2.1 Partial differential decomposition method and theoretical hypothesis

Obviously, tax and fee reduction policies has significant spillover effects. Based on this, this study introduced to Dixit-Stiglitz monopoly competition model (monopoly competition D-S model), and added tax and fee reduction policies adjustment around the core-edge theory. The core-edge theoretical model is an invariant effect function of substitution elasticity (CES) nested in the Cobb-Douglas (CD) function. On the basis of this framework structure and for the convenience of subsequent mathematical model derivation, the following assumptions are proposed: the first, there are two open regions. The free flow of production factors across borders can be realized, which is the key to explore the influence of externalities; The second, through combing relevant literature, it is found that developed regions have good infrastructure, which can attract more investment and significantly promote technological progress [12]. Therefore, technological progress in developed regions can generate spillover, while relatively poor regions mainly absorb the technology spillover from developed regions as the driving force of their own innovation and development due to the limitation of economic development level.

It is assumed that there are two types of productive economies in the economy, which include labor-intensive and capital-intensive sectors. The production function calculates output in terms of input factors, included labor L and capital K. Then it is assumed that the output of labor-intensive sectors is P_L, and that of capital-intensive sectors is P_K. At this point, the region’s gross income can be expressed as P_Y = F(P_L, P_K), and the CES production function after substituting each factor of production is (1)

In the above formula, γ and 1 − γ are the proportion of the contribution of labor and capital to the total output respectively, and κ is the elasticity of substitution of the two production factors. In addition, it is assumed that the final output is only related to the factor input and its corresponding technology level, and the technology level of labor-intensive industry is N_L, and the technology level of capital-intensive industry is N_K, then the following two expressions can be obtained: (2) (3)

Eqs (2) and (3) describe the maximum benefits that can be obtained by each production department when the input of a certain factor is kept constant. In addition, considering the regions with backward economic development level and assuming that the developed regions have external technology spillovers, the influence coefficient is set as β. Existing studies have confirmed that the intensity of technology spillover is closely related to the technology gap between rich and poor regions [13]. Obviously, the relationship between the two can be represented as , N_i refers to the level of technological development of developed areas, refers to the level of technological progress of backward areas, which can be expressed as follows: (4) (5) λ, λ′ and additional items ε, ε′ are the parameters that distinguish the impact of technological progress such as independent research and development from other factors on the output of two economies. Through the theoretical inference and research hypothesis analysis above, the determinants of technological development level in backward areas can be summarized as follows: First, local fiscal and tax policies. This study focuses on tax reduction policies, assuming tax reduction T_i; Second, developed areas of technology spillover β. Based on the above assumptions, the level of technological progress in economically backward areas can be expressed as: (6)

Considering that local fiscal and tax policies can affect the level of scientific and technological development [14, 15], and no matter whether the government reduces tax on a certain technological research project, the existing level of the technology is not equal to 0, so the function condition in Eq (6) can be set: f(0) > 0 and any T_i ≥ 0, there is f′(T_i) > 0. In addition, measures the technological difference reflected by two economies at different levels of development, where α indicates the effect intensity of the gap affecting the technology spillover. In general, there is , and δ_i controls the relevant influencing factor. Therefore, the profit function of the different development sectors can be expressed in the following form: (7) (8) (9) (10)

The parameters of the above equation are: P_i and . They are the prices of output products of different types (such as labor-intensive and capital-intensive) in the two economies respectively. Where μ is the wage rate and φ is the rent rate. In an economy, that can realize free flow of factors. Although the economic development level of each region is different, the wage rate and the rent rate can eventually be in a stable state of equilibrium. ω(x) represents the independent research and development costs of different factor production departments under the influence of government tax and fee reduction policies. From what has been discussed above, as far as backward areas are concerned, the first-order partial derivatives of T_i are respectively solved for Eqs (9) and (10) above, and the results are substituted into Eq (6) to obtain: (11)

Substitute Eq (2) into the first derivative of Eq (1), and express it as the ratio of price levels of two factors in relatively backward areas, and Eq (12) can be obtained: (12)

Substituting Eq (12) into Eq (11), we get: (13)

Further, Acemoglu proposed that technological development can adjust the marginal output of capital or labor input and influence its selection bias, so as to cause the technological bias of capital or labor [16]. The long-term investment and development of a regional bias will lead to technological progress such as independent research and development in the region [17]. Based on the research of Acemoglu and Lerner, this study takes the derivative of Eqs (1) and (2), and substitutes the corresponding factor selection for each development region, so that the marginal output of capital and labor can be obtained, namely [16, 18]: (14) (15) (16) (17)

