temporal decompositions in the contemporaneous, short-term, and long-term periods.The empirical results have implications for elucidating the resilience of regional migration systems and sustainable regional development.
The remainder of this paper is organised as follows.Section 2 presents a literature review on the ageing population, spatiotemporal dependence of migration systems, and the impact of socioeconomic variables in migration systems.Section 3 describes the data and methodology and provides a mathematical expression for the space-time elasticity of supply and demand.Section 4 estimates the elasticity of supply and demand, and cross elasticity as well as the coefficients of the regional characteristics of origin and destination regions.Section 5 discusses the current findings alongside related studies.Finally, conclusions are drawn and future research directions are recommended in Section 6.

Ageing Population and Socioeconomic Development
Many studies focused on regions with severe population ageing, such as Europe and Japan, showed that ageing has adversely affected economic growth, savings, and investment [38,39].Other studies focused on East Asia confirmed the role of demographic dividends [40,41].Bloom et al. [42][43][44] proposed that the role of the demographic dividend depends on a certain economic and social policy environment; the demographic structure and economic development of China and India can be attributed to the demographic dividend.
Since China became an ageing society in 2000, the problem of its population ageing has attracted increasing attention.Many research results showed that the sufficient supply of labour is one of the factors of China's rapid economic growth, and that the contribution of population structure, caused by the increase in the proportion of working-age population and the decline of the population-dependency ratio, to China's economic growth is in the range of 1/6-1/3 [45][46][47].As the population is ageing, China's economic growth will face the threat of a shortage of labour [48,49].

Spatiotemporal Dependence of Migration Systems
Migration is a spatiotemporally path-dependent process [50,51].The time dependence of migration flows has been proven to have a positive effect by dynamic panel data models [52][53][54].Curry [55] was the first to identify the interaction between spatial dependence and distance decay among gravity flows.Recently, different spatial dependencies among migration flows have been quantified in spatial econometric interaction models [56][57][58].Rogers et al. and Tobler [51,59] pointed out that both the age structure and the spatial pattern of migration reveal migration rules.The use of origin, destination, and network effects have been proposed for the evaluation of the influence of regional factors on migration flows [60,61].Intrigued by the growing availability of large data sets, coupling time and space dimensions is a major focus of panel data analyses [62][63][64][65].In contrast, network autocorrelation within cross-sectional or panel migration flows can be modeled by eigenvector spatial filtering techniques, decomposing the error terms into a network autocorrelated component and a nonnetwork component [66][67][68].However, the eigenvectors need to be chosen using a stepwise procedure, which complicates the multiple regression and ignores the explicit spillover effects in geography.The characteristics of the flow field of interprovincial migration were analyzed by revealing the regional differentiation of contemporary Chinese migration [69,70].The significant spatial dependence through spatial econometric models and spatial dynamic panel models in Chinese interprovincial migration flows have been explored [71].
The elasticity of supply and demand refers to the degree of response of supply or demand sides to changes in price in microeconomics or to changes in the unemployment rate, real wages, inflation, and taxes in labour economics, respectively [72][73][74][75].Many studies have attempted to predict future trends by building mathematical models or setting labour participation rates based on historical data of labour supply [74,76,77].Peterman [78] investigated the gap between the microeconometric estimates of Frisch labour supply elasticity (0-0.5) and the macroeconometric values of 2-4 and demonstrated that the estimates of macro elasticity are sensitive to the estimation procedures and the exclusion of the elderly.However, few studies have considered the time-space dependence of the migration system and the influence of various factors on labour supply and demand elasticity [72].

