2.1 Research hypothesis and validation model

Drawing on Giulietti et al. (Reference Giulietti, Wahba and Zenou2018) and Patacchini and Zenou (Reference Patacchini and Zenou2008), the study first develops a dyad model to express the process through which social network ties affect individual pension asset allocation decisions. Theoretical hypotheses are then derived.

To begin with, there is one and only one strong ties association for all individuals, and they all treat each other as the only strong ties association. They belong to the same strong ties combination, and the ties associations outside the strong ties combination belong to the weak ties. Individuals have only two states: purchased commercial pension insurance products (i.e., state 1) and not purchased (i.e., state 0), so that at any moment the aggregate consists of three types of strong ties combinations:

d ₂-Individuals in this combination have all purchased commercial pension insurance products;

d ₁-Only one individual in this combination has purchased a commercial ped ₂nsion insurance product;

d ₀-None of the individuals in this combination have purchased commercial pension insurance products.

At the time of t, d _0t, d _1t, and d _2t denote the ratio of the number of d ₀, d ₁, and d ₂ combinations to the population respectively. Then it has

(1)$$d_{0t} + d_{1t} + d_{2t} = \displaystyle{1 \over 2}$$

by the definition of dyad model. Let the purchase rate of commercial pension insurance products in population at this time be m _t and m _t ∈ [0, 1], then the ratio of population who do not purchase commercial pension insurance at this time is 1 − m _t. There are:

(2)$$\left\{{\matrix{ {m_t = 2d_{2t} + d_{1t}} \cr {1-m_t = 2d_{0t} + d_{1t}} \cr } } \right.\;$$

Individuals who do not purchase commercial pension insurance products may either not be exposed to the appropriate information or may receive the appropriate information, but the information is not sufficient to motivate their purchase behavior. The influence process of social ties interaction in an individual's commercial pension insurance purchasing decision is reflected in the social activities of individuals who meet each other. Individuals' social activities can be categorized into two types: strong ties social interaction and weak ties social interaction. At t time, the probabilities of matching individuals with strong and weak ties are ω and λ respectively, which are set as exogenous constants. The probability of individuals receiving positive information about commercial pension insurance products from the weak ties network depends on the purchase rate of commercial pension insurance m _t. The probability of individuals receiving positive word-of-mouth from the strong ties network depends on the satisfaction rate of individuals who have already purchased the commercial pension insurance product (depicted by 1 − q, q is the commercial pension insurance unsubscribe rate).

From time t to t + dt, the net inflows for each combination are:

(3)$$\left\{{\matrix{ {\mathop {d_{2t}}\limits^ \bullet{ = } \omega ( {1-q_t} ) d_{1t} + \lambda m_td_{1t}-2q_td_{2t}} \hfill \cr {\mathop {d_{1t}}\limits^ \bullet{ = } 2q_td_{2t} + 2\lambda m_td_{0t}-\omega ( {1-q_t} ) d_{1t}-\lambda m_td_{1t}-q_td_{1t}} \hfill \cr {\mathop {d_{0t}}\limits^ \bullet{ = } q_td_{1t}-2\lambda m_td_{0t}} \hfill \cr } } \right.$$

If there exists an equilibrium in the constructed system, the flows in equation (3) are all zero at the system equilibrium state$( {d_2^\ast , \;d_1^\ast , \;d_0^\ast } )$, which yields:

(4)$$m^\ast{ = } \displaystyle{{q^2} \over {2\lambda ^2d_0^\ast }} + \displaystyle{{\omega q-\omega -q} \over \lambda }$$

(5)$$d_2^\ast{ = } \displaystyle{{[ {\omega ( {1-q} ) + \lambda m} ] } \over {2q}}d_1^\ast $$

(6)$$d_1^\ast{ = } \displaystyle{{2\lambda m^\ast } \over q}d_0^\ast $$

Combining equations (1), (2), and (4), we have the polynomial:

(7)$$\phi ( {d_0} ) = 4\lambda ^2\omega q( {q-1} ) d_0^2 + 2\lambda q^2[ {\omega ( {q-1} ) -\lambda } ] d_0 + q^4$$

Solving the unique solution of equation (7) yields $d_0^\ast$, and forms the expression for the function of m* on λ, ω based on equation (4).

Thus, there is proposition I:

There will always exist a stable equilibrium N in which no one buys commercial pension insurance products and only d ₀ combinations exist in the market. Thus, $m^\ast{ = } d_2^\ast{ = } d_1^\ast{ = } 0, \;d_0^\ast{ = } ( 1/2)$.

When the turnover in the market goes beyond zero, i.e., 0 < m* < 1, there exists an internal steady-state equilibrium I, if equation (8) is satisfied.