And the technological selectivity bias of independent research and development in different economies are: (18) (19)

In the above formula, , , G′ represent the variables of relatively backward economies, while MP_L, NP_K, G represent the relative variables of relatively developed economies. By importing Eq (13) into Eq (18), we can get: (20)

Finally, after transforming Eq (19) into Eq (20), we can get: (21)

From Eq (21) above, it can be seen that fiscal tax and fee reduction policies (η(T′)) can affect the technological bias of independent research and development in a region. At the same time, the more developed economies (G) have technology spillover to the underdeveloped regions (G′). The specific action mode is determined by the factor substitution elasticity of the two economies (κ, κ′) and the related technology spillover intensity coefficient (α). Based on this, research hypothesis H1 of this study can be put forward:

1. H1: Fiscal tax and fee reduction policies are closely related to the technological preference of the region, and act on the surrounding areas by spillover effect, which have certain externality.

Further, the Eq (21) can be transformed into: (22)

From Eq (22) above, it can be seen that local fiscal competitive expenditure strategy will act on the selection of labor and capital factors, specifically affecting the elasticity of substitution of labor and capital factors (κ, κ′). At the same time, focusing on Eq (22), the coefficient κ on the right of the equal sign changes with the variation of the coefficient coefficient on the left, which reflects the imitation and assimilation of the production types of factors in developed economies in underdeveloped regions. Along with the selective agglomeration of factors in the two economies, the spillover expansion of any of the advantages of the adjacent areas will occupy the choice of the advantages of the independent research and development of the surrounding areas. Based on this, hypothesis H2 can be put forward:

1. H2: Tax and fee reduction policies will affect the selection of labor and capital factors for independent research and development in this region. At the same time, this advantageous factor will lead the technological research and development direction of the whole region by being imitated by neighboring regions.

However, combining with the analysis hypothesis H1, H2 and the corresponding assumptions, it reveals the existence of another case that regional tax and fee reduction policies can directly affect the choice of local independent research and development technology elements, and the advantage of technological progress factor will be closely related to the adjacent area of fiscal policy. This reveals that tax policy overflow is not a one-way diffusion, but is a bidirectional regulation effect. Based on the above analysis, hypothesis H3 can be put forward:

1. H3: The choice of tax and fee reduction policies on the advantageous elements of independent research and development is not a single spillover. The externality of tax and fee reductions on independent R&D is not one-way, but has a two-way effect of spillover and feedback. The developed regions not only serve as the starting point of technology spillover, but also receive feedback from the underdeveloped regions.

2.2 Spatial Durbin Model (SDM) method

Tax and fee reduction policies have a significant impact on independent innovation, and the scope of impact is not limited to fixed geographical space or administrative divisions. Due to the externality of economy, the specific ways of policy effect and innovation are also different. Is it the expression of direct effect or indirect effect? Is there regional feedback? In order to verify the externalities and spatial spillovers of tax and fee reduction policies, and examine their impact on independent research and development, this study uses the Spatial Durbin Model (SDM) proposed by Lesage and Pace for spatial model fitting [19].

There are three general spatial models: Spatial Lag Model, Spatial Error Model and Spatial Durbin Model [20]. We need pay more attention to the parameters in the three model, because the transformation of parameters will define the model. The traditional effect estimation model generally selects Panel Data Model, Differences-in-Differences, Regression Discontinuity and other policy evaluation methods, but the traditional methods cannot achieve the spatial correlation between variables. The traditional measurement model is believed that individuals are independent of each other, but in real life, we should recognize that individuals are interconnected and communicate with each other [21]. For example, the coastal provinces are economically developed and geographically close, which shows the spatial connection; Due to the existence of geographical distance, there will be a certain relationship between individuals, which is represented by the spatial weight matrix. The spatial model, we use different weight matrices such as adjacency and distance to define the geographical relationship between individuals. Different matrix calculation methods are different, and the matrices can be nested. The explained variables are not only influenced by spatial distance, but also by the degree of correlation between economic activities. The model sets as follows: (23)

