Research on multivariate statistical analysis methods of rural economic dynamics in the context of digital countryside

For vector regression models: (23)yt=c+ϕ1yt−1+ϕ2yt−2+…+ϕpyt−p+εt$${y_t} = c + {\phi_1}{y_{t - 1}} + {\phi_2}{y_{t - 2}} + \ldots + {\phi_p}{y_{t - p}} + {\varepsilon_t}$$

Among them: (24)E(εt)=0 E(εtεr)={ σ2 t=τ 0 t≠τ$$\begin{array}{*{20}{l}} {E\left( {{\varepsilon _t}} \right) = 0} \\ {E\left( {{\varepsilon _t}{\varepsilon _r}} \right) = \left\{ {\begin{array}{*{20}{c}} {{\sigma ^2}}&{t = \tau } \\ 0&{t \ne \tau } \end{array}} \right.} \end{array}$$

If we extend it further, we need to consider the dynamic interactions between multiple variables, i.e., let y_t be a (n × 1)-vector [27]. By setting the lag order of the regression to p, the model is written as VAR(p): (25)yt=c+Φ1yt−1+Φ2yt−2+…+Φpyt−p+εt$${y_t} = c + {\Phi_1}{y_{t - 1}} + {\Phi_2}{y_{t - 2}} + \ldots + {\Phi_p}{y_{t - p}} + {\varepsilon_t}$$

Here c represents a (n × 1) vector of constant terms. Φ_j is a (n × n) matrix of autoregressive coefficients, j = 1, 2, …, p. ε_t is a (n × 1) white noise vector characterized by: (26)E(εt) = 0 E(εtετ) = { Ω t=τ 0 t≠τ$$\begin{array}{rcl} E\left( {{\varepsilon_t}} \right) &=& 0 \\ E\left( {{\varepsilon_t}{\varepsilon_\tau }} \right) &=& \left\{ {\begin{array}{*{20}{l}} \Omega &{t = \tau } \\ 0&{t \ne \tau } \end{array}} \right. \\ \end{array}$$

where Ω is a (n × n)-symmetric positive definite matrix.

Let c_i denote the ith element of vector c and let ϕij1$$\phi_{ij}^1$$ denote the ith row and jth column elements of the matrix. Then the VAR model is written in the form of a single equation as: (27)y1t = c1+ϕ111yl(t−1)+ϕ121y2(t−2)+…+ϕ1n1yn(n−1) +ϕ112y1(t−2)+ϕ122y2(t−2)+…+ϕ1n2yn(n−2) +…+ϕ11py1(t−p)+ϕ12py2(t−p)+…+ϕ1npyn(t−p)$$\begin{array}{rcl} {y_{1t}} &=& {c_1} + \phi_{11}^1{y_{l(t - 1)}} + \phi_{12}^1{y_{2(t - 2)}} + \ldots + \phi_{1n}^1{y_{n(n - 1)}} \\ &&+ \phi_{11}^2{y_{1(t - 2)}} + \phi_{12}^2{y_{2(t - 2)}} + \ldots + \phi_{1n}^2{y_{n(n - 2)}} \\ &&+ \ldots + \phi_{11}^p{y_{1(t - p)}} + \phi_{12}^p{y_{2(t - p)}} + \ldots + \phi_{1n}^p{y_{n(t - p)}} \\ \end{array}$$

VAR(p) system, each variable regresses on the constant term and its pnd-order lagged value simultaneously on the pth-order lagged values of VAR(p) the other variables in the system.

For VAR(p) process y_t, the VAR(p) process is covariance smooth if its first-order moments E(yt)$$E\left( {{y_t}} \right)$$ and second-order moments Eyty′t−j$$E{y_t}{y'_{t - j}}$$ are independent with respect to moment t. After determining that the VAR(p) process covariance is smooth, we deform the equation by taking the expectation, and the mean μ value obtained for this process is: (28)μ=c+Φ1μ+Φ2μ+…+Φpμ$$\mu = c + {\Phi_1}\mu + {\Phi_2}\mu + \ldots + {\Phi_p}\mu$$

Organize to get: (29)μ=(In−Φ1−Φ2−…−Φp)−1c$$\mu = {\left( {{I_n} - {\Phi_1} - {\Phi_2} - \ldots - {\Phi_p}} \right)^{ - 1}}c$$