The Impact of Socioeconomic Variables in Migration Systems
In terms of the impact of age structure on migration systems in the context of socioeconomic sustainable development, attention has been paid to labour and dependency ratio [79,80].Scholars have conducted a lot of research on the ageing population and migration, including qualitative and quantitative studies on the impact of migration on regional age structure and exploring the interaction between migration and age structure, education level, and labour [6][7][8][9][10].Golini [8] estimated the impact of migration on the working age population ratio, population size, and GDP in Italy and found that migration affects the economic gap between the North and South.In the studies of the interaction between age structure and migration systems, scholars have explored the impact of ageing on the intensity of migration and its causes, as well as the different impacts of migration on age structure [9][10][11].In developed countries, immigration has been expected to slow down the decline in their labour forces and to reduce their old-age dependency ratio since it had a higher labour force participation rate, paid proportionately more in indirect taxes, and provided the necessary impetus for social and economic sustainability [9,11].
Population size is often chosen as the basic explanatory variable to characterize regional mass in spatial interaction models [81].In addition, regional economic development may inspire people to move and enhance China's interprovincial migration [82,83].Since the late 1970s, unbalanced regional economic development in China has led to huge migration flows [84][85][86].It has been found that people would migrate to seek better job opportunities and to improve their quality of life [87,88].For individuals, the migration of family members can increase absolute income and improve the relative socioeconomic status of the families in local areas [89].

Data Sources
The panel data of China's interprovincial migration from 1985 to 2015 were collected from three national population censuses and three 1% population sample surveys under the National Bureau of Statistics of China [90,91]; these censuses and surveys excluded Hong Kong, Macao, and Taiwan.The data of Chongqing Municipality and Hainan Province before their establishment were obtained by Kriging interpolation from neighbouring provinces.The total number of interprovincial migration flows was 5580 (=31 provinces × 30 provinces × 6 time periods; one time period equals 5 years), without considering intra-provincial migration flows.The explanatory variables for origins and destinations included regional gross domestic product (GDP), population size, labour population ratio, total dependency ratio, real wages, and railway time distance between the capital cities of the two provinces (Table 1).
To avoid endogeneity, regional socioeconomic variables were selected from the China Statistical Yearbook at the beginning of the corresponding period, lagging migration flows by five years [56,92].The shortest railway travel time by railway was obtained from the China Railway Timetable during the same time period [93][94][95][96][97][98].Considering that the GDP data provided by China's provincial statistical bureaus is incomplete, we chose to use GDP from 1980, which is complete, as the base period price for the calculation of constant price GDP to eliminate the impact of price fluctuations.Real wage is the ratio of the average wage level of an urban employee to the consumer price index (CPI).The labour population ratio was the proportion of the working-age population aged 15-64 years to the total population, representing the age structure.The total dependency ratio was the ratio of the nonworking population (children younger than 14 years old and the elderly older than 65 years) to the actual labour population, which is one of the indicators used to measure the impact of age structure on socioeconomic development.Based on changes in the age structure of the population, the total dependency ratio has great socioeconomic significance [10,99].
In this study, we use five explanatory variables based on previous migration studies.Since the late 1970s, China has continued to generate migration flows under an imbalanced regional economic development [19,83].The increase of regional GDP might decrease the intention to leave but attract more foreign people to settle down.Regional population size has been demonstrated as an important factor during interprovincial migration system in China [83].A region with a large population size may generate substantial migration flows from or to the region itself.According to census and survey data, the labour force occupies a dominant position in the total migrant population [100][101][102][103][104][105].When the working-age population of an area increases, labour tends to achieve higher income level and less social competition by migrating to other areas where labour is in short supply.An increase in the total dependency ratio in a region would increase the costs of living, and people tend to move to other regions to seek higher incomes and to reduce the strenuousness of their lives.More migrants choose to leave their hometowns for higher wages and better lives, while fewer people are willing to leave places with higher incomes.

Spatial Dynamic Panel Data Model
In the migration network shown in Figure 1 used to measure the impact of age structure on socioeconomic development.Based changes in the age structure of the population, the total dependency ratio has great soc economic significance [10,99].
In this study, we use five explanatory variables based on previous migration studi Since the late 1970s, China has continued to generate migration flows under an imb anced regional economic development [19,83].The increase of regional GDP might d crease the intention to leave but attract more foreign people to settle down.Regional po ulation size has been demonstrated as an important factor during interprovincial mig tion system in China [83].A region with a large population size may generate substant migration flows from or to the region itself.According to census and survey data, the bour force occupies a dominant position in the total migrant population [100][101][102][103][104][105].Wh the working-age population of an area increases, labour tends to achieve higher incom level and less social competition by migrating to other areas where labour is in short su ply.An increase in the total dependency ratio in a region would increase the costs of l ing, and people tend to move to other regions to seek higher incomes and to reduce t strenuousness of their lives.More migrants choose to leave their hometowns for high wages and better lives, while fewer people are willing to leave places with higher incom

