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Rivista
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2444-8656
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01 Jan 2016
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# The Economic Model of Rural Supply and Demand Under the Data Analysis Function Based on Ordered Probit

###### Accettato: 28 Apr 2022
Dettagli della rivista
Formato
Rivista
eISSN
2444-8656
Prima pubblicazione
01 Jan 2016
Frequenza di pubblicazione
2 volte all'anno
Lingue
Inglese
Introduction

Agriculture is the main body in the traditional rural economy[1]. Rural economy and agricultural economy are often used interchangeably as equivalent concepts. With the continuous expansion of the rural economic field, traditional agriculture can no longer cover the connotation and extension of the rural economy. The rural economy covers agriculture in a broad sense and the secondary and tertiary industries that occur in rural areas. Therefore, the agricultural economic growth can be measured by the agricultural output value and the non-agricultural income of the rural part-timers [2]. Rural economic growth is affected by factors such as rural industrial structure, human capital, rural culture, and policies that benefit farmers and the intermediate inputs of agricultural production. Most of the related studies do not make a strict distinction between rural economic growth and agricultural economic growth. These studies lack relatively systematic econometric empirical studies on the influencing factors of rural economic growth. Therefore, this paper intends to construct a variable system that affects rural economic growth based on relevant research. We used a binary discrete choice model to perform regression analysis on the survey sample. The article analyzes the influence of each variable and explores the countermeasures to develop the rural economy.

Ordered Probit Model Establishment

Because the ordinal dependent variable is not continuous, the latent variable method is usually used to estimate the model. Respondents’ selection of ordinal variables implies a selection of an interval of continuous net utility values. We assume that y* is an unobservable continuous latent variable [3]. We have a total of D + 1 sorts, which have the following correspondences: $y={0y* <φ01φ0≤y*≤φ12φ1≤y*≤φ2⋮⋮Dy*≥φD+1$ y = \left\{{\matrix{0 & {{y^*}\, < {\varphi _0}} \cr 1 & {{\varphi _0} \le {y^*} \le {\varphi _1}} \cr 2 & {{\varphi _1} \le {y^*} \le {\varphi _2}} \cr \vdots & \vdots \cr D & {{y^*} \ge {\varphi _{D + 1}}} \cr}} \right.

If y* ∈ [φd− 1, φd), then y = d. We assume φ0 = −∞, φ1 = 0, φD = +∞, then φ = (φ2 …, φD+1) is also a parameter to be estimated. Bayesian methods treat both parameters and latent variables as random variables. The hierarchical model structure is just right for the MCMC method [4]. Therefore, this paper chooses the Bayesian method to estimate the model. First, analyze an ordinary ordinal regression: $p(yi=d)=p(φd−1≤y*<φd)=p(φd−1≤Xiβ+ri≤φd)=p(cφd−1≤cXiβ+cri≤cφd)$ \matrix{{p\left({{y_i} = d} \right) = p\left({{\varphi _{d - 1}} \le {y^*} < {\varphi _d}} \right) =} \hfill \cr {p\left({{\varphi _{d - 1}} \le {X_i}\beta + {r_i} \le {\varphi _d}} \right) = p\left({c{\varphi _{d - 1}} \le c{X_i}\beta + c{r_i} \le c{\varphi _d}} \right)} \hfill \cr}

Assuming that the latent variable y* is known to obtain y, we can obtain: p(β, σ2 | y*) = p(β, σ2 | y*, y). We can first obtain the conditional sampling distribution of the parameters with the latent variable as the dependent variable. Then we get the conditional sampling distribution of y*, φ based on the correspondence between y and y* and other parameters [5]. The second-level models of the hierarchical model are all ordinary continuous variable models. This paper assumes that the one-layer model error obeys a normal distribution (Probit model). First, assuming that y* is known, the likelihood probability is: $p(D|δ,β,ρ)=(2π)−n/2|A|exp{−12υ1′υ1}$ p\left({D|\delta,\beta,\rho} \right) = {\left({2\pi} \right)^{- n/2}}\left| A \right|\exp \left\{{- {1 \over 2}\upsilon _1^{'}{\upsilon _1}} \right\}

D = {y*, X, W}, υ1 = Ay*, X = [Δ ΔX1], $δ=[β0jβ1j]$ \delta = \left[{\matrix{{{\beta _{0j}}} \cr {{\beta _{1j}}} \cr}} \right] , A = InρW. where |A| is due to the Jacobian: $J=|∂r∂y*|=|∂(Ay*−X˜δ−Xβ)∂y*|=|∂(Ay*)∂y*|=|A′|=|A|.$ J = \left| {{{\partial r} \over {\partial {y^*}}}} \right| = \left| {{{\partial \left({A{y^*} - \tilde X\delta - X\beta} \right)} \over {\partial {y^*}}}} \right| = \left| {{{\partial \left({A{y^*}} \right)} \over {\partial {y^*}}}} \right| = \left| {A'} \right| = \left| A \right|.