The system equation is written in the departure form as: (30)(yt−μ)=Φ1(yt−1−μ)+Φ2(yt−2−μ)+…+Φp(yt−p−μ)+εt$$\left( {{y_t} - \mu } \right) = {\Phi_1}\left( {{y_{t - 1}} - \mu } \right) + {\Phi_2}\left( {{y_{t - 2}} - \mu } \right) + \ldots + {\Phi_p}\left( {{y_{t - p}} - \mu } \right) + {\varepsilon_t}$$

Definition: (31)ξt=[ yt−μ yt−1−μ ⋯ yt−p+1−μ]$${\xi_t} = \left[ {\begin{array}{*{20}{c}} {{y_t} - \mu }&{{y_{t - 1}} - \mu }& \cdots &{{y_{t - p + 1}} - \mu } \end{array}} \right]$$ (32)F=[ Φ1 Φ2 Φ3 ⋯ Φp−1 Φp In 0 0 ⋯ 0 0 0 In 0 ⋯ 0 0 ⋮ ⋮ ⋮ ⋯ ⋮ ⋮ 0 0 0 ⋯ In 0]$$F = \left[ {\begin{array}{*{20}{c}} {{\Phi_1}}&{{\Phi_2}}&{{\Phi_3}}& \cdots &{{\Phi_{p - 1}}}&{{\Phi_p}} \\ {{I_n}}&0&0& \cdots &0&0 \\ 0&{{I_n}}&0& \cdots &0&0 \\ \vdots & \vdots & \vdots & \cdots & \vdots & \vdots \\ 0&0&0& \cdots &{{I_n}}&0 \end{array}} \right]$$ (33)Vt=[ εt 0 ⋯ 0]$${V_t} = \left[ {\begin{array}{*{20}{l}} {{\varepsilon_t}}&0& \cdots &0 \end{array}} \right]$$

Thus VAR(p) is further written in matrix form as: (34)ξt=Fξt−1+Vt$${\xi_t} = F{\xi_{t - 1}} + {V_t}$$

F(VtVt′)={ Q t=τ 0 t≠τ$$F\left( {{V_t}{V_t}^\prime } \right) = \left\{ {\begin{array}{*{20}{l}} Q&{t = \tau } \\ 0&{t \ne \tau } \end{array}} \right.$$ of them. (35)Q=[ Ω 0 ⋯ 0 0 0 ⋯ 0 ⋮ ⋮ ⋯ ⋮ 0 0 ⋯ 0]$$Q = \left[ {\begin{array}{*{20}{c}} \Omega &0& \cdots &0 \\ 0&0& \cdots &0 \\ \vdots & \vdots & \cdots & \vdots \\ 0&0& \cdots &0 \end{array}} \right]$$

The equation implies: (36)ξt+s=vt+s+Fvt+s−1+F2vt+s−2+…+Fs−1vt+1+Fsvt$${\xi_{t + s}} = {v_{t + s}} + F{v_{t + s - 1}} + {F^2}{v_{t + s - 2}} + \ldots + {F^{s - 1}}{v_{t + 1}} + {F^s}{v_t}$$

The VAR model is covariance smooth if the eigenvalues of the F test are all within the unit circle, i.e., is for the roots of the characteristic equation |Inλp−Φ1λp−1−Φ2λp−2−…−Φp|=0$$\left| {{I_n}{\lambda^p} - {\Phi_1}{\lambda^{p - 1}} - {\Phi_2}{\lambda^{p - 2}} - \ldots - {\Phi_p}} \right| = 0$$ fall within the unit circle, i.e., |λ|<1$$\left| \lambda \right| < 1$$ [28].

Granger Causality Test Given an information set A_t, which contains at least (Xt,Yt)$$\left( {{X_t},{Y_t}} \right)$$, if “past and present changes can affect future outcomes, while future changes do not affect the past” holds, X_t is said to be the Granger cause of Y_t if the use of X_t past is a better predictor of Y_t than when it is not utilized, and the same is true anyway.