Spatial Dynamic Panel Data Model
In the migration network shown in Figure 1, the migration flow from the origin (O to the destination (D1) was spatially dependent on several flows.For example, flows fro the neighbouring areas (e.g., O2) of the origin (O1) to the destination (D1) would exhi origin-based spatial dependence; similarly, flows from the origin (O1) to the neighbouri areas (e.g., D2) of the destination (D1) would show destination-based spatial dependen flows from the neighbouring areas (e.g., O3) of origin O1 to the neighbouring areas (e.D2) of destination D1 would present flow-based spatial dependence [56,106].Based on the traditional gravity model, we constructed a spatial econometric interaction model that captured the network autocorrelation phenomenon between migration flows in one time period [56,107,108].
where y is an N (= n 2 ) × 1 column vector of regional gross migration flows from an n×n flow matrix for n regions; X o and X d are the explanatory variables at origins and destinations, respectively, formed by an n×k explanatory variable matrix X and an n×1 vector of one (ι n ) through the Kronecker product ( ×1 column vector from the n×n matrix of distances between origins and destinations; W o , W d , and W w represent the origin-, destination-, and flow-based network weight matrices, respectively, constructed by the identity matrix I n and the n × n spatial weight matrix W through Kronecker product operations The matrix W defines the spatial adjacency relationship of different geographic objects based on the simple first-order contiguity criteria.For Hainan, an island that once belonged to Guangdong Province, we designated Guangdong as its neighbour.The model simultaneously considers three types of network effects between the migration flows through W o y, W d y, and W w y.The error term vector ε is assumed to be identically distributed with a zero mean and variance σ 2 I N ; α is the intercept to be estimated; β o and β d reflect the impacts associated with the origin-and destination-specific characteristics, respectively; γ denotes the impact of distance between different regions; and ρ o , ρ d and ρ d represent the degrees of origin-, destination-, and flow-based spatial dependence, respectively.Any migration flow is affected not only by the origins/destinations and neighbouring locations [56,[107][108][109], but also by the previous and surrounding flows over time and space.Therefore, a spatial dynamic panel model can be constructed as follows [71]: where Y is the NT × 1 dependent variable vector of migration flows for each time period;

Interpreting the Space-Time Effects of Migration Processes
The change in the th element in the area  ( ) would affect its outflows and inflows as well as the surroundings; this produces secondary outflows and inflows, potentially including feedback to the original area [57,61].Therefore, all the effects of the change in  on the outflows of area i are labelled origin effects, including (1) the direct impact on the outflows of the area, and (2) changes in other regional factors caused by the change in  that may have feedback effects on the region itself.Similarly, the destination effects of the change in  on the inflows to area  can be obtained.Owing to spillovers, changes in  could potentially affect all other flows that are not related to area ; these can be labelled as network effects.In addition, the intraregional effects arising from the change in  can also be obtained.The total effect is the summation of all the above effects.
By introducing the time dimension into the spatial econometric interaction model, we obtained the dynamic origin, destination, network, and intraregional effects across different time periods as follows.