(8)$$\lambda > \displaystyle{{q^2} \over {\omega ( {1-q} ) + q}}$$

As is shown in Figure 1, the condition for the existence of an internal steady-state equilibrium I is not related to the probability of matching with a strong tie for an individual (ω), and depends only on the unsubscribe rate q and the probability that an individual matches with a weak tie (λ). In the theoretical model, both parameters are exogenous, but the condition for the existence of an internal steady-state equilibrium is more difficult to achieve when λ is extremely small or q is extremely large.

Figure 1.

Contour plot of the values taken by the expression on the right side of equation (8). Contour plot shows the change in value of (q ²/ω(1 − q) + q). Different color blocks indicate different value for (q ²/ω(1 − q) + q) when ω and q take different values. The dark blue block represents the smallest value of (q ²/ω(1 − q) + q) while the bright orange block represents the biggest value of ${( q^2/\omega ( 1-q) + q)} {{q^2} \over {\omega ( {1-q} ) + q}}$. The vertical axis is the value of ω, increasing from 0 to 1 from the bottom to the top, while the horizontal axis is the value of q, increasing from 0 to 1 from left to right. As the color block changes from dark blue to bright orange, the value of (q ²/ω(1 − q) + q) gets larger and larger. As shown in contour plot, the value of (q ²/ω(1 − q) + q) is independent of the value of ω and positively related to the value of q.

When internal steady-state equilibrium is reached, we have:

(9)$$m^\ast{ = } \displaystyle{1 \over 2} + \displaystyle{{\sqrt {{( {\lambda + \omega -\omega q} ) }^2 + 4q\omega ( {1-q} ) } } \over {2\lambda }}-\displaystyle{{\omega ( {1-q} ) + 2q} \over {2\lambda }}$$

(10)$$d_0^\ast{ = } \displaystyle{{q^2} \over {\lambda \left[{\sqrt {{( {q\omega -\omega -\alpha } ) }^2 + 4q\omega ( {1-q} ) } + ( {\lambda + \omega -q\omega } ) } \right]}}$$

Proposition II:

When equation (8) is satisfied, (∂m*/∂ω) > 0 holds constantly, by taking the partial derivative of equation (9). The condition of (∂m*/∂λ) > 0 is λ − ω(1 − q) ≥ 0 while ω and λ are parameters external to the theoretical model.

From proposition II, it is possible to further compare the condition for the existence of an internal steady-state equilibrium with (∂m*/∂λ) > 0, i.e., the value of

(11)$$\omega ( {1-q} ) -\displaystyle{{q^2} \over {\omega ( {1-q} ) + q}}$$

Figure 2 shows that when q is extremely large and ω is extremely small, then equation (11) is constantly less than 0, i.e., as long as the existence of λ satisfies the condition of existence of internal steady-state equilibrium, then λ must meet the condition of (∂m*/∂λ) > 0; and when q is extremely small, the ω is extremely large, then equation (11) is constantly greater than 0. The value of (∂m*/∂λ) also may not be greater than zero even if there exists an internal steady-state equilibrium.

Figure 2.

Contour plot of equation (11). Contour plot shows the change in the value of ω(1 − q) − (q ²/ω(1 − q) + q) (equation [11]). Different color blocks indicate different value for equation (11) when ω and q take different values. The dark blue block represents the smallest value of equation (11) while the bright orange block represents the biggest value of equation (11). The vertical axis is the value of ω , increasing from 0 to 1 from the bottom to the top, while the horizontal axis is the value of q, increasing from 0 to 1 from left to right. As the color block changes from dark blue to bright orange, the value of equation (11) gets larger and larger. As shown in contour plot, the value of equation (11) is jointly determined by the value of ω and the value of q. The value of equation (11) is constantly greater than 0 only if q = 0.

Therefore, the value of (∂m*/∂λ) depends entirely on the value of λ. The larger λ is, the larger the value is, and the higher probability (∂m*/∂λ) > 0 holds. When λ is extremely small, then either the internal steady-state equilibrium may not be reached or even if it is reached, it may not satisfy the condition of (∂m*/∂λ) > 0. λ is defined as the probability of matching an individual's weak ties, and its magnitude is determined by the frequency of the individual's participation in the social activities of the weak ties.

Based on proposition II and the inferences based on proposition II, two hypotheses of the study could be proposed.