Since the 1970s, the rapid development of geographic information technology has promoted the increasingly rich spatial data. The development of spatial data has caused many scholars to pay attention to and investigate the spatial location factors in the field of regional development. The spatial effect has led to great doubts about the independence assumption of variables and the reliability of regression parameters in traditional econometrics, and spatial econometrics came into being. Anselin put forward the classic definition of spatial econometrics, and put the spatial interaction and spatial structure of economic activities into the consideration of econometrics, and established a spatial model. He believed that the spatial effect of the spatial relationship between the variables could be divided into spatial correlation and spatial heterogeneity. The introduction of spatial effect is the main difference between spatial econometrics and traditional econometrics. Spatial correlation refers to the mutual influence of various variables in different locations, that is, the observed values in different spatial locations are consistent, such as spillover and proximity effects, which lead to the influence of a variable on other variables around. Spatial Durbin Model (SDM) is a combination and extension of Spatial Lag Model and Spatial Error Model, which can be established by adding corresponding constraints to the Spatial Lag Model and Spatial Error Model. It is an enhanced SAR model (Spatial Lag Model) by adding spatial lag variables. The SDM is characterized by taking into account the spatial lag correlation between the explanatory variable and the explained variable. In order to analyze all the effects of the explanatory variables in detail, they are divided into direct effects and indirect effects according to their sources. Among them, the direct effect can be divided into two types, one is the direct influence of the explanatory variable on the explained variable, and the other is the feedback effect caused by the independent variable influencing the dependent variable in the adjacent region; The indirect effect is the spatial spillover effect of explanatory variables, which can also be divided into two types. One is the influence of the independent variables of neighboring regions on the dependent variables of the region, and the other is the influence of the independent variables of neighboring regions on the dependent variables of the region.

The above equation is the concrete construction form of static Spatial Durbin Model, and on the basis of testing whether the above equation has significant direct and indirect effects, the dynamic Spatial Durbin Model can be further constructed. On the one hand, in order to ensure the comprehensiveness of spatial panel data information. On the other hand, partial endogenity can be overcome to some extent by introducing the lag period of explained variables. The specific setting of dynamic Spatial Durbin Model is as follows: (24)

Eq (24) explains the acting conditions and spillover externalities of the effect of tax and fee reduction policies on independent research and development, and illustrates the dynamic spatial effect of the connection between the effect of tax and fee reduction policies (E_it) and independent research and development (Y_it). Where, i and t are the parameters of cities and years selected for analysis in this study. ω_it is the spatial weight matrix set in this chapter and C_it is other control variables. μ_i and λ_i represent the fixed effect of city and time respectively, and ε_it is a random error term. Y_t−1 is the explained variable lag, and the dynamic coefficient of ρ is measured to independent research and development level of spatial autocorrelation coefficient. β represents the direct effect coefficient of tax and fee reduction policies, and γ is the spatial spillover effect coefficient after the implementation of the policy. And the parameters of each control variable are to be set as δ. Then the above parameters are the parameters to be estimated in this study.

2.3 Variables and data sources

2.3.1 Selection of variables.

(1) Explanatory variables. Tax and fee reduction variables: Most literature focuses on changes in tax burden rates and the reduction of fees [22]. However, China has a wide range of taxes and diverse charging methods, and there is currently no effective way to account for all changes. Considering that fiscal revenue includes tax revenue and non tax revenue, we start from the concept of small finance and choose the change in fiscal revenue as the variable of tax and fee reduction, while retaining the corresponding symbols of the data to cope with negative changes. In order to mitigate data fluctuations and control the heteroscedasticity, the variable is logarithmized, and the difference form of the indicator in the previous and last two years is used.

It should be emphasized that the policy effect of reducing taxes and fees is often directly reflected through fiscal or tax competition among local governments [23, 24]. Most studies choose the proportion of public financial expenditure to measure the intensity of local competition [25], but economic research data are limited by personnel, methods and tools, and often have deviations from the actual information. This study uses DMSP/OLS night light data to describe local financial Porter’s generic strategies, to replace the main explanatory variables for robustness testing [26]. Night light data, as a description index of regional economy such as infrastructure and economic development level, can overcome the false information in economic research data and avoid the endogeneity of reverse causality between economic variable data.

DMSP/OLS Night Lights Raw Data: DMSP originated from the U.S. Department of Defense’s Weather Satellite Program. The data selected in the paper are the non-radiometric calibrated version IV published by NGDC. We use the fixed target area method as the correction method for acquiring light images, and use the Lambert equiangular cone form for re projection of DMSP/OLS night light data. We also adjust the resampling range to 1 kilometer, and then crop it to the average intensity of a non-radiometric calibration form with a spatial resolution of 1 kilometer for extraction and denoising. The time span of the nighttime light data we can collect is 2000–2013, involving 243 Prefecture-level city nationwide. Finally, the composite average value of the light intensity collected by each province is taken as the substitute variable for the robustness test.