First set the prior distribution of parameters belonging to layer one only. The conjugate prior distribution of the normal mean when the variance is known is also normal, so we set the prior distribution of β to be a normal distribution: $β~N(c,T−1)$ \beta \sim N\left({c,{T^{- 1}}} \right)

We guarantee that the error covariance matrix is positive definite. ρ must fall within a feasible interval $(κW min−1, κW max−1)$ \left({\kappa _{W\,\min}^{- 1},\,\kappa _{W\,\max}^{- 1}} \right) , where $κW min−1$ \kappa _{W\,\min}^{- 1} , $κW max−1$ \kappa _{W\,\max}^{- 1} , represents the smallest and largest eigenvalues of W. We generally choose the uniform distribution as the prior distribution, namely: $π(ρ)~U(κW min−1, κW max−1)∝1$ \pi \left(\rho \right)\sim U\left({\kappa _{W\,\min}^{- 1},\,\kappa _{W\,\max}^{- 1}} \right) \propto 1

Since layer 2 provides some useful information about β0j and β1j, the prior distribution can be obtained using this information. Since the errors of the two models are not independent of each other, we need to combine them to estimate [6]. First analyze the $δ=[β0jβ1j]$ \delta = \left[{\matrix{{{\beta _{0j}}} \cr {{\beta _{1j}}} \cr}} \right] data generation process, namely: $δ=C−1(Z˜α+u)$ \delta = {C^{- 1}}\left({\tilde Z\alpha + u} \right) $Z˜=[Z000Z1]$ \tilde Z = \left[{\matrix{{{Z_0}} & 0 \cr 0 & {{Z_1}} \cr}} \right] , $C=[Im−λ0MIm−λ1M]$ C = \left[{\matrix{{{I_m} - {\lambda _0}M} & {} \cr {} & {{I_m} - {\lambda _1}M} \cr}} \right] , $α=[a0a1]$ \alpha = \left[{\matrix{{{a_0}} \cr {{a_1}} \cr}} \right] , $u=[u0u1]$ u = \left[{\matrix{{{u_0}} \cr {{u_1}} \cr}} \right] , Z0 and Z1 contain constant terms. The prior distribution of parameter δ can be obtained from the likelihood function of the layer-two random intercept model: $π(δ|α,λ0,λ1,∑)∝|∑|−m/2|C|exp{−12υ2′(∑−1⊗Im)υ2}$ \pi \left({\delta \left| {\alpha,{\lambda _0},{\lambda _1},\sum} \right.} \right) \propto {\left| \sum \right|^{- m/2}}\left| C \right|\exp \left\{{- {1 \over 2}\upsilon _2^{'}\left({{\sum ^{- 1}} \otimes {I_m}} \right){\upsilon _2}} \right\}

Among them, υ2 = . which is: $δ|α,λ0,λ1,∑~N(C−1Z˜α, [C′(∑−1⊗Im)C]−1)$ \delta \left| {\alpha,{\lambda _0},{\lambda _1},\sum \sim N\left({{C^{- 1}}\tilde Z\alpha,\,{{\left[{C'\left({{\sum ^{- 1}} \otimes {I_m}} \right)C} \right]}^{- 1}}} \right)} \right.