2.2 Setting of the empirical model

The sample for the empirical study is drawn from all over the country, and it is highly likely that data from respondents at the same level and with similar characteristics are correlated and cannot satisfy the assumptions of independence and chi-square for the dependent variable, the random effects is therefore introduced to establish a mixed model to fit the coefficients. The study uses a Markov chain Monte Carlo (MCMC) Bayesian approach to estimate the GLMM, and thus has the following three advantages in the empirical evidence: firstly, the actual distribution of the dependent variable in the sample of the empirical study provides rich priori information for the empirical model, which to some extent reflects the characteristics of the overall distribution. The fitting process of maximum likelihood estimation ignores this prior information, but a Bayesian model that reflects this information in the prior setting will be more helpful in improving the accuracy of the econometric model. Secondly, the assumption of independent identical distribution of random effects in the mixed model can introduce the likelihood function about the covariance of unknown parameters and random effects, but in general, it is difficult to directly product the integral about random effects in the likelihood function, and the MCMC method can solve this difficulty well by simulating the idea of estimating parameters. Finally, in the fitting process, the Bayesian model is completely simulated by the MCMC method based on sample random sampling of the coefficients, and its coefficient estimation results are the true reflection of the exact probability of the coefficients, and its coefficient estimation is more accurate.

Combining equations (1), (5), and (6), we have:

(12)$$m^\ast{ = } [ {\lambda m^\ast{ + } \omega ( {1-q} ) + q} ] \displaystyle{{2\lambda m^\ast } \over {q^2}}d_0^\ast $$

Equation (12) represents the relationship between the commercial pension insurance purchase rate and the individual weak ties matching probability (λ), the individual strong ties matching probability (ω), and the commercial pension insurance unsubscribe rate (q) at equilibrium. From the definition of q, the commercial pension insurance unsubscribe rate can be considered as a constant with respect to the current commercial pension insurance purchase rate to simplify the measurement process, and the function then degenerates to m* = f(λ, ω), whose linear form is:

(13)$$m_i = \beta _0 + \beta _1\lambda _i + \beta _2\omega _i + \delta {\rm X}_i^{\rm ^{\prime}} + \varepsilon _i$$

where ${\rm X}_i^{\rm ^{\prime}}$ are the remaining control variables that affect the purchase rate of commercial pension insurance g.

For estimation purpose, it is necessary to convert explanatory variables in equation (13) to generalized linear form. The probability of not purchasing commercial pension insurance is set as the control group, and the logarithm of odds ratio for purchasing probability of commercial pension insurance is θ _i = ln(m _i/1 − m _i), where m _i is the probability of purchasing commercial pension insurance for individual i. Then the generalized linear form of the econometric model can be expressed as:

(14)$$\theta _i = \beta _{0, i} + \beta _{1, i}\lambda _i + \beta _{2, i}\omega _i + \delta {\rm X}_i^{\rm ^{\prime}} + \varepsilon _i$$

where x is the N × p dimensional fixed-effects explanatory variables matrix, N is the sample size, p is the number of explanatory variables, and β represents the fixed-effects coefficients. z is the N × q dimensional random-effects variable matrix, q is the grouping dimension of the random-effects term, and α represents the random effects of the corresponding options. N × 1 column residual vector e represents the portion of the model that is not yet explained by the fixed and random effects. Hypotheses 1 and 2 can be tested by comparing each coefficient estimate.

2.3 Variable description

The study selects the CHARLS as the empirical database. CHARLS has been widely used for empirical evidence on the consumption behavior of middle-aged and elderly people, and the validity of its data has been confirmed by many influential empirical studies (Li et al., Reference Li2018; Pan and Chee, Reference Pan and Chee2020; Wang et al., Reference Wang2019). The basic empirical evidence of this study will be based on the cross-sectional data of wave 4 conducted in 2018, which is the latest data revealed by CHARLS, and further robustness tests will be done with the data of waves 2 and 3, which were conducted in 2013 and 2015, respectively.

The dependent variable of the study comes from the commercial pension insurance purchasing behaviors of the sample individuals, including life insurance, commercial pension insurance, and other pension insurance in the statistics of CHARLS. Although there are differences in the product design and purchase methods of these types of insurance, they are all the pension insurance purchase behaviors of individuals in addition to the basic pension insurance, which more comprehensively reflect the commercial-type pension insurance purchase behaviors of individuals.

Exogenous variables ω and λ in the theoretical model measuring the probability of matching an individual with the strong and weak ties are determined by the frequency with which individuals are associated with strong and weak ties social interaction. This is the core explanatory variable in this paper; drawing on Pan and Chee (Reference Pan and Chee2020), the treatment of the strong and weak ties social interaction variables was generated.