The Granger causality test model is: if X_t, Y_t is a smooth process, for the model: (37){ Xt=c1+∑j=1pαjXt−j+∑j=1qβjYt−j+ε1t Yt=c2+∑j=1pγjYt−j+∑j=1qδjXt−j+ε2t$$\left\{ {\begin{array}{*{20}{l}} {{X_t} = {c_1} + \sum\limits_{j = 1}^p {{\alpha_j}} {X_{t - j}} + \sum\limits_{j = 1}^q {{\beta_j}} {Y_{t - j}} + {\varepsilon_{1t}}} \\ {{Y_t} = {c_2} + \sum\limits_{j = 1}^p {{\gamma_j}} {Y_{t - j}} + \sum\limits_{j = 1}^q {{\delta_j}} {X_{t - j}} + {\varepsilon_{2t}}} \end{array}} \right.$$

ε₁, ε₂ for white noise. The following cases exist: 1)

If β_j = δ_j = 0(j = 1, 2, ⋯, q), then X_t, Y_t are independent of each other.

Obviously, 3) and 2) show a one-way bootstrapping relationship (causality), 4) is a two-way feedback relationship (causality), and 1) is an independent relationship.

The Granger causality F test is: for Xt=c1+∑j=1pαjXt−j+∑j=1qβjYt−j+ε1t$${X_t} = {c_1} + \sum\limits_{j = 1}^p {{\alpha_j}} {X_{t - j}} + \sum\limits_{j = 1}^q {{\beta_j}} {Y_{t - j}} + {\varepsilon_{1t}}$$ perform the test by setting H₀ : β_j = 0, H₁ : β_j ≠ 0. j = 1, 2, ⋯, q Estimate the 0LS for this model, noting the residual sum of squares as ESS_X(q, p) [29]. Estimate model Xt=c1+∑j=1pαjXt−j+εt$${X_t} = {c_1} + \sum\limits_{j = 1}^p {{\alpha_j}} {X_{t - j}} + {\varepsilon_t}$$ again, noting the residual sum of squares as ESS_X(p). Constructor statistic: (38)FX=[ESSX(p)−ESSX(q,p)]/pESSX(q,p)/(n−p−q−1)~F(p,n−p−q−1)$${F_X} = \frac{{\left[ {ES{S_X}(p) - ES{S_X}(q,p)} \right]/p}}{{ES{S_X}(q,p)/(n - p - q - 1)}}\sim F(p,n - p - q - 1)$$

Given confidence level α, look up the critical value F_α and consider Y to be the Granger cause of X if F > F_α, reject H₀, and accept H₁.

Similarly, the original and alternative hypotheses are H₀ : δ_j = 0, H₁ : γ_j ≠ 0 for the model Yt=c2+∑j=1pγjYt−j+∑j=1qδjXt−j+ε2t$${Y_t} = {c_2} + \sum\limits_{j = 1}^p {{\gamma_j}} {Y_{t - j}} + \sum\limits_{j = 1}^q {{\delta_j}} {X_{t - j}} + {\varepsilon_{2t}}$$ test. j = 1, 2, ⋯, q The 0LS estimation of this model is recorded as the residual sum of squares as ESS_Y(q, p), and then the 0LS estimation of model Yt=c2+∑j=1pγjYt−j+εt$${Y_t} = {c_2} + \sum\limits_{j = 1}^p {{\gamma_j}} {Y_{t - j}} + {\varepsilon_t}$$ yields a residual sum of squares of ESS_Y(p) and an F-statistic of: (39)FY=[ESSY(p)−ESY(q,p)]/pESSY(q,p)/(n−p−q−1)~F(p,n−p−q−1)$${F_Y} = \frac{{\left[ {ES{S_Y}(p) - E{S_Y}(q,p)} \right]/p}}{{ES{S_Y}(q,p)/(n - p - q - 1)}}\sim F(p,n - p - q - 1)$$

If H₀ is rejected and H₁ is accepted, consider X to be the Granger cause of Y.

