Higher education innovation and reform model based on hierarchical probit

The ordered probit model uses observable ordered response data to establish a model to study the changing law of unobservable latent variables. It is a special case of the restricted dependent variable model [7]. The happiness degree variable studied in this article has no specific sample data, so it is also a kind of latent variable, and its influence equation is expressed in linear form as follows. (1) inci*=α0+xi′α+ui inc_i^* = {\alpha _0} + x_i^\prime \alpha + {u_i} Where i is the education level. inci* inc_i^* represents the degree of happiness. x_i is a set of explanatory variable vectors that may affect the degree of happiness. α is the corresponding unknown coefficient. α₀ is a constant term. u_i is a random disturbance term with a mean value of 0 and a variance of δ². Although the sample in this article inci* inc_i^* cannot be directly observed, it falls into four connected segments (denoted as 0, 1, 2, and 3 respectively) and can be observed. We use the variable y_i to denote. The relationship between y_i and inci* inc_i^* is: (2) y={0,0<inci*≤1.51,1.5<inci*≤22,2<inci*≤2.53,inci*>2.5 y = \left\{ {\matrix{ {0,} \hfill & {0 < inc_i^* \le 1.5} \hfill \cr {1,} \hfill & {1.5 < inc_i^* \le 2} \hfill \cr {2,} \hfill & {2 < inc_i^* \le 2.5} \hfill \cr {3,} \hfill & {inc_i^* > 2.5} \hfill \cr } } \right. The probability that y_i takes each value j is: P(yi=0|xi)=P(ui≤1.5−a0−xi′a)=F(λ1−xi′β)P(yi=1|xi)=P(1.5−a0−xi′a<ui≤2−xi′α)=F(λ2−xi′β)−F(λ1−xi′β)P(yi=2|xi)=P(2−a0−xi′a<ui≤2.5−xi′α)=F(λ3−xi′β)−F(λ2−xi′β)P(yi=3|xi)=P(ui>2.5−a0−xi′a)=1−F(λ3−xi′β) λ1=(1.5−a0)/δ, λ2=(2−a0)/δ, λ3=(2.5−a0)/δ, β=α/β, F(⋅) \matrix{ {P\left( {{y_i} = 0|{x_i}} \right) = P\left( {{u_i} \le 1.5 - {a_0} - x_i^\prime a} \right) = F\left( {{\lambda _1} - x_i^\prime \beta } \right)} \hfill \cr {P\left( {{y_i} = 1|{x_i}} \right) = P\left( {1.5 - {a_0} - x_i^\prime a < {u_i} \le 2 - x_i^\prime \alpha } \right) = F\left( {{\lambda _2} - x_i^\prime \beta } \right) - F\left( {{\lambda _1} - x_i^\prime \beta } \right)} \hfill \cr {P\left( {{y_i} = 2|{x_i}} \right) = P\left( {2 - {a_0} - x_i^\prime a < {u_i} \le 2.5 - x_i^\prime \alpha } \right) = F\left( {{\lambda _3} - x_i^\prime \beta } \right) - F\left( {{\lambda _2} - x_i^\prime \beta } \right)} \hfill \cr {P\left( {{y_i} = 3|{x_i}} \right) = P\left( {{u_i} > 2.5 - {a_0} - x_i^\prime a} \right) = 1 - F\left( {{\lambda _3} - x_i^\prime \beta } \right)} \hfill \cr {\;\;\,\;\;\;\;\;\;\;\;\;\;\;\;\;{\lambda _1} = \left( {1.5 - {a_0}} \right)/\delta ,\quad {\lambda _2} = \left( {2 - {a_0}} \right)/\delta ,\quad {\lambda _3} = \left( {2.5 - {a_0}} \right)/\delta ,\quad \beta = \alpha /\beta ,\quad F( \cdot )} \hfill \cr } The above equations are the ε ≡ u/δ distribution function, respectively. If we write yi*=(inci*−a0)/δ y_i^* = (inc_i^* - {a_0})/\delta , then the ordered models (1) and (2) have the same probability distribution P(y_i = j|x_i) as the following model. (3) yi*=xi′β+εi, yi={0,yi*≤λ11,λ1<yi*≤λ22,λ2<yi*≤λ33,yi*>λ3 y_i^* = x_i^\prime \beta + {\varepsilon _i},\quad {y_i} = \left\{ {\matrix{ {0,} \hfill & {y_i^* \le {\lambda _1}} \hfill \cr {1,} \hfill & {{\lambda _1} < y_i^* \le {\lambda _2}} \hfill \cr {2,} \hfill & {{\lambda _2} < y_i^* \le {\lambda _3}} \hfill \cr {3,} \hfill & {y_i^* > {\lambda _3}} \hfill \cr } } \right. Among them ε is a disturbance term with a mean value of zero and a variance of 1, and its distribution function is F(·). Model (3) is a classic ordered response model. It is the standardization of happiness models (1) and (2), and its threshold λ₁, λ₂, λ₃ is also an unknown parameter to be estimated. The log-likelihood function is: (4) lnL(β,λ1,λ2,λ3)=∑i=1n∑j=03l{yi=j}ln[F(λj+1−xi′β)−F(λj−xi′β)] \ln L\left( {\beta ,{\lambda _1},{\lambda _2},{\lambda _3}} \right) = \sum\limits_{i = 1}^n \sum\limits_{j = 0}^3 l\left\{ {{y_i} = j} \right\}\ln \left[ {F\left( {{\lambda _{j + 1}} - x_i^\prime \beta } \right) - F\left( {{\lambda _j} - x_i^\prime \beta } \right)} \right] Among them, λ₄ = ∞, λ₀ = −∞, λ₁, λ₂, λ₃ is the same as above and 1{·}is an indicator function. It is 1 when the conditions in the brackets are satisfied. Otherwise, it is 0. From the above derivation, it can be seen that the estimated β differs from the coefficient α in the original happiness model (1) and (2) by a positive multiple, but the probability of happiness falling into each small section remains unchanged. Therefore, the estimated results of the standardized model (3) can be used to analyze the influence of various factors on the degree of happiness. When ε obeys the standard normal distribution, F in the above likelihood function is replaced by the cumulative distribution function of the standard normal distribution, which is the parameter estimation of the Probit model [8]. When ε does not obey the standard normal distribution, such estimation will lead to inconsistent estimators of the parameters. The correction method is to use the semi-parametric method.