(2)Control variables. A series of other control variables are mainly to measure the level of local economic development, because the level of economic development in a region usually determines the endogenous driving force of independent research and development [27]. The specific indicators selected in this study are: economic growth rate, population dependency ratio, population growth rate, opening level, consumer price index, unemployment rate. Among them, the economic growth rate has a strong correlation with the level of independent research and development of a country or region; The degree of urbanization can represent the overall development level of a city, and can also be directly used to evaluate the economic development level and its derivative can partly represent the growth rate; Talents are regarded as the basis of innovation, so population variables, such as dependency ratio and Population growth, are introduced in this paper; As a description of the development level of an economy, indicators such as openness level, consumer price index and unemployment rate all reflect the realization basis of independent research and development, which also need to be introduced; The degree of openness of a region to the outside world determines the impact of foreign investment on the economy, and we have chosen foreign direct investment/per capita GDP to represent it. Detailed variables and data are shown in Table 1. And to control the heteroscedasticity, the logarithm of the above variables is taken.

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Table 1. Variable description and statistical characteristics.

https://doi.org/10.1371/journal.pone.0288535.t001

(3) Explained variables. The ratio of R&D expenditure to operating income is used to interpret the level of independent research and development. Furthermore, this article replaced the dependent variable in the robustness test. Decompose the Malmquist productivity index to obtain an indicator that characterizes the level of independent innovation—the technological progress index. The technological progress index can reflect R&D investment, output capacity, and the independent innovation situation of enterprises. Malquist index is a measure of the output input ratio, used to assess production efficiency and can be decomposed into several sub efficiency indicators. The calculation method of Malmquist index can be expressed by a formula: Malmquist index can be decomposed into technical progress index and technical efficiency change index. Among them, the technological progress index is the ratio of productivity indices at two time points, and the technological efficiency change index is the ratio of productivity indices at two time points divided by the technological progress index. The Malmquist technology progress index is selected in this manuscript, and the specific calculation is: . x represents the R&D expenditure for period t, which is the indicator value of scientific and technological progress investment. y represents the output profit value of period t, which is the indicator value of scientific and technological progress output, and is the distance function. See Table 1 for detailed data description.

3.1 Spatial autocorrelation test

Spatial autocorrelation statistics are a fundamental property used to measure geographic data: the degree of interdependence between data at a certain location and data at other locations. Usually, this dependency is called spatial dependency. Due to the influence of spatial interaction and diffusion, geographic data may no longer be independent of each other, but rather related. Spatial autocorrelation is an important starting point for the setting of spatial models. It is a method to calculate the correlation of a variable to a spatial location and an effective way to measure the aggregation degree of adjacent areas in space. The spatial autocorrelation test can be divided into global and local spatial autocorrelation according to the size of the set spatial range. At present, in the field of spatial research, there are many methods for testing and estimating spatial autocorrelation. However, the mainstream economic works and research literatures show that most scholars generally agree on the estimation method of Moran’s I index, which can make the most effective statistics on spatial distribution characteristics and differences. Moran’s I index covers not only global Moran’s I index, but also local Moran’s I index, which is also known as LISA. We use the Moran’s I index for spatial autocorrelation test, where the spatial weight matrix selects the spatial adjacency matrix (0–1 matrix). The spatial weight of adjacent provinces is 1, and the spatial weight of non adjacent provinces is 0. The Moran’s I index is an indicator used to measure spatial correlation, and its core is the spatial adjacency matrix. Before setting the spatial model, we first tested the spatial correlation and specifically calculates the global Moran’s I index for the level of independent research and development, which was estimated by GEODA in this study.

Fig 2 shows the line chart of Moran’s I index and corresponding P value in each year from 1998 to 2019. It can be seen from the data in the figure that the value of Moran’s I index can be positive or negative, and the corresponding P value is within the significance level of 5%, which indicates that there is spatial correlation between independent research and development. The positive or negative spatial autocorrelation needs to be further verified. In addition, on the basis of measuring local Moran’s I index, we also draw a scatter plot of Moran’s I index, as shown in Fig 1 in each year from 1996 to 2019 (due to space limitations, please refer to the attached Moran’s I Index chart for the year 1998–2019). The Fig 1 shows that the spatial similarity of independent research and development of various provinces in China is obvious in a certain period, while the spatial heterogeneity in a certain period also indicates that there is no consistency in the spatial dependence characteristics, and the matching types of neighboring regions have different attributes. Figs 1 and 2 show that the model is spatially dependent, and the Moran’s I exponent symbol is not uniform,which indicate that the spatial autocorrelation may be positive, and there is a negative possibility.