Interpreting the Space-Time Effects of Migration Processes
The change in the rth element in the area i (x r i ) would affect its outflows and inflows as well as the surroundings; this produces secondary outflows and inflows, potentially including feedback to the original area [57,61].Therefore, all the effects of the change in x r i on the outflows of area i are labelled origin effects, including (1) the direct impact on the outflows of the area, and (2) changes in other regional factors caused by the change in x r i that may have feedback effects on the region itself.Similarly, the destination effects of the change in x r i on the inflows to area i can be obtained.Owing to spillovers, changes in x r i could potentially affect all other flows that are not related to area i; these can be labelled as network effects.In addition, the intraregional effects arising from the change in x r i can also be obtained.The total effect is the summation of all the above effects.
By introducing the time dimension into the spatial econometric interaction model, we obtained the dynamic origin, destination, network, and intraregional effects across different time periods as follows.
where Q is the NT × NT matrix, which implies a spatiotemporally dependent process.Furthermore, we rewrite Q −1 as a triangular matrix [64]: where D s is the time-space diffusion filter for different time periods.We only need to calculate B −1 and C to analyse the partial derivative impacts on migration flows for any time horizon.Specifically, the change in x r i in the t-th time period has varying influence on migration flows in the t + s-th time period.The total effect (TE) of migration flows on a transitory change in the r-th variable at the t + s-th time period is as follows: . . .
where Jo i is an n × n matrix of zeros with the ith column equal to ι n , and Jd i is an n × n matrix of zeros with the ith row equal to ι n .When TE is known, the scalar instantaneous total effect (te) in different time periods can be calculated as te = 1 n 2 ι n 2 .TE.ι n .Similarly, scalar measures of origin effects (oe), destination effects (de), intraregional effects (ie), and network effects (ne) can be based on oe = 1 IE.ι n , and ne = 1 n 2 ι n 2 .NE.ι n , where OE, DE, IE, and NE are the n 2 × n matrices defined as follows [71]: Sustainability 2023, 15, 8001 where Jo i ( Jd i ) is the n × n matrix adjusting the element (i, i) in Jo i (Jd i ) to zero; Ji i is the n × n matrix of zeros with a '1' in element (i, i); Jn i is the n × n matrix with the elements of the i-th row and the i-th column of 0, while the others are 1.
At the same time, for the accumulated total effects (CTE) during period t ∼ t + T, the calculation formula is expressed as follows: . . .
When T = 0, only one period is included in this time period (s = 0).The corresponding time spillover is 0, and spillover exists in the form of spatial effects.When T→+∞, the cumulative spatiotemporal diffusion is ∑ T s=0 D s = (B + C) −1 .The corresponding instantaneous total effects tend to 0, being the long-run overall effect toward an equilibrium state.Similarly, accumulated origin effects (COE), destination effects (CDE), intraregional effects (CIE), and network effects (CNE) can be obtained in T time periods.

Model Parameter Estimates
The Bayesian MCMC sampling method, based on the spatial econometrics toolbox (jplv7) in MATLAB, was used to estimate the parameters of the spatial dynamic panel data model [110].We sampled 300,000 draws, with the first 290,000 omitted for burn-in, to ensure the accuracy of the estimates.In addition, we initialised three chains to verify whether the MCMC samples converged.Table 2 lists the parameter estimates for the model.The significant and positive origin-, destination-, and flow-based spatial dependencies represent three different migration paths.In the origin-based path, the migration flows would promote flows from the neighbouring origins to the destination.Similarly, the destination-based path from the origin to the neighboring destinations is also strengthened.Comparatively, the extent of flows from neighbouring origins to neighbouring destinations is relatively weaker.The central and western regions in China have always been the main origins of the outflow, further supporting the largest spatial dependence (ρ o = 0.4167).Migration flows exhibit robust time dependence [111,112].The significant and positive time-dependent coefficient (φ = 0.5990) also indicates that past flows would have a strong impact on existing and future migration.Significant and negative network-dependent parameters indicate that spatiotemporal interactions have opposing effects on the diffusion of migration.The minimum parameter (θ w = −0.2339)might reflect that current migration flows might be greatly suppressed by flows between past neighboring origins and neighboring destinations, reflecting the self-regulation of a complex migration system.

Analysis of the Elasticity of Supply and Demand
Table 3 shows the elasticity estimates of migration flows for explanatory variables in the spatial dynamic panel data model.The 't 0 ' rows are the contemporaneous effect of migration flows to changes in regional characteristics; the rows between t 1 and t 5 are the period-by-period impact on migration flows to changes in regional characteristics at different subsequent periods; the last rows ' . . . . . .' mean the cumulated long-run effects.The 'Mean' columns are the means of the instantaneous period-by-period effects, and those 'Cumulative' are the cumulated effects from the initial to the current period.At the same time, the elasticity of the supply and demand, as well as cross elasticity, were also listed.Over time, the elasticities of each time period gradually decrease to zero, and the corresponding cumulative elasticities tend to be in long-run equilibrium in the future.

