4.1. Unit Root Test and Cointegration Test
Although the panel data of the PVAR model has reduced the correlation between variables, its timing characteristics determine the possibility of non-stationary stochastic fluctuations; therefore, a unit root test for serial smoothness is performed before establishing the PVAR model. The results are shown in
Table 2.
At a significance level of 5%, the variables odep, lnpcons, lnpgdp, and sav are all non-stationary series. After the first difference, Δodep and Δsav are stationary time series; after the second difference, Δ(Δlnpgdp) and Δ(Δlnpcons) are stationary, that is, odep, lnpgdp, lnpcons, and sav are all integrated of order 2, that is, I(2). In cases where integration of order 2 emerges, the cointegration test analysis of variables is carried out to test whether there is a long-term equilibrium relationship between variables.
In general, the panel data vector autoregressive model (PVAR) is more effective than the panel data vector error correction model (PVEC), and the panel data vector error correction model (PVEC) should be established when there is a cointegration relationship in the panel data; if there is no cointegration relationship, the panel data vector autoregressive model (PVAR) is more effective [
27].
The cointegration test is conducted separately for the two groups formed by the following variables:
lnpgdp,
odep, and
lnpcons;
lnpgdp,
odep, and
sav. This is to test whether there is a cointegration relationship between the two groups of data. This paper adopts the panel cointegration test method, and the results of the group and panel statistics are shown in
Table 3 and
Table 4.
As can be seen from
Table 3, the four statistical groups of Gt, Ga, Pt, and Pa are not significant at the 5% significance level, thus indicating that there is no cointegration relationship between the three variables of {
lnpgdp,
lnpcons,
odep} (i.e., there is no long-run equilibrium relationship). Similarly,
Table 4 shows that there is no cointegration relationship between the three variables {
lnpgdp,
sav,
odep}.
Therefore, this paper establishes a panel vector autoregressive (PVAR) model to empirically analyze the effects of aging on consumption and economic growth, particularly the effects of aging on savings and economic growth for the two sets of variables {lnpgdp, lnpcons, odep} and {lnpgdp, sav, odep}, using panel data from 31 provinces and cities in China from 2000 to 2020, respectively.
4.2. Selection of Lag Order
In this paper, AIC, BIC, and HQIC statistics are used to determine the optimal autoregressive lag order; the optimal lag order of the model is determined based on the order in which AIC, BIC, or HQIC takes the minimum value. When the three are inconsistent, BIC/HQIC tends to choose the more compact model, whereas AIC tends to choose the more complex model; BIC/HQIC is usually superior to AIC [
28]. The results are shown in
Table 5 and
Table 6.
As seen in
Table 5, the PVAR model with the three variables,
lnpgdp,
lnpcons, and
odep, provides the lowest AIC, BIC, and HQIC statistics when the lag order of the variables is three; this indicates that the lag order should be selected as three to establish the PVAR (3) model of “aging-consumption-economic growth”. Similarly, we can see from
Table 6 that the PVAR model with the three variables,
lnpgdp,
sav, and
odep, has the lowest statistics for AIC, BIC, and HQIC when the lag order of the variables is three; therefore, the lag order of three was also selected to establish the PVAR model of “aging–savings–economic growth”.
4.3. PVAR Estimation
As the PVAR model contains both time and individual effects, this paper eliminated them before establishing the PVAR model by applying the cross-sectional mean difference to each variable to eliminate the time effect. Then, the forward mean difference was used to eliminate the individual effects (the Helmert process transformation). A bias in coefficient estimation, caused by the time effect and the individual effect, was thus avoided. In this section, the model was estimated using the Generalized Method of Moments (GMM) [
29]; this involved using the gross regional product per capita as the dependent variable, and the other variables and their third-order lag were used as independent variables, respectively.
4.3.1. PVAR Estimation of “Aging-Consumption-Economic Growth”
A PVAR model including third-order lags was developed to analyze the dynamic effects of aging on residential consumption and economic growth. Panel data concerning the old-age dependency ratio, log of per capita residential consumption expenditure, and log of per capita gross regional product for 31 provinces in China from 2000 to 2020 were used. The estimated results are shown in
Table 7.
As can be seen from
Table 7, the direct relationship between economic growth and aging is not significant when considering household consumption. In terms of the relationship between
lnpgdp and
odep, in light of the changes in the old-age dependency ratio, the dynamic response of the logarithm of the per capita regional product was −0.007 in the first phase and 0.0001 and −0.004 in the second and third phases, respectively; however, the coefficients were not significant. This shows that the direct relationship between economic growth and an aging population is not significant.
An aging population will reduce consumption. In terms of the relationship between lnpcons and odep, in light of changes in the old-age dependency ratio, the dynamic response of the logarithm of per capita household consumption expenditure is negative: −0.008 in the first phase and −0.004 and −0.001 in the second and third phases, respectively. Moreover, the coefficient of the first two phases is significant at a significance level of 5% and 10%. This indicates that as the population continues to age, residents’ consumption will be inhibited, but the effect of this inhibition will gradually decline.