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Fig 1. Moran’s I index from 1996 to 2019.

https://doi.org/10.1371/journal.pone.0288535.g001

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Fig 2. Moran’s I index and corresponding P value from 1998 to 2019.

https://doi.org/10.1371/journal.pone.0288535.g002

In order to test the spatial characteristics and determine the rationality of the Spatial Lag Model and Spatial Error Model, we also add the robust LM test to demonstrate the adaptability of the spatial model. As shown in Table 2, the adjacent weight matrix is selected for estimation in the study, which respectively corresponds to the joint OLS estimation, spatial fixed effect, time fixed effect and bidirectional fixed effect models. The specific estimated parameters are detailed in the Table 2.

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Table 2. No interaction effect estimation results and spatial diagnostic test.

https://doi.org/10.1371/journal.pone.0288535.t002

As shown in Table 2 above, both the traditional LM estimate and the robust LM test estimate reject the null hypothesis. This indicates that the spatial model should contain the spatial lag explained variable and the spatial error term. The specific details are as follows: when setting up the traditional LM test, the model estimates significantly reject the original hypothesis without spatial lag terms (the dependent variable) and spatial error terms. However, when using the robust LM estimation test, which states that the robust LM test with both time and space fixed effects do not pass the hypothesis test, all other types reject the null hypothesis without spatial lag dependent variables except for Eq (4). This can be shown that the spatial delay explained variable and the spatial error term should be set in the estimation of the spatial model. In addition, the likelihood ratio LR test is used to test the non-significant joint null hypothesis of spatial fixed effect. The results were as follows: the null hypothesis of joint non-significance of spatial fixed effect was rejected (the estimated value of this test: 429.89, degrees of freedom: 31, P <0.01); The null hypothesis of the combination of time fixed effect and non-significance was also rejected (the estimated value of this test: 198.68, degrees of freedom: 18, P <0.01). According to the above results, the spatial model should adopt the bidirectional fixed effect model, that is that the spatial model should be extended into the form of bidirectional fixed effect model with space fixed effect and time fixed effect.

After selecting the bidirectional fixed effect model as the basic form of the spatial model, the Spatial Durbin Model should be constructed initially, and then the Spatial Lag Model or Spatial Error Model should be selected through relevant hypothesis testing. Among them, if all the null assumptions are rejected after estimating the parameters of the model, then we should adopt the Spatial Durbin Model for subsequent studies. On the contrary, we should choose whether to build Spatial Lag Model or Spatial Error Model through specific test indicators. The above two kinds of tests are both subject to chi-square distribution, and LR test or Wald test is usually used in the form of test. The advantages and disadvantages of the two kinds of tests are obvious. In order to ensure the accuracy and integrity of the test, the calculation and estimation of the two kinds of tests are carried out in this study. The estimation results are shown in Table 3. Since the static space panel is set in the first step, we further construct ML estimation for the static space panel, and test its spatial fixed effect, time fixed effect and bidirectional fixed effect respectively, which correspond to columns (1)—(3) in Table 3 below. Among them, the Wald and LR estimates of the three static spatial models all passed the significance test, and all refused to be transformed into Spatial Lag or Spatial Error Model. In addition, the estimated value of teta, which is the difference parameter between the fixed-effect model and the random-effect model is 0.99 and very significant. Combined with Hausman’s estimation results (estimated value: 33.687, degree of freedom: 13, p = 0.0013<0.01), it is shown that the spatial model should not exclude fixed effect. To sum up the estimation results, the Spatial Durbin Model of bidirectional fixed effect is used for further analysis in this study.