Labour Population
Table 3 shows that the supply and demand elasticities of labour population on interprovincial migration flows were the largest among all explanatory variables.A 1% increase (or decrease) in the proportion of the working-age population in a typical region would result in a 2.168% increase (or decrease) in the outflows and a 2.2721% increase (or decrease) in the inflows in the same period.Over time, the degree of these period-by-period influences gradually weakened.Finally, a 1% increase in the proportion of the working-age population would accumulate a nearly 4.90% increase in the outflows and a 4.32% increase in the inflows in the long run.
The cross elasticity of labour population has a huge impact on the evolution of the migration system.A 1% increase in the proportion of the working-age population in a typical region would result in a 12.49% increase in the migration flows of other regions in the entire network in the contemporaneous period, which would exceed its cumulative supply elasticity in the long run.However, as time goes by, the cross elasticity continues to decrease, indicating that the influence of the labour population on other regions gradually weakens as the working-age people emigrate.In the long run, its cross elasticity would eventually result in a 7.71% increase in the entire migration network.
Total elasticity represents the overall impact of changes in the age structure on migration.According to Table 3, China's interprovincial migration is extremely sensitive to age structure.If the proportion of the labour population in a typical region increases (or decreases) by 1%, interprovincial migration flows would increase (or decrease) by 16.93% over the same period, reaching 17.46% in the first time period.Subsequently, the transitory impact turns into negative, causing the accumulated impact decreases to an equilibrium state of 16.93% in the long run.Overall, a small change in the age structure of the population in a region would have a significant impact on the overall migration flows.

Total Dependency Ratio
The increase in the total dependency ratio has greater cross elasticity than its elasticity of supply and demand, thereby generating a significant impact on the migration system.A 1% increase in the total dependency ratio of a typical region would cause other immigrants throughout the network to increase by 2.68% at the contemporaneous time period.Over time, the cross elasticities gradually decreased.Owing to the interaction of time and space, changes in the total dependency ratio in one area would have an attenuated impact on migration flows to other areas in a certain period, finally reaching to an equilibrium state of 1.83% change in the long run.
Generally, if the total dependency ratio of a typical area increases (or decreases) by 1%, the migration flows would increase (or decrease) by 3.50% during the same period and result in a change of 3.36% in the long run.For a typical region, the impact of a change in the total dependency ratio would still have an important effect on the current and future population mobility even if the change was less than that in the age structure.
In terms of labour supply, total dependency ratio significantly inhibits advanced and rationalised manufacturing structures [113].China's effective labour supply and labour mobility would be negatively affected by ageing to varying degrees, posing huge challenges to Chinese society in the future.However, with respect to demand, the total dependency ratio has an increasing impact on the migration flows.In the long run, a 1% increase in the total dependency ratio might create an increase of 1.34% in migration flows with respect to demand, producing an impact 7.2 times that of supply.
The total dependency ratio has a negative correlation with household consumption (especially the child dependency ratio) [29].This means that an increase in the total dependency ratio in a region, coupled with a certain lag in the migration system, would increase the pressure on the local labour force and the costs of living.In the short term, people tend to move to other regions to seek higher incomes and to reduce the strenuousness of their lives.As a result, this has an impact on migration.At the same time, owing to the decrease in the absolute amount of the labour force in the area and a certain degree of migration, a large amount of labour demand is generated, reducing the competition for local resources.The government should implement certain policies to attract and retain migrants.This increases immigration and results in the dynamic balance of the system.
In recent years, the total dependency ratio in China has been fluctuating between 34% and 45% due to the incredibly low birth rate and the resulting reduction in the child dependency ratio, as shown in Figure 3.The total dependency ratio reached a turning point in 2010.The decline in the child dependency ratio was offset by an increase in the elderly dependency ratio, leading to a continuous increase in the total dependency ratio.Because the absolute labour supply has been determined and will continue to grow negatively for a long time, the misalignment of labour supply and demand might cause a scarcity of low-end labour resources and a surplus of high-end labour resources [114].Although individuals continue to move into the labour force in the dynamic migration system, the total dependency ratio still exhibits an upward trend.In recent years, the total dependency ratio in China has been fluctuating between 34% and 45% due to the incredibly low birth rate and the resulting reduction in the child dependency ratio, as shown in Figure 3.The total dependency ratio reached a turning point in 2010.The decline in the child dependency ratio was offset by an increase in the elderly dependency ratio, leading to a continuous increase in the total dependency ratio.Because the absolute labour supply has been determined and will continue to grow negatively for a long time, the misalignment of labour supply and demand might cause a scarcity of lowend labour resources and a surplus of high-end labour resources [114].Although individuals continue to move into the labour force in the dynamic migration system, the total dependency ratio still exhibits an upward trend.