Then we set the prior distribution of each model parameter of layer 2 in turn. In this paper, the setting of the prior distribution of α is as follows: $α~N(ca,Ta−1)$ \alpha \sim N\left({{c_a},T_a^{- 1}} \right)

Due to the problem of model identification, we need to fix σ0,0 = 1. This will result in Σ not obeying a known distribution [7]. Suppose there are Q seemingly uncorrelated regressions. Suppose $∑=[σ11γ′γ∑G]$ \sum = \left[{\matrix{{{\sigma _{11}}} & {\gamma '} \cr \gamma & {{\sum _G}} \cr}} \right] , where γ represents the covariance of the residuals of the other dimension equations Q − 1 with the residuals of the first equation. We assume U = ε1, G = ε2, L, εQ. ΣG Represents the variance-covariance matrix of G in (Q − 1) × (Q − 1) dimensions. It can be known from the conditional marginal distribution of the normal joint distribution: G|U ~ N((γ/σ11) U, ΣGγγ / σ11), let Φ = ΣGγγ / σ11, then: ${∑|σ11=1}=[1γ′γΦ+γγ′]$ \left\{{\sum \left| {{\sigma _{11}} = 1} \right.} \right\} = \left[{\matrix{1 & {\gamma '} \cr \gamma & {\Phi + \gamma \gamma '} \cr}} \right]

Then the parameter to be estimated becomes γ, Φ. The prior distribution is set as: $γ~N(γ¯, B−1)$ \gamma \sim N\left({\bar \gamma,\,{B^{- 1}}} \right) $Φ~IW(υ, Ω)$ \Phi \sim IW\left({\upsilon,\,\Omega} \right)

The prior distribution setting principle of each parameter of layer two can be consistent with that of layer one. Then there are: $π(λ0)~U(κM min−1, κM max−1)∝1$ \pi \left({{\lambda _0}} \right)\sim U\left({\kappa _{M\,\min}^{- 1},\,\kappa _{M\,\max}^{- 1}} \right) \propto 1 $π(λ1)~U(κM min−1, κM max−1)∝1$ \pi \left({{\lambda _1}} \right)\sim U\left({\kappa _{M\,\min}^{- 1},\,\kappa _{M\,\max}^{- 1}} \right) \propto 1

Knowing the Bayesian method, there is the following relation $p(θ|D)=p(D|θ)p(θ)p(D)∝p(D|θ)[(θ)$ p\left({\theta |D} \right) = {{p\left({D|\theta} \right)p\left(\theta \right)} \over {p\left(D \right)}} \propto p\left({D|\theta} \right)[\left(\theta \right)

Then the joint posterior distribution of each parameter of layer 1 is: $p(β, δ, ρ|D)∝p(D|β, δ, ρ)×π(δ)×π(β)×π(ρ)$ p\left({\beta,\,\delta,\,\rho |D} \right) \propto p\left({D|\beta,\,\delta,\,\rho} \right) \times \pi \left(\delta \right) \times \pi \left(\beta \right) \times \pi \left(\rho \right)

Then the fully conditional posterior distribution of each parameter can be calculated one by one: $∝ exp{−12[Xβ−(Ay*−Xδ)]′[Xβ−(Ay*−Xδ)]}×exp{−12(β−c)′T(β−c)}$ \matrix{{\propto \,\exp \left\{{- {1 \over 2}\left[{X\beta - \left({A{y^*} - X\delta} \right)} \right]'\left[{X\beta - \left({A{y^*} - X\delta} \right)} \right]} \right\} \times} \hfill \cr {\exp \left\{{- {1 \over 2}\left({\beta - c} \right)'T\left({\beta - c} \right)} \right\}} \hfill \cr}

The fully conditional posterior distribution of β: $β|δ,ρ~N(c*, T*)$ \beta |\delta,\rho \sim N\left({{c^*},\,{T^*}} \right) $c*=T*(X′(Ay*−X˜δ)+Tc)$ {c^*} = {T^*}\left({X'\left({A{y^*} - \tilde X\delta} \right) + Tc} \right) , T* = (XX + T)−1.

Equation (18) uses an informative prior distribution. We need to set the parameters in the prior distribution reasonably. If you set c = 0 and specify β the prior variance is very large. The covariance between the inner parameters is 0. For example : T−1 = Ik 1010. Assuming that the mean of ρ and δ is known, c* is exactly the least-squares estimate of β. This shows that using an uninformative prior distribution [8]. We can rely on uninformative priors for estimation. $β|ρ,δ~N(μβ, ∑β)$ \beta |\rho,\delta \sim N\left({{\mu _\beta},\,{\sum _\beta}} \right) $μβ=(X′X)−1 (X(Ay*−X˜δ))$ {\mu _\beta} = {\left({X'X} \right)^{- 1}}\,\left({X\left({A{y^*} - \tilde X\delta} \right)} \right) , Σβ = (XX)−1. The kernel of the conditional posterior distribution of ρ is expressed as follows $p(ρ|β,δ)∝p(D|β, δ, ρ)π(ρ)∝|A|exp[−12υ1′υ1]$ p\left({\rho |\beta,\delta} \right) \propto p\left({D|\beta,\,\delta,\,\rho} \right)\pi \left(\rho \right) \propto \left| A \right|\exp \left[{- {1 \over 2}\upsilon _1^{'}{\upsilon _1}} \right]