Regarding the distribution of as unknown, the Hermit sequence fk(ε)=1c(∑s=0kγsεs)2ϕ(ε) {f_k}(\varepsilon ) = {1 \over c}{\left( {\sum\limits_{s = 0}^k {\gamma _s}{\varepsilon ^s}} \right)^2}\phi (\varepsilon ) is used to approximate the density function of ε. Where c is the standardization factor in ensuring that the integral of f_k(ε) is equal to 1. The coefficient F is also the parameter to be estimated. ϕ(·) is the density function of the standard normal distribution. The γ_s in the likelihood function is replaced by the cumulative distribution function f_k(·) that approximates the distribution. The estimation of parameter β is obtained by solving the problem of maximizing the quasi-likelihood function. It can be proved that this estimator is consistent and asymptotically normal under weaker conditions. When k = 2, the estimation is the same as the parameter estimation of the ordered Probit model, so the likelihood ratio test can be used to select k or explain the necessity of semiparametric estimation.

If the degree of non-employment education is not in the research sample, the decision-making selection of education participation will lead to sample selection problems [9]. To eliminate potential sample selection bias, an ordered sample selection model should be used to estimate the impact of various factors on happiness. Suppose the education participation equation is: (5) Di=1{γ0+zi′γ−υi>0} {D_i} = 1\{ {\gamma _0} + z_i^\prime \gamma - {\upsilon _i} > 0\} The resulting equation is model (3), that is, only when people participate in education (D_i = 1), the happiness level falls into a certain section j. Here vector z is a vector of factors that affect the selection. In the model z contains at least one variable that is not included in x. Because the two types of factors contained in z are not all contained in x, the model satisfies the recognition conditions [10]. At this time, the probability that the degree of happiness of education involved in education falls in each section can be similarly deduced as: (6) P(yi=j,Di=1|xi,zi)=G(λj+1−xi′β,γ0+zi′γ)−G(λj−xi′β,γ0+zi′γ) P\left( {{y_i} = j,{D_i} = 1|{x_i},{z_i}} \right) = G\left( {{\lambda _{j + 1}} - x_i^\prime \beta ,{\gamma _0} + z_i^\prime \gamma } \right) - G\left( {{\lambda _j} - x_i^\prime \beta ,{\gamma _0} + z_i^\prime \gamma } \right) Where G(,)is the joint distribution function of ε and υ. At this time, the parameters β, λ₁, λ₂, λ₃ and γ₀, γ in the sample selection ordered response models (3) and (5) can be estimated by the maximum likelihood method. Its log-likelihood function lnL(β, γ₀, γ, λ₁, λ₂, λ₃) is: (7) ∑i=1n(1−Di)ln[1−G2(γ0+zi′γ)]+∑i=1n∑j=03Di1{yi=j}ln[G(λj+1−xi′β,γ0+zi′γ)−G(λj−xi′β,γ0+zi′γ) \sum\limits_{i = 1}^n \left( {1 - {D_i}} \right)\ln \left[ {1 - {G_2}\left( {{\gamma _0} + z_i^\prime \gamma } \right)} \right] + \sum\limits_{i = 1}^n \sum\limits_{j = 0}^3 {D_i}1\{ {y_i} = j\} \ln [G\left( {{\lambda _{j + 1}} - x_i^\prime \beta ,{\gamma _0} + z_i^\prime \gamma } \right) - G\left( {{\lambda _j} - x_i^\prime \beta ,{\gamma _0} + z_i^\prime \gamma } \right) Where λ₄ = ∞, λ₀ = −∞, λ₁, λ₂, λ₃ is the same as above. G2(γ0+zi′γ)=1−P(υi≥γ0+zi′γ)=G(∞,γ0+zi′γ) {G_2}\left( {{\gamma _0} + z_i^\prime \gamma } \right) = 1 - P\left( {{\upsilon _i} \ge {\gamma _0} + z_i^\prime \gamma } \right) = G\left( {\infty ,{\gamma _0} + z_i^\prime \gamma } \right) When ε and υ follow a normal joint distribution with zero mean, unit variance, and correlation coefficient ρ, the joint distribution function G in the above likelihood function is replaced by the two-variable Gaussian distribution function Φ(ε, υ;ρ). The estimation of models (3) and (5) is the estimation of the Probit model. The difference between log-likelihood function (4) and (7) is that the first sum in (6) reflects the contribution of the unemployed education level subsample (D_i = 0) to the likelihood function, while (4) ignores this item. If the sample does not have a selection problem, the first sum does not appear.

When ε and υ do not obey the normal joint distribution, applying the maximum likelihood estimation of Φ(ε, υ;ρ) will result in inconsistent parameter estimators. To obtain the consistent estimation of the parameters, the restriction on the joint distribution G(,) of ε and υ should be relaxed, and the semiparametric method should be adopted.