Other Explanatory Variables
Population size is often the basic explanatory variable used to characterise regional mass in traditional gravity models.Migration flows are directly proportional to the population size of the origins and destinations if other explanatory variables remain unchanged.The significant and positive supply and demand elasticities in Table 3 indicate that population size exerts strong push and pull effects on China's interprovincial migration flows from 1985 to 2015.Comparatively, its supply elasticity is significantly greater than its corresponding demand elasticity, meaning that the promotion effects brought about by the increase in population size are stronger than the agglomeration effects.In addition, a change in the population size in a typical region might also affect migration flows in other regions through the network over space and time.If the population size of a typical region increases by 1%, migration flows between other regions would increase by 1.58% during the same time period, which exceeds the growth of outflows from and inflows to the region itself.Over time, the cross elasticity gradually decreased, eventually resulting in an impact of 0.93% on interprovincial population movement.The continuing increase in the number of immigrants exacerbates regional job competition, thus gradually decreasing the number of immigrants, with their attenuating willingness coupled with time lag.

Other Explanatory Variables
Population size is often the basic explanatory variable used to characterise regional mass in traditional gravity models.Migration flows are directly proportional to the population size of the origins and destinations if other explanatory variables remain unchanged.The significant and positive supply and demand elasticities in Table 3 indicate that population size exerts strong push and pull effects on China's interprovincial migration flows from 1985 to 2015.Comparatively, its supply elasticity is significantly greater than its corresponding demand elasticity, meaning that the promotion effects brought about by the increase in population size are stronger than the agglomeration effects.In addition, a change in the population size in a typical region might also affect migration flows in other regions through the network over space and time.If the population size of a typical region increases by 1%, migration flows between other regions would increase by 1.58% during the same time period, which exceeds the growth of outflows from and inflows to the region itself.Over time, the cross elasticity gradually decreased, eventually resulting in an impact of 0.93% on interprovincial population movement.The continuing increase in the number of immigrants exacerbates regional job competition, thus gradually decreasing the number of immigrants, with their attenuating willingness coupled with time lag.
According to Table 3, regional GDP encourages people to migrate to a certain extent.Its elasticity of supply was negative: within the same period (T = 0), a 1% increase in regional GDP would decrease outflows by 0.21% on average.The overall cumulative supply elasticity would mean that the development of the regional economy may inhibit 0.58% of outflows in the long run, approximately 2.7 times that of the contemporaneous period.The positive demand elasticity of GDP indicates that the improvement in the regional economic level would attract additional immigrants; its demand elasticity is significantly weaker than the supply elasticity in the contemporaneous period or in the long run.Comparatively, its cross elasticity is weaker than its supply elasticity in absolute values but larger than its demand elasticity in the contemporaneous period, indicating nonnegligence effects on the other regions.Overall, regional GDP might initially promote migration flows, but it has lagged negative effects over time.
Table 3 shows that the estimated elasticities of real wages were no longer prominent, reflecting that wage levels were not as important as expected during the migration process.Real wages would reflect the income and consumption levels of residents.Developed areas with higher wage levels often face higher consumption and the serious hardship of survival, making them less attractive to immigrants.At the same time, with the support of national policies, remote and border areas with higher wages, such as Tibet, are not the main destinations of Chinese interprovincial migrants; this is in contrast with the expectation that high-wage areas generally attract additional immigrants [115].In general, an increase in real wages reduces residents' motivation to emigrate and encourages increased immigration.