The prior distribution of δ also depends on the parameters to be estimated at layer 2, so: $p(δ|β,ρ,λ0,λ1,α,∑)∝p(D|β, δ, ρ)×π(δ|λ0, λ1, α, ∑)∝exp(−12[X˜δ−(Ay*−Xβ)]′[X˜δ−(Ay*−Xβ)])×∝exp(−12(Cδ−Z˜α)′((∑−1⊗Im))(Cδ−Z˜α))$ \matrix{{p\left({\delta |\beta,\rho,{\lambda _0},{\lambda _1},\alpha,\sum} \right) \propto p\left({D|\beta,\,\delta,\,\rho} \right) \times \pi \left({\delta |{\lambda _0},\,{\lambda _1},\,\alpha,\,\sum} \right)} \hfill \cr {\propto \exp \left({- {1 \over 2}\left[{\tilde X\delta - \left({A{y^*} - X\beta} \right)} \right]'\left[{\tilde X\delta - \left({A{y^*} - X\beta} \right)} \right]} \right) \times} \hfill \cr {\propto \exp \left({- {1 \over 2}\left({C\delta - \tilde Z\alpha} \right)'\left({\left({{\sum ^{- 1}} \otimes {I_m}} \right)} \right)\left({C\delta - \tilde Z\alpha} \right)} \right)} \hfill \cr}

Which is: $δ|β,ρ,λ0, λ1, α, ∑~N(∑δμδ, ∑δ)$ \delta |\beta,\rho,{\lambda _0},\,{\lambda _1},\,\alpha,\,\sum \sim N\left({{\sum _\delta}{\mu _\delta},\,{\sum _\delta}} \right) $μδ=(X˜′)(Ay*−Xβ)+C′(∑−1×Im)C(C−1Z˜α)$ {\mu _\delta} = \left({\tilde X'} \right)\left({A{y^*} - X\beta} \right) + C'\left({{\sum ^{- 1}} \times {I_m}} \right)C\left({{C^{- 1}}\tilde Z\alpha} \right) , $∑δ=(X˜′X˜+C′(∑−1⊗Im)C)−1$ {\sum _\delta} = {\left({\tilde X'\tilde X + C'\left({{\sum ^{- 1}} \otimes {I_m}} \right)C} \right)^{- 1}} is the posterior distribution of variable intercept coefficients containing layer one and layer two information [9]. The conditional posterior distribution of the second-level parameters is expressed as follows $α|δ, λ0, λ1, ∑~N[cα*, Tα*]$ \alpha |\delta,\,{\lambda _0},\,{\lambda _1},\,\sum \sim N\left[{c_\alpha ^*,\,T_\alpha ^*} \right]

Where $cα*=Tα*(Z˜′(∑−1⊗Im)Cδ)+Tαcα)$ \left. {c_\alpha ^* = T_\alpha ^*\left({\tilde Z'\left({{\sum ^{- 1}} \otimes {I_m}} \right)C\delta} \right) + {T_\alpha}{c_\alpha}} \right) , $Tα*=(Z˜′(∑−1⊗Im)Z˜+Tα)−1$ T_\alpha ^* = {\left({\tilde Z'\left({{\sum ^{- 1}} \otimes {I_m}} \right)\tilde Z + {T_\alpha}} \right)^{- 1}} . Similarly, we use the uninformative prior distribution, then c*α is exactly the parameter result of the least-squares estimation of α, namely: $μα=(Z˜′(∑−1⊗Im)Z˜)−1 (Z˜′(∑−1⊗Im)Cδ))$ {\mu _\alpha} = {\left({\tilde Z'\left({{\sum ^{- 1}} \otimes {I_m}} \right)\tilde Z} \right)^{- 1}}\,\left. {\left({\tilde Z'\left({{\sum ^{- 1}} \otimes {I_m}} \right)C\delta} \right)} \right) , $∑α=(Z˜′(∑−1⊗Im)Z˜)−1$ {\sum _\alpha} = {\left({\tilde Z'\left({{\sum ^{- 1}} \otimes {I_m}} \right)\tilde Z} \right)^{- 1}} . γ and Φ can similarly derive their conditional posterior distributions. $γ|δ, α, λ0, λ1~N(−(υ11)−1υ10,(υ00−υ01(υ11)−1υ10)u11−1)$ \gamma |\delta,\,\alpha,\,{\lambda _0},\,{\lambda _1}\sim N\left({- {{\left({{\upsilon ^{11}}} \right)}^{- 1}}{\upsilon ^{10}},\left({{\upsilon ^{00}} - {\upsilon ^{01}}{{\left({{\upsilon ^{11}}} \right)}^{- 1}}{\upsilon ^{10}}} \right)u_{11}^{- 1}} \right)