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Table 3. Static space model checking.

https://doi.org/10.1371/journal.pone.0288535.t003

From Table 3 above, it can be seen that the spatial model selected in this study can be further extended into a static Spatial Durbin Model with double fixed effect. In addition to testing the spatial properties of the model constructed in this study, the estimated value in Table 3 above shows that the spatial lag term of independent research and development of the explained variable is positive, which indicates that the independent research and development of local provinces can affect the independent research and development level of neighboring provinces with certain positive externalities. At the same time, it focuses on the policy effect of reducing taxes and fees. The local province is affected by the effect of local tax and fee reduction policies, which is reflected in the improvement of independent research and development level, while the neighboring provinces are affected by negative externalities, which restrains their independent research and development process. It is worth emphasizing that the specific estimated parameters of the static Spatial Durbin Model do not represent the marginal effect. Focusing on the analysis of time lag in the following study, the effect can be decomposed into short-term and long-term coefficients. In this study, the tax and fee reduction policies are estimated to have positive externalities and the policy shows positive income effect in the local area and negative substitution effect in the neighboring area. This makes the regional independent research and development show a point with the clustering, which is gradually form a central point of the agglomeration phenomenon.

3.2 Dynamic Spatial Durbin model estimation

The spatial lag items of the main variables listed in Table 3 above have basically passed the significance test, which indicates that the policy effect of local finance and the process of independent research and development present a strong spatial dependence. The effect of tax and fee reduction policies and the level of independent research and development of neighboring provinces will affect that of their own provinces. In order to effectively control the endogeneity of the time trend item and to more carefully describe the impact of the policy effect of reducing taxes and fees on the level of independent research and development, this chapter chooses the dynamic Spatial Durbin Model for further analysis, and its specific estimates are shown in Table 4 below. As it can be seen, the estimated value of LR is 2× (-3022.15–3196.55) = -12437.4, the degree of freedom is 2, p<0.01, which indicates that the static space panel can be extended to the dynamic space panel model for estimation. The estimated parameters of the specific dynamic Spatial Durbin Model are shown in Table 4 below.

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Table 4. Dynamic Durbin model estimation of spatial interaction.

https://doi.org/10.1371/journal.pone.0288535.t004

As shown in Table 4, various parameter estimates of the dynamic Spatial Durbin Model are estimated in the above table. Among them, by comparing the Spatial Static Durbin model, it can be found that after the introduction of the time lag term and the explained variable of the space lag term, the sign of the variable coefficient of the effect of tax and fee reduction policies has been greatly shifted. This indicates that the tax and fee reduction policies of this province have an externality effect, which presents regional restraining effect on the independent research and development process of neighboring provinces, while the local fiscal policy strategies only aim to improve the independent research and development level of this region. Furthermore, whether space and time lag items are added or not, the coefficient of independent research and development level is basically positive, which indicates that the specific effect of local independent research and development level is easy to form a regional pattern, which can accelerate the process of regional independent innovation from point to area. At the same time, after introducing the spatial weight matrix of geographical factors, economic factors and political decentralization factors into the dynamic Spatial Durbin Model, the policy effect of reducing taxes and fees and the process of independent research and develpoment do not show the sign fluctuation, which further guarantees the accuracy of parameter estimation in this chapter.

3.3 Partial differential decomposition of dynamic space effects

In the model estimation of spatial metrology, the direct coefficient interpretation of dynamic Spatial Durbin Model does not represent the marginal effect. The estimated value of each parameter in Table 4 above can not be used to describe the marginal change quantity. Therefore, partial differential decomposition of the coefficient estimates above is performed in this section to obtain the specific action form of policy effects. The results are shown in Table 5 below.

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Table 5. Dynamic spatial effect decomposition.

https://doi.org/10.1371/journal.pone.0288535.t005

As the Table 5 estimates, regardless of what kind of spatial weight matrix, the building such as adjacency matrix, geographic distance matrix and economic distance matrix, fiscal decentralization matrix (corresponding to the Table 5 column (1)—(4)), those do not affect the dynamic space doberman model of partial differential coefficient decomposition, and the results are significant. In the short term, the direct effect is positive, while the indirect effect is negative. However, in the long run, no matter the direct effect or the indirect effect, no matter the long-term effect or the short-term effect, the estimated results of the effect of tax and fee reduction policies are negative. That is to say, the tax and fee reduction policies of this province can not only act on the local independent research and development, but also affect this of neighboring provinces through external spillover.

In the short term, no matter the adjacency matrix, geographical distance matrix, economic distance matrix or fiscal decentralization matrix, the policy effect of reducing taxes and fees significantly promotes local independent research and development, which is in line with the essence of national innovation development strategy. However, when focusing on indirect effects, no matter the adjacency matrix, geographic distance matrix, economic distance matrix, or