Similar to the semi-parametric estimation method of Probit model (3), the semi-parametric estimation of sample selection ordered Probit model (3) and (5) regards the joint distribution of and as unknown. We use the Hermite function sequence. (8) f*(ε,υ;θ)=1ψR(θ)τR(ε,υ;θ)2ϕ(ε)ϕ(υ) {f^*}(\varepsilon ,\upsilon ;\theta ) = {1 \over {{\psi _R}(\theta )}}{\tau _R}{(\varepsilon ,\upsilon ;\theta )^2}\phi (\varepsilon )\phi (\upsilon ) Approximate the joint density function of ε and υ. Where τ_R(ε, υ; θ) is the R = (R₁, R₂) order polynomial about ε, υ, and θ is the parameter vector to be estimated constructed by R₁, R₂ unknown coefficients. ψ_R(θ) is the standardization factor in ensuring that the integral of f* is equal to 1. From this, the approximation function G*(ε, υ; θ) of the joint distribution function G of ε and υ can be obtained. We replace Gin the likelihood function (7) with the approximate distribution function G*(ε, υ; θ). The semi-parametric estimator of the parameter vector (β, γ₀, γ, λ₁, λ₂, λ₃, θ) is obtained by solving the quasi-likelihood maximization problem. It can be proved that as long as R₁, R₂ increases with sample size, n this estimator is consistent and asymptotically normal. The difference from the semi-parametric estimation of the model (3) is that the model corresponding to R₁, R₂ that does not increase simultaneously has no nesting. For example, the two approximation models corresponding to (R₁, R₂) = (2, 2) and are not nested, so the likelihood ratio test is not suitable for selecting approximation models of different orders. However, we can choose R₁, R₂ using AIC or BIC criteria [11]. This can also explain the necessity of semi-parametric estimation.

Considering that happiness is an ordered dependent variable, we use an ordered probit model to estimate. To ensure the consistency of parameter estimation, we use the Hermit sequence to simulate the residual distribution to modify the ordered Probit model. This results in a consistent estimate of the parameters.

We use a semi-parametric estimation method based on the ordered probit model to test the impact of education on happiness. When k≥ 4, according to the likelihood ratio test (LRtest), it can be judged that the semi-parametric estimation result is better than the parameter estimation result, and the estimated parameter does not change significantly, so k=4 is selected. The estimated results are shown in Table 2.

Results (1) and (2) are estimated results for the entire sample. Results (1) show that the improvement of education level can significantly promote residents’ happiness, and there are significant differences in happiness between urban and rural residents. Due to Chinese long-term urban-rural dual structure, urban and rural residents show significant differences in economic conditions and ideological concepts. So is there a significant difference in the effect of education on the happiness of urban and rural residents? We add the intersection of education and town, as shown in result (2). The coefficient of the cross term is negative, and the test is significant [12]. This shows that education is not as effective in improving the happiness of urban residents as it does for rural residents. This may be because the popularization of education for urban residents is earlier than that for rural residents, and the popularity is also higher. According to the sample data, the proportion of rural senior high school students and above accounted for 10.3%, while the proportion in urban areas was 32.05%. According to the principle of vagueness, the scarcity of high education makes rural residents happier than urban residents.

Further estimates are made on the urban and rural samples separately. Results (3)(4) are estimated results using OP non-parametric model. It can be seen that the coefficient of the education variable in the urban sample is about half of the coefficient in the rural sample, and there is a big difference between the two.

From the results of the above model, it can be seen that education can significantly improve individuals’ subjective well-being. For those who wish to receive education, education itself is happy. Education is willing to improve people's cognition, helps cultivate correct world outlook, values, and enhances self-evaluation of happiness. On the other hand, existing research proves that education indirectly impacts happiness through economic and other factors. From the description of sample statistics, we can see that the proportion of people with high school education and above is only 20.54%. We classify this type of people as those who have received higher education [13]. We mainly explore the mechanism of higher education's influence on individual subjective well-being and still adopt the non-parametric estimation of the orderly corresponding probit model. It can be seen from the previous results that this method is superior to ordinarily ordered probit. The results of judging the impact path by introducing cross-terms are shown in Table 3.

The estimation result (5) uses the binary variable higher education (edu2) instead of education (Edu), and the coefficient is still significantly positive. This also shows the robustness of the model. The estimation results (6)–(9) were added to the cross items of education and income (logarithm), health, work status, and status, and the statistical tests of each cross item were significant. The coefficient and significance of higher education in the estimation result (6) have not changed much. This shows that higher education cannot promote the individual happiness of residents by increasing personal income. As a result, the signs of higher education coefficients of (7), (8), and (9) became negative and no longer significant. This shows that higher education can improve the individual's subjective well-being by improving the individual's health, work status, and status.