Among them, $u=(Cδ−Zα)=(u0u1)$ u = \left({C\delta - Z\alpha} \right) = \left({\matrix{{{u_0}} \cr {{u_1}} \cr}} \right) , $V=[u00u01u10u11]−1=[υ00υ01υ10υ11]$ V = {\left[{\matrix{{{u_{00}}} & {{u_{01}}} \cr {{u_{10}}} & {{u_{11}}} \cr}} \right]^{- 1}} = \left[{\matrix{{{\upsilon ^{00}}} & {{\upsilon ^{01}}} \cr {{\upsilon ^{10}}} & {{\upsilon ^{11}}} \cr}} \right] , $u00=u0′u0,u01=u0′u1,u11=u1′u1$ {u_{00}} = u_0^{'}{u_0},{u_{01}} = u_0^{'}{u_1},{u_{11}} = u_1^{'}{u_1} . The conditional posterior Distribution of Φ is: $Φ|δ,α,λ0,λ1~IW(n2,υ11)×((υ00−υ01(υ11)−1υ10)−1υ10)χn2−12)$ \Phi |\delta,\alpha,{\lambda _0},{\lambda _1}\sim IW\left({{n_2},{\upsilon ^{11}}} \right) \times \left({\left. {{{\left({{\upsilon ^{00}} - {\upsilon ^{01}}{{\left({{\upsilon ^{11}}} \right)}^{- 1}}{\upsilon ^{10}}} \right)}^{- 1}}{\upsilon ^{10}}} \right)\chi _{n2 - 1}^2} \right)

It can be seen that the conditional posterior distribution of γ, Φ is a distribution of known form: $p(λ0|λ1, δ, ∑)∝p(δ|λ0, λ1, δ, ∑)π(λ0)∝|C(λ1)|exp[−12υ1′(λ1)(∑−1⊗Im)υ1(λ1)]$ p\left({{\lambda _0}|{\lambda _1},\,\delta,\,\sum} \right) \propto p\left({\delta |{\lambda _0},\,{\lambda _1},\,\delta,\,\sum} \right)\pi \left({{\lambda _0}} \right) \propto \left| {C\left({{\lambda _1}} \right)} \right|\exp \left[{- {1 \over 2}\upsilon _1^{'}\left({{\lambda _1}} \right)\left({{\sum ^{- 1}} \otimes {I_m}} \right){\upsilon _1}\left({{\lambda _1}} \right)} \right] $p(λ1|λ0, δ, ∑)∝p(δ|λ0, λ1, δ, ∑)π(λ1)∝|C(λ0)|exp[−12υ2′(λ0)(∑−1⊗Im)υ2(λ0)]$ p\left({{\lambda _1}|{\lambda _0},\,\delta,\,\sum} \right) \propto p\left({\delta |{\lambda _0},\,{\lambda _1},\,\delta,\,\sum} \right)\pi \left({{\lambda _1}} \right) \propto \left| {C\left({{\lambda _0}} \right)} \right|\exp \left[{- {1 \over 2}\upsilon _2^{'}\left({{\lambda _0}} \right)\left({{\sum ^{- 1}} \otimes {I_m}} \right){\upsilon _2}\left({{\lambda _0}} \right)} \right]

Finally, this paper obtains the conditional posterior distribution of y* and ϕ based on the real ordinal y and the other parameters mentioned above [10]. First, we can directly find that y* obeys the multivariate truncated normal distribution according to the corresponding relationship as follows: $p(y*|y,δ,β,ρ,ϕ)=K−1(2π)−n/2|∑|−1/2exp{−12υ1′υ1}IR(y*)$ p\left({{y^*}|y,\delta,\beta,\rho,\phi} \right) = {K^{- 1}}{\left({2\pi} \right)^{- n/2}}{\left| \sum \right|^{- 1/2}}\exp \left\{{- {1 \over 2}\upsilon _1^{'}{\upsilon _1}} \right\}{I_R}\left({{y^*}} \right)