The method of cointegration test is suitable for testing the existence of only one cointegration relationship between variables. Take two variables x and y as an example, first, we team two variables for single integrality test, such as two variables single integrality is the same, the cointegration relationship is established. Let’s assume that x and y are first-order single-integrated series, i.e., x ~ I(1) and y ~ I(1), then the steps of EG two-step test are as follows:

First, the following model is estimated by applying the least squares method: (40)yt=β0+β1xt+εt$${y_t} = {\beta_0} + {\beta_1}{x_t} + {\varepsilon_t}$$

And calculate the corresponding residual series et=yt−(β^0+β^1xt)$${e_t} = {y_t} - \left( {{{\hat \beta }_0} + {{\hat \beta }_1}{x_t}} \right)$$

Second, test the smoothness of the residual series in the model with the following tests: (41)Δet=δet−1+∑i=1mγiΔet−i+εt$$\Delta {e_t} = \delta {e_{t - 1}} + \sum\limits_{i = 1}^m {{\gamma_i}} \Delta {e_{t - i}} + {\varepsilon_t}$$ (42)Δet=α+δet−1+∑i=1mγiΔet−i+εt$$\Delta {e_t} = \alpha + \delta {e_{t - 1}} + \sum\limits_{i = 1}^m {{\gamma_i}} \Delta {e_{t - i}} + {\varepsilon_t}$$ (43)Δet=α+βt+δet−1+∑i=1mγiΔet−i+εt$$\Delta {e_t} = \alpha + \beta t + \delta {e_{t - 1}} + \sum\limits_{i = 1}^m {{\gamma_i}} \Delta {e_{t - i}} + {\varepsilon_t}$$

If the original hypothesis H₀ : δ = 0 is rejected by the DF test (or ADF test), which means that the residual series is smooth, then there is a cointegration relationship between x and y and the cointegration equation is y_t = β₀ + β₁x_t + ε_t.

Some scholars are concerned about the impact of population changes and changes in the unemployment rate on the rate of economic growth. In addition, some scholars believe that changes in the level of per capita income, labor productivity and other growth will also have a greater impact on future economic growth. In terms of future economic development, there is still much room for growth in per capita income level, productivity, factor prices, etc. Obviously, China’s GDP growth rate has a close relationship with population and productivity, and this paper believes that the three variables of per capita output growth, per capita income growth and labor productivity growth can be used to study the GDP growth rate.

Per capita output growth is the growth rate of labor productivity of employees, labor productivity growth is considered to be the growth rate of labor productivity per hour, and per capita income growth has a simultaneous change pattern with the change in per capita GDP growth rate. The vector autoregressive model can take each endogenous variable as the lagged value of all endogenous variables in the system, which is the rule in the unstructured model, according to which the dynamic correlation between different variables in the model can be mined. The author utilizes the GDP data of a village for a total of 12 months from January 2013 to December 2013 for empirical analysis. The multivariate regression model of the dynamic changes in the rural economy is established as: (44)Y=β0+β1X1+β2X2+β3X3+β4X4+β5X5+β6X6+μ$$Y = {\beta_0} + {\beta_1}{X_1} + {\beta_2}{X_2} + {\beta_3}{X_3} + {\beta_4}{X_4} + {\beta_5}{X_5} + {\beta_6}{X_6} + \mu$$

Where Y is the rural GDP, X1 is the monthly average price index of primary agricultural products, X2 is the monthly average price index of general agricultural products, X3 is the monthly data of the dollar exchange rate, X4 is the monthly data of the consumer price index of farmers, X5 is the monthly data of the price index of agricultural means of production, and X6 is the monthly data of the money supply M2. μ is the random error term.

The smoothness of the ln GDP, ln DI, lnFDI and ln L series was tested by ADF, and the results of the ADF unit root test for the series are shown in Table 3 (c, t, n in the type of the test (c, t, n) denote that the unit root test equation contains a constant term, a time trend, and a lagged order, respectively, and 0 denotes that it does not contain it. p-value is the probability value of MacKinnon’s one-sided test. (*, **, *** denote the rejection of the original hypothesis at 10%, 5%, and 1% significance levels, respectively, and the variables are stable at the corresponding significance levels). The results show that lnGDP, lnDI, ln FDI and ln L are non-stationary series, and after first-order differencing respectively, only Δln FDI and Δln L are stationary and there is no unit root at 10% and 1% significance level respectively. Further second order differencing of the variables, all the variables reject the original hypothesis at 1% significance level and there is no unit root, therefore, these variables are I(2) series.