Respondents were first categorized according to their responses to the question “Have you done any of the following social activities in the past month?” The social interaction of the sample was categorized according to their responses to the question. The question options included “hanging out, and socializing with friends,” “playing mahjong, chess and cards, and going to the community room,” “offering help to relatives, friends or neighbors who do not live with you,” “dancing, working out, and practicing qigong, etc.,” “participating in club activities,” “taking part in volunteer activities or charity activities,” “caring for a sick or disabled person who does not live with you,” “attending school or training courses,” “speculating in stocks (funds and other financial securities),” “using the Internet,” and “other.” Based on the meaning of strong and weak ties in our context, it is easy to see that visiting friend's house, socializing with friends, providing help to relatives, friends or neighbors, and frequent Internet chatting are considered as strong ties, while the rest are weak ties.

Second, the social interaction of individuals is classified by combining the social interaction intensity and social interaction frequency. There are three types of strong ties social interaction of individuals, when individuals have at least one high-frequency strong ties social interaction, they belong to high-frequency strong ties interaction; when individuals have at least one low-frequency strong ties social interaction and do not have any high-frequency strong ties social interaction, they belong to low-frequency strong ties interaction; when the sample has neither high-frequency strong ties social interaction nor low-frequency strong ties social interaction, they belong to no strong ties interaction. Following a similar logic, the types of weak ties social interaction of individuals are classified as high-frequency weak ties social interaction, low-frequency weak ties social interaction, and no weak ties social interaction.

By setting individuals with no social behavior at all as a control group, the intensity and frequency of social interactions of individuals are distinguished into eight completely independent types, and dummy variables corresponding to them are generated for assignment, as shown in Table 1.

Table 1.

Matrix of social interactions by intensity and frequency

In addition to being influenced by social interaction, the study added control variables from three dimensions: causal demographic factors, enabling factors, and neediness factors, drawing on the Anderson model (Pan et al., Reference Pan2020). The causal demographic factors include age, gender, education level, health level, and marital satisfaction; the enabling factors contain income and assets; and the demand factors cover number of children and child satisfaction.

2.4 Empirical testing strategy

2.5 Data collation

The sample size in wave 4 that meets the age requirement (less than or equal to 60 years old) is 9190. After a 1% winsorize, personal savings, wage income, and other income are proceeded to standardize standard deviation. The descriptive statistics of the treated sample are shown in Table 2.

Table 2.

Descriptive statistics of the sample data

Overall, the gender distribution within the sample is relatively even, with an average age of 52.856 years, and the urban–rural distribution is more relevant to the actual scenario, with only 41.4% of individuals in the sample coming from urban areas. The statistical description of the sample also indicates that there is still great room for improvement in the breadth of pension insurance coverage, with only about 8% of the individuals in the sample having purchased commercial pension insurance, in contrast to the 86.1% coverage rate of basic pension insurance. The low level of coverage also indicates that there are still some groups in need of protection in China that are still outside the financial pension protection system.

2.6 Benchmark model

The MCMC method is a Bayesian approach, which requires setting prior knowledge for the fixed effects, random effects, and the residual part of the model. The benchmark model does not contain random effects and only requires setting prior knowledge for the other two components.

2.6.2 Stability test

The stability test of the model mainly examines the convergence of the MCMC process in the model. The autocorrelation degree test and the Gelman–Rubin method are used to determine whether the Markov chain has reached a steady state, which subsequently determines whether the coefficients of the benchmark model are robust.

1. 1. Degree of autocorrelation

The number of iterations of the Markov chain in the model reaches 70,000. The fixed-effects autocorrelation coefficients reported in Table 3 indicate that the degree of autocorrelation of model 1 Markov chain is already at a low level (<0.1) even after only 200 lags, and after 10,000 lags, the correlation estimated by the fixed-effect coefficients completely disappears. This indicates that the MCMC process has excellent ergodicity, and the coefficient estimation results based on it are robust.

1. 2. Gelman–Rubin method

Table 3.

Autocorrelation of fixed effect

Since the MCMC method estimates the posterior distribution of the model by sampling, the Gelman–Rubin method is suggested to determine the convergence of the chain. The potential scale reduction factor (psrf) mainly reflects the consistency of different coefficients estimated by sampling for the same setup model, and if the difference between two sampled coefficient estimates is too large, the model is considered to be inconsistent in its estimation of the coefficients. The multivariate potential scale reduction factor (Multivariate psrf) of model 1 is only 1.02, indicating that the estimated coefficients of the model exhibit excellent consistency and multiple MCMC processes point to the same coefficient estimates, i.e., the coefficient estimates of the model are robust.

In summary, the coefficient estimation MCMC process of model 1 has excellent ergodicity and convergence, and the coefficient estimation results of the model built on the stochastic process have stable reproducibility.