Where $K=∫a1b1∫a2b2…∫anbnf(y*|…)dy*$ K = \int\limits_{{a_1}}^{{b_1}} {\int\limits_{{a_2}}^{{b_2}} {\ldots \int\limits_{{a_n}}^{{b_n}} {f\left({{y^*}| \ldots} \right)d{y^*}}}} , f ( ) is the density function of the ordinary multivariate normal distribution N(μy*, Σ): $y*~TN(A−1(Δβ0j+Xβ), ∑, α, b)$ {y^*}\sim TN\left({{A^{- 1}}\left({\Delta {\beta _{0j}} + X\beta} \right),\,\sum,\,\alpha,\,b} \right)

Where Σ = (A A)−1, a, b is the value of ϕ determined by the specific position of y in the ordered variable, which represents the lower and upper limits of y*.

If you know φd, you can get the interval of φ : [φd, φ+d]. At the same time, we can find the maximum $y*:yd max*$ {y^*}:y_{d\,\max}^* and minimum $y*:yd mn*$ {y^*}:y_{d\,{\mathop{\rm mn}\nolimits}}^* corresponding to each ordinal number according to the correspondence between y and y*. We strictly control the increasing relationship of φ. This means that φd obeys a uniform conditional posterior distribution: $φ~U[max(yd max*, φd−1), min(yd min*, φd+1)$ \varphi \sim U\left[{\max \left({y_{d\,\max}^*,\,{\varphi _{d - 1}}} \right),\,\min \left({y_{d\,\min}^*,\,{\varphi _{d + 1}}} \right)} \right.

If the fully conditional posterior distribution of the parameters obeys a known type of posterior distribution, then we can use Gibbs sampling for iterative sampling. However, the conditional posterior distributions of ρ and λ are not known distributions, and the posterior mean and variance need to be obtained by numerical integration [11]. But obviously, this approach is more complicated. We use random walk Metropolis sampling. This provides an efficient way to sample parameters whose distributions are unknown. The specific sampling steps are as follows:

Step 1 Given β(0), δ(0), ρ(0), ϕ(0), we draw $y(1)*$ y_{\left(1 \right)}^* from a truncated normal distribution as in (29).

Step 2 Given $y(1)*$ y_{\left(1 \right)}^* , β(0), δ(0), ρ(0),, we sample ϕ(1) from a uniform distribution as in (30).

Step 3 Given $y(1)*$ y_{\left(1 \right)}^* , ρ(0), δ(0), we draw β(1) from the multivariate normal distribution N(μβ, Σβ).

Step 4 Given $y(1)*$ y_{\left(1 \right)}^* , β(1), δ(0), we draw ρ(1), from distribution p(ρ|) using the random walk Metropolis method.

Step 5 Given $y(1)*$ y_{\left(1 \right)}^* , β(1), ρ(1), α0(0), λ1(0), γ(0), Φ(0), we draw δ(1) from the multivariate normal distribution Nδ μδ, Σδ).

Step 6 Given δ(1), λ0(0), λ1(0), γ(0), Φ(0), we draw α1 from the multivariate normal distribution N[μα, Σα].

Step 7 Given δ(1), α1, λ1(0), γ(0), Φ(0), we draw λ0(1) from distribution p(λ0|) using the M-H method.

Step 8 Given δ(1), α1, λ0(1), γ(0), Φ(0), we draw λ1(1) from distribution p(λ1|) using the M-H method.

Step 9 Given δ(1), α1, λ0(1), λ1(1), Φ(0), we sample γ(1) from the normal distribution as in (25).

Step 10 Given δ(1), α1, λ0(1), λ1(1), γ(1), we sample Φ(1), from the distribution as in (26).

Then continue to iterate until the Markov chain of each parameter converges.