Discussion
Compared with previous studies on interprovincial migration in China [47,48,[69][70][71], this study highlights the crucial role of age structure and its spatiotemporal interaction in migration.The mechanism of interregional migration is also confirmed: larger population sizes and higher economic levels provide the basis for emigration, and labour is the largest driver of migration, while distance decay effects make people choose to move to spatially proximate areas; this is consistent with previous studies [47,48,71].In addition, we focus on the significant network autocorrelation in China's interprovincial migration; this means that the choice of destination is not only influenced by some factors, such as the economic level of the destination, but also by the relevant ones of its surrounding areas.
These findings are also supported by international migration studies.Compared with Latin America, where the political and economic environment is unstable, China's economic development and the demographic dividend have led to the effective redistribution of labour resources [42][43][44].In developed European countries, labour supply and demand and wage levels were seen as dominant factors in migration processes [3,4,8,9].When facing the problem of an ageing population, the competition for labour and talents between regions becomes increasingly prominent.It is necessary to pay critical attention to the change of age structure and formulate coping policies that will help carry out the regional allocation and efficient use of increasingly scarce labor resources.
Underdeveloped areas with large populations can take advantage of labour and land costs to develop labour-intensive industries.Areas in need of labourers should adjust immigration policies to promote industrial transformation and upgrading in order to develop modern service industries and advanced manufacturing.At present, with the slowdown of China's economic growth and the transformation of industrial structure, a large number of jobs still accumulate in some developed regions; meanwhile, the cost of living has become an obstacle to migration and further development of regional economy [48,49].Seeking a balance among migration, economic growth, and the cost of living will be the focus of attention in the near future.

Conclusions
Globally, as the problem of an ageing population is becoming increasingly prominent, the change in age structure plays a crucial role in regional socioeconomic growth and sustainable development.Considering the complex space and time interactions in migration processes, this study explored the path-dependent migration processes through a spatial dynamic panel data model.The migration flows of 31 provinces in mainland China between 1985 and 2015 were used as the dependent variable, labour population ratio and total dependency ratio were explanatory variables used to proxy for the status of the ageing population, and regional socioeconomic characteristics, including regional GDP, population size, real wages, and railway travel time between provincial capitals were used as the control variables.In addition, we also incorporated temporal autocorrelation, spatial autocorrelation, and space-time diffusion effects of the dependent variable of migration flows to the inherent mechanisms in migration.The Bayesian MCMC approach was employed to compute the space-time dependencies of the migration processes.Based on the parameter estimates, we obtained the supply, demand, and cross elasticities of age structure (labour population ratio and total dependency ratio) as well as other regional characteristics, and decomposed them into contemporaneous, short-term, and long-run effects in time dimension.
First, migration flows are highly sensitive to the age structure of a population.A small change in the labour force might have a huge impact on mobility, not only in the elasticities of supply and demand but also in the cross elasticities through the network, thereby affecting the entire migration system.As a reflection of the relationship between the economically dependent groups (i.e., children, students, and retirees) and the working-age population, the elasticities of supply and demand of the total dependency ratio exhibits a significant trend different from that of labor population ratio, reflecting the economic burden and strenuous life of the working forces.By observing these two variables, it can further formulate effective policies for sustainable social development in terms of migration and employment.Second, regional economic variables (GDP and real wages) at origins have a negative impact on the migration system, as expected, which further verifies the common belief that people migrate to improve their livelihoods.However, their total elasticities in the long run are very small, reflecting that the responses of migration to the changes in regional economic development and individual income levels are not as significant as our expectations.
Finally, cross elasticities play an important role in migration processes, counting for over 70% in the contemporaneous period and over 40% in the long run, which can only be estimated by incorporating the complex spatiotemporal dependencies among migration flows in the space-time models.In addition, the decomposition of cross elasticities makes it possible to understand the spillover effects of labour movement over space and time, which cannot be observed in traditional gravity models.The significant impact of cross elasticities on the migration system reflects the stability of the system in the process of sustainable development of society and provides a reference for policy making.
This study attempts to apply a spatial dynamic panel data model to quantitatively analyse the spatiotemporal responses of internal migration flows to changes in the age structure and other regional socioeconomic characteristics.It would lay the groundwork for identifying the interaction mechanisms of migration systems and provide theoretical guidance on regional sustainable development.However, in this study we only considered the global spillover effects of the lagged dependent variables over space and time; did not include model uncertainty in selecting different explanatory variables and model structures.
In the future, we will focus on labour resources and population ageing to identify the elderly or working-age population in migration systems at the city or county levels and include model uncertainty for further analysis.
, the migration flow from the origin (O 1 ) to the destination (D 1 ) was spatially dependent on several flows.For example, flows from the neighbouring areas (e.g., O 2 ) of the origin (O 1 ) to the destination (D 1 ) would exhibit origin-based spatial dependence; similarly, flows from the origin (O 1 ) to the neighbouring areas (e.g., D 2 ) of the destination (D 1 ) would show destination-based spatial dependence; flows from the neighbouring areas (e.g., O 3 ) of origin O 1 to the neighbouring areas (e.g., D 2 ) of destination D 1 would present flow-based spatial dependence [56,106].