Since all the variables are second-order single-integrated series, cointegration test can be performed and Johansen cointegration test is adopted. Combining the five indicators of LR, FPE, AIC, SC and HQ, the optimal number of lags of the VAR model is determined to be 3. At this time, the inverse of the modes of all the roots of the VAR(3) model are within the unit circle, and the AR root diagram of the VAR(3) model is shown in Figure 1. The model has stability. The lag order chosen for the cointegration test should be 2 (it is equal to the optimal lag order of the unconstrained VAR model minus 1).

Figure 1.

The AR root diagram of the VAR(3) model

The results of Johansen cointegration test are shown in Table 4. The results of the tests for the maximum characteristic root and trace statistic simultaneously indicate that there are 2 cointegration relationships between lnGDP, ln DI, ln FDI and ln L at the 1% significance level. The cointegration vector about ln GDP is regularized to obtain the standardized cointegration vector and the cointegration equation.

The standardized cointegration vector is shown in Table 5. From the table and the cointegration equation, it can be seen that the coefficients of lnDI, lnFDI and ln L are more significant, and they have an impact on lnGDP, and there is a long-run stabilizing relationship between the variables. The sign of each coefficient indicates that investment (DI), foreign direct investment (FDI) and the number of social employees (L) move positively with GDP, which is consistent with economic significance. The degree of influence of DI, FDI and L on GDP varies significantly, and the internal rural investment has the greatest impact on GDP, with a 1% increase in rural investment triggering a 0.365521% increase in GDP. This is followed by FDI. The smallest impact is on the number of people working in society, with GDP rising by 0.092311% for every 1% increase in the number of people working.

The above cointegration test indicates that there is a long-term stable relationship between rural internal investment, foreign direct investment, the number of social workers and GDP, but not necessarily constitute a causal relationship, so it is necessary to further test the Granger causality between the variables, Granger causality test as shown in Table 6 (**, *** indicates the rejection of the original hypothesis at the 10%, 5%, 1% significance level, respectively). The analysis shows that intra-rural investment, foreign direct investment and the number of people working in the society are Granger causes of GDP growth, and GDP is only a Granger cause of the increase in intra-rural investment. Increase in intra-rural investment creates environment, conditions and opportunities to attract more labor and foreign direct investment, it is the Granger cause of increase in the number of employees as well as foreign direct investment. Foreign direct investment contributes to an increase in rural capital, which has an accelerating effect on the growth of intra-rural investment and is the Granger cause of the increase in intra-rural investment. The increase in the number of employees can lead to the growth of the regional economy, which in turn leads to an increase in foreign direct investment, and the number of employees is the Granger cause of the increase in foreign direct investment at the 10 per cent significance level.

The impulse corresponding analysis graph is shown in Fig. 2, (a) from the 3rd period onwards the first-class agricultural products have some pulling effect on the gross product, but the sustained effect is not long, and in the 4th period this pulling effect will begin to weaken until the 8th period when the pulling effect will begin again. (b) General agricultural products out show a fluctuating negative effect on the GDP of rural areas. (c) Dollar exchange rate shocks cause inverse changes in rural GDP, with the negative effect gradually appearing from period 4 onwards. (d) Farmers’ consumption of rural GDP shows a large positive impact of the shock, but it is worth noting that the positive effect has a tendency to weaken gradually. (e) Agricultural means of production also shows a positive impact, and there is a gradual strengthening of the trend, mainly due to the continuous adjustment of the economic growth mode and industrial structure within the countryside, the tertiary industry plays a pivotal role in promoting economic growth in rural areas. (f) Money supply has a relatively stable positive impact on GDP in rural areas, but the impact is not significant.

Figure 2.

Esponse to cholesky one S.D. innovations±2 S.E.

The results of LNGDP variance decomposition (%) are shown in Table 7. From the table, it can be seen that the change in GDP growth in rural areas in the long run is affected by the impact of its own disturbance term in a gradually decreasing trend, from the initial 100% to 19.57%. Whereas, all the other economic variables’ disturbance terms have an increasing effect on the GDP growth. About 20% of the GDP of rural areas is determined by itself, about 35% by the total value of the tertiary industry, about 20% by the net export, and the lag effect of both of them on the GDP is very obvious, about 10% by fixed investment, about 6% by the consumption of villagers, and 5% to 7% by the fiscal expenditure and the number of students enrolled in general higher education, which is a full manifestation of the fact that the amount of the net export and the gross value of tertiary industry has a significant impact on rural GDP, which is highly consistent with the previous analysis.