We sample the following process for t = 0, 1, 2, …, loop:

Let ρt be the value at time t. We generate candidate values ρ* of parameters from some proposed distribution Q(ρ*| ρt) = q(ρ*ρt);

We randomly sample from the uniform distribution U[0, 1] and denote it as ru;

Calculate the probability of acceptance. We prevent the denominator from approaching 0. The log determinant is easier to compute. So we make the acceptance probability as: $α(ρt, ρ*)=min[1, ln(p(ρ*|)Q(ρt|ρ*))ln(p(ρt|)Q(ρ*|ρt))]$ \alpha \left({{\rho _t},\,{\rho ^*}} \right) = \min \left[{1,\,{{\ln \left({p\left({{\rho ^*}|} \right)Q\left({{\rho _t}|{\rho ^*}} \right)} \right)} \over {\ln \left({p\left({{\rho _t}|} \right)Q\left({{\rho ^*}|{\rho _t}} \right)} \right)}}} \right] Where $ln(p(ρ|y*,β,β0j))=ln|A|−12υ′1υ1$ \ln \left({p\left({\rho |{y^*},\beta,{\beta _{0j}}} \right)} \right) = \ln \left| A \right| - {1 \over 2}\upsilon {'_1}{\upsilon _1} ;

Judgment if α (ρt, ρ*) > ru, then accept the transfer. (ρt+ 1, ρ*). Otherwise, the transfer will not be accepted.

This paper also adopts the standard normal distribution generally used in the research, that is, the candidate value generation process is ρ* = ρt + c z, z ~ N(0, 1). We restrict each candidate value to be between $(κW min−1, κW max−1)$ \left({\kappa _{W\,\min}^{- 1},\,\kappa _{W\,\max}^{- 1}} \right) . and $q(ρt−ρ*)q(ρ*−ρt)=q(−cz)q(cz)=1$ {{q\left({{\rho ^t} - {\rho ^*}} \right)} \over {q\left({{\rho ^*} - {\rho ^t}} \right)}} = {{q\left({- cz} \right)} \over {q\left({cz} \right)}} = 1 , so: $α(ρt, ρ*)=min[1, ln(p(ρ*|)ln(p(ρt|)]$ \alpha \left({{\rho _t},\,{\rho ^*}} \right) = \min \left[{1,\,\,{{\ln \left({p\left({{\rho ^*}|} \right)} \right.} \over {\ln \left({p\left({{\rho _t}|} \right)} \right.}}} \right]

Because a fixed c is likely to cause sampling to fall into an inappropriate always-reject or always-accept state, we must adjust c based on the acceptance rate. When the acceptance rate is low, we recommend re-sampling by reducing the variance of the normal random bias of the distribution. It is generally stipulated that we make c = c/1.1 when the acceptance rate is less than 40%. Conversely, we set c = 1.1* c when the acceptance rate is greater than 60%.

Variable setting

The article studies the demographic variables of respondents’ income Y, part-time job degree X1, age X2, gender X3, education level X4, and non-agricultural skills X5. Investigate the input characteristic variables of the per capita agricultural machinery and equipment X6 and the per capita arable land area X7. It also includes soft environmental variables such as rural public culture X8, agricultural direct subsidy X9, and other agricultural-friendly policies X10. Each variable is assigned according to the discrete variable processing method.

Empirical Analysis of Binary Discrete Choice Model
Goodness of fit and parameter estimation

We use the Probit function module in SPSS13.0 for analysis. By default, the system analyzes the frequency table when each variable takes different values. Specify the frequency variable here. We chose to transform with the inverse of the cumulative standard normal distribution function [12]. At this point, we are required to calculate the observed value, expected value, residual, and probability of each independent variable with different value levels. In this way, the goodness of fit and parameter estimation of the Probit equation are obtained. In the Pearson goodness-of-fit test, the chi-square value, degrees of freedom, and model value were 96.988, 119, and 0.999, respectively. This indicates that the goodness of fit of the model is good.

Table 1 shows the optimal parameter estimates after 20 iterations. At this point, we obtain the binary Pro-plus discrete choice model as: $Pr ob(Yi=1|X′i)=Φ(1.298+1.841 X1+0.061 X2+0.351 X3+0.139 X4−0.039 X5+0.250 X6+0.355 X7−0.287 X8+0.576 X9+0.561 X10)$ \matrix{{\Pr \,ob\left({{Y_i} = 1|X{'_i}} \right) = \Phi \left({1.298 + 1.841\,{X_1} + 0.061\,{X_2} + 0.351\,{X_3} + 0.139\,{X_4} - 0.039\,{X_5} +} \right.} \hfill \cr {\left. {0.250\,{X_6} + 0.355\,{X_7} - 0.287\,{X_8} + 0.576\,{X_9} + 0.561\,{X_{10}}} \right)} \hfill \cr}