Figure 1 .
Figure 1.Chart showing the spatial dependence structure of migration flows on a cross section.

Figure 1 .
Figure 1.Chart showing the spatial dependence structure of migration flows on a cross section.
and (I T ⊗ W w )Y are three spatial lags of origin-, destination-, and flow-based dependent vectors at each time period, respectively; Y −1 is the first-order time lag of migration flows; and (I T ⊗ W o )Y −1 , (I T ⊗ W d )Y −1 , and (I T ⊗ W w )Y −1 are three corresponding spatial diffusions of the time lagged migration flows, respectively.X o and X d are the NT × k origin/destination-specific covariates with the associated k×1 parameter vectors β o and β d .Similarly, g is an NT ×1 column vector from the T time periods of the n×n matrix of distances between origins and destinations associated with the coefficient of γ. (ρ o , ρ d , ρ d ) and (θ o , θ d , θ d ) represent the degrees of spatial dependence and diffusion effects, respectively.Here, we consider the heterogeneity of migration flows through N (= n 2 ) individual random effects (µ N ) and T temporal random effects (ξ T ).The NT × 1 vector ε represents normally distributed error terms.The model contained significantly richer information and increased the accuracy of the model parameters to a certain extent.The complex relationship between panel migration flows is shown in Figure2.

Figure 2 .
Figure 2. The chart for temporal dependence, spatial dependence, and space-time diffusion structure of panel migration flows.(Gray arrows indicate temporal dependence, solid black arrows indicate spatial dependence, and black dotted arrows indicate space-time diffusion structure of panel migration flows.)

Figure 2 .
Figure 2. The chart for temporal dependence, spatial dependence, and space-time diffusion structure of panel migration flows.(Gray arrows indicate temporal dependence, solid black arrows indicate spatial dependence, and black dotted arrows indicate space-time diffusion structure of panel migration flows.)

Sustainability 2023 ,
15,  x FOR PEER REVIEW 12 of 18 local resources.The government should implement certain policies to attract and retain migrants.This increases immigration and results in the dynamic balance of the system.

Figure 3 .
Figure 3. Age dependency ratio in China from 2001 to 2019.

Figure 3 .
Figure 3. Age dependency ratio in China from 2001 to 2019.

Table 1 .
Description of explanatory variables.

Table 2 .
Parameter estimates for the spatial dynamic panel data model.

Table 3 .
Space-time elasticity estimates for the spatial dynamic panel data model.