4.3.2. PVAR Estimation of “Aging-Savings-Economic Growth”
A PVAR model including third-order lags was developed to analyze the dynamic effects of aging on national savings and economic growth. Panel data on the old-age dependency ratio, saving rate, and log of per capita gross regional product for 31 provinces in China from 2000 to 2020 were used. The estimated results are shown in
Table 8.
As can be seen from
Table 8, the direct relationship between economic growth and aging is not significant when national savings are considered. After the addition of the variable sav, in light of changes in the old-age dependency ratio
odep, the dynamic response of the logarithm of per capita regional product (GDP) in the first phase is −0.009, and in the second and third phases, it is 0.001 and 0.002, respectively; however, the coefficients are not significant, thus indicating that the direct relationship between economic growth and an aging population is not significant. These findings are similar to the results concerning household consumption.
The direct relationship between national savings and aging is also not significant. In terms of the relationship between sav and odep, the dynamic response of national savings rate was 0.08 in the first phase, −0.076 in the second phase, and 0.159 in the third phase, thus indicating that an aging population has a certain lagged positive effect on national savings; however, the coefficients in the third phase are not significant. This indicates that as people continue to age, the impact on national savings is not significant.
4.4. Analysis of Impulse–Response Function
To test the dynamic relationship between the variables, this paper uses the impulse–response function to study the effect of endogenous variable shocks on the variables and other endogenous variables. The Cholesky orthogonal decomposition of the impulse–response function is very sensitive to the ranking of the variables, and the change in demographic structure reflects the change in the working population; this leads to changes in the level of per capita income, which, in turn, affects consumption and savings. Conversely, economic growth does not immediately cause changes in the demographic structure, which changes relatively slowly; therefore, in the Cholesky decomposition of the impulse–response function, odep, which represents the demographic variables, is ranked first, followed by lnpgdp, the per capita gross regional product, lnpcons, the per capita consumption expenditure, or sav, the national saving rate. The two sets of variables are thus {odep, lnpgdp, lnpcons} and {odep, lnpgdp, sav}.
In this paper, impulse–response function plots are obtained by giving a standard deviation shock to the variables; this was achieved by using Monte Carlo simulations 500 times, and 95% confidence intervals are given.
4.4.1. Impulse–Response Function Analysis of “Aging-Consumption-Economic Growth”
The impulse–response function plot is shown in
Figure 1, which is obtained via a Monte Carlo simulation of the variables {
odep,
lnpgdp,
lnpcons}; this was achieved using a PVAR model with the old-age dependency ratio, log of per capita gross regional product, and log of per capita consumption expenditure.
From
Figure 1, the impact of a shock of an orthogonalized innovation, regarding the effect of aging on economic growth, is 0 in the first period; then, for the remaining periods, the value is negative, and the negative effect is more stable, thus indicating that the shock of an aging population has no effect on economic growth in the same period. However, the negative effect of aging on economic growth is persistent and stable, and aging has a dragging effect on economic growth.
The impulse–response function plot of the old-age dependency ratio on the log of per capita consumption expenditure (third row, first column) shows that the impact of an orthogonalized innovation (standard deviation) shock on the degree of aging and on per capita consumption expenditure has been consistently negative, showing a downward and then upward trend; however, the upward response is weaker and still negative, and it finally produces a very small negative effect, thus indicating that the shock of aging has a continuously negative effect on per capita consumption expenditure in China, to some extent. Nevertheless, the impact of the negative effect gradually becomes weaker.
In addition, considering the indirect path of an aging population → consumption level → economic growth, the graph showing the impulse–response function of the old-age dependency ratio and the logarithm of per capita consumption expenditure (third row, first column) and the graph showing the impulse–response function of the logarithm of per capita consumption expenditure and the logarithm of per capita regional GDP (second row, third column) show that an orthogonalized innovation, with regard to aging, first has a negative effect on the level of residential consumption, which, in turn, has a negative influence on economic growth. Thus, the indirect path of “aging → consumption level → economic growth” shows that the negative effect of aging on economic growth is partly due to the transmission of the negative effect of aging on the household consumption level. In conclusion, an aging population has a negative effect on both consumption and economic growth when considering household consumption, and regarding the indirect path of aging → consumption level → economic growth, aging is not conducive to an increase in consumption level, and thus, it has a negative effect on economic growth.
4.4.2. Impulse–Response Function Analysis of Aging-Savings-Economic Growth
The impulse–response function plot is shown in
Figure 2, which is obtained by the Monte Carlo simulation of the variables {
odep,
lnpgdp,
sav} using a PVAR model with the old-age dependency ratio, national savings rate, and log of per capita gross regional product.
As shown in
Figure 2, considering national savings and using the impulse–response function plot (second row, first column) of the old-age dependency ratio on the log of per capita regional GDP, it is evident that the impact of an orthogonalized innovation shock, regarding the degree of aging on economic growth, continues to be negative; it is followed by an upward trend, but it remains negative, thus indicating that the shock of an aging population leads to a negative change in economic growth. Nevertheless, the impact of the negative effect gradually becomes weaker.
From the impulse–response function plot concerning the old-age dependency ratio on the national savings rate (third row, first column), it is evident that the impact of an orthogonalized innovation shock, with regard to an aging population, on economic growth is zero in the first period, then it becomes positive before decreasing; this produces a very small negative effect, thus indicating that when facing the shock of an aging population, the national saving rate is not affected in the current period, and it has a positive effect on national savings in the short term. However, aging has a negative effect on national savings in the medium and long term.
In addition, considering the indirect path of aging → national savings → economic growth, two pairs of impulse–response function plots (the old-age dependency ratio on the national savings rate (third row, first column) and the savings rate on the log of per capita gross regional product (second row, third column)) show that the shock of orthogonalized innovations on the national saving rate leads to a positive change in economic growth, with a clear upward trend, thus indicating that national saving is beneficial to economic growth. Regarding the indirect impact path, an orthogonalized innovation shock, with regard to aging, has a small positive effect on national savings first, and then, to some extent, it has a positive effect on economic growth; therefore, the indirect path of aging → national savings → economic growth shows that aging has a positive effect on economic growth to some extent, but the overall effect of aging on economic growth is negative, thus indicating that the positive effect of aging on economic growth is small, and an increase in national savings is not enough to directly offset the negative effect of an aging population on economic growth.
In conclusion, regarding the national saving path, aging has a negative effect on economic growth, with an elevated effect on national savings in the short term, followed by a negative effect in the long term; this produces a small positive cumulative effect. In the indirect path of aging → national savings → economic growth, the small positive effect concerning national savings is not enough to offset the direct negative effect of aging on economic growth.
4.5. Analysis of Variance Decomposition
In order to examine the degree of interaction between aging, economic growth, residential consumption, and national savings more precisely, this paper obtains variance decomposition via a Monte Carlo simulation that operated 500 times. Moreover, the contribution of structural shocks to the fluctuation of endogenous variables was also analyzed.
4.5.1. Analysis of Variance Decomposition of “Aging-Consumption-Economic Growth”
The variance decomposition is obtained via a Monte Carlo simulation of the variables {
odep,
lnpgdp,
lnpcons} using the PVAR models established by the old-age dependency ratio, the logarithm of per capita regional GDP, and the logarithm of per capita consumption expenditure. The results of the analysis of variance for the 10th and 20th projection periods are presented in
Table 9.
Regarding the path considering household consumption, the results of the variance decomposition in
Table 9 show that the old-age dependency ratio has a greater impact on its own shock impact, contributing 94.48% of its own variance in the 10th period and slightly decreasing to 89.16% in the 20th period.
The old-age dependency ratio also has stronger explanatory power in terms of the per capita regional GDP growth rate, contributing 17.80% to its variance in the 10th period, thus indicating that 17.80% of the change in economic growth can be directly explained by aging. In the 20th period, it rises to 19.32%, thus indicating that the per capita regional GDP has the greatest impact on its own shock, reaching 68.22% and 67.22% in the 10th and 20th periods, respectively.
The old-age dependency ratio has less explanatory power with regard to the growth rate of per capita consumption expenditure, contributing 8.58% to its variance in the 10th period, thus indicating that 8.58% of the change in per capita consumption expenditure can be directly explained by an aging population. It decreases slightly to 8.31% in the 20th period, thus indicating that per capita consumption expenditure has the greatest impact on its own shock, reaching 69.69% and 60.58% in the 10th and 20th periods, respectively.
4.5.2. Analysis of Variance Decomposition of “Aging-Savings-Economic Growth”
The variance decomposition was obtained from the Monte Carlo simulation of the variables {
odep,
lnpgdp,
sav} using the PVAR models established by the old-age dependency ratio, the logarithm of per capita regional GDP, and the national savings rate. The results of the analysis of the variance decomposition for the 10th and 20th projection periods are presented in
Table 10.
Regarding national savings, the results of the variance decomposition in
Table 10 show that the old-age dependency ratio
odep has a greater impact on its own shock, contributing 97.04% of its own variance in the 10th period and slightly decreasing to 96.34% in the 20th period.
The explanatory power of the old-age dependency ratio on the per capita regional GDP growth rate reaches 4.96% of its variance in the 10th period, thus indicating that 4.96% of the economic growth change can be directly explained by an aging population. It rises to 5.64% in the 20th period, thus indicating that the per capita regional GDP has the greatest impact on its own shock, reaching 75.09% and 70.67% in the 10th and 20th periods, respectively.
The explanatory power of the old-age dependency ratio on the national savings rate contributes 2.05% to its variance in the 10th period, thus indicating that 2.05% of the change in national savings can be explained by an aging population, whereas the change is maintained at 2.03% in the 20th period. National savings have the greatest impact on their own shock, reaching 84.76% and 84.98% in the 10th and 20th periods, respectively.