Probit model regression coefficients, standard deviation estimates

Regression Coe£ Standard Error Coeff./S.E
Intercept 1.398 0.815 1.593
X1 1.814 0.777 3.78
X2 0.071 0.181 0.338
X3 0.351 0.397 0.885
X4 0.139 0.435 0.337
X5 −0.039 0.707 −0.055
X6 0.35 0.743 0.388
X7 0.355 0.778 0.533
X8 −0.387 0.385 −1.008
X9 0.577 0.417 1.385
X10 0.571 0.354 1.585
Impact Analysis

The constant term in the binary Probit discrete choice model is 1.298. This indicates that the rural economy has an initial growth momentum without other variables. Among the respective variables, the coefficient of concurrent employment is the largest (1.841), followed by the coefficients of direct agricultural subsidies and other policies that benefit farmers, and the coefficients of non-agricultural skills and rural public culture are negative [13]. The binary Probit discrete choice model can obtain the observed value, expected value, residual, and probability of each independent variable at different value levels (Table 2).

Parameter estimation of the logistic regression equation

B SE Wald DF Sig Exp(B) 95%CI for EXP(B)
Lower Lower
−1414 1152 1508 1 218 0.243
X1 5333 1712 10848 1 1 207087 8783 48778
X2 3837 3 0.278
X2 −1280 1075 1418 1 234 0.278 0.034 2287
X2 0755 1073 505 1 477 2128 0.275 17084
X2 −1573 1382 1287 1 255 0.207 0014 3110
X 3 881 785 1288 1 257 2437 523 11352
X 4 −0347 841 171 1 778 707 137 3772
X 5 −0801 1375 0.435 1 508 0.407 0028 5800
X 6 1133 1070 1121 1 280 3105 381 25285
X7 178 1112 23 1 880 1183 0.134 10457
X8 −0805 707 2232 1 135 0.405 0.123 1327
X9 1537 825 3474 1 72 4753 824 23433
X10 0.775 573 1844 1 175 2148 0712 7478
Conclusion

This paper uses Probit and Logistic binary discrete choice models to analyze the survey samples. The results show that the degree of part-time job is the decisive factor of rural economic growth. The impact of human capital and agricultural production and management methods is not significant. Therefore, we must continue accelerating the transformation and upgrading of the rural economy from traditional agriculture with a single structure to three industries. Spare no effort to promote the process of agricultural industrialization. While encouraging farmers to “go out,” farmers are trained by strengthening rural education. Migrant workers are encouraged to return to their hometowns to optimize rural human capital to start businesses. By consolidating and improving the land contract responsibility system and circulation mechanism, we can mobilize farmers’ enthusiasm for production and improve the efficiency of land resource allocation. We will continue to increase capital investment in “Sannong” while integrating capital channels, optimizing the structure of expenditures, and improving the supervision mechanism. This will improve the efficiency of the use of funds.

#### Parameter estimation of the logistic regression equation

B SE Wald DF Sig Exp(B) 95%CI for EXP(B)
Lower Lower
−1414 1152 1508 1 218 0.243
X1 5333 1712 10848 1 1 207087 8783 48778
X2 3837 3 0.278
X2 −1280 1075 1418 1 234 0.278 0.034 2287
X2 0755 1073 505 1 477 2128 0.275 17084
X2 −1573 1382 1287 1 255 0.207 0014 3110
X 3 881 785 1288 1 257 2437 523 11352
X 4 −0347 841 171 1 778 707 137 3772
X 5 −0801 1375 0.435 1 508 0.407 0028 5800
X 6 1133 1070 1121 1 280 3105 381 25285
X7 178 1112 23 1 880 1183 0.134 10457
X8 −0805 707 2232 1 135 0.405 0.123 1327
X9 1537 825 3474 1 72 4753 824 23433
X10 0.775 573 1844 1 175 2148 0712 7478

#### Probit model regression coefficients, standard deviation estimates

Regression Coe£ Standard Error Coeff./S.E
Intercept 1.398 0.815 1.593
X1 1.814 0.777 3.78
X2 0.071 0.181 0.338
X3 0.351 0.397 0.885
X4 0.139 0.435 0.337
X5 −0.039 0.707 −0.055
X6 0.35 0.743 0.388
X7 0.355 0.778 0.533
X8 −0.387 0.385 −1.008
X9 0.577 0.417 1.385
X10 0.571 0.354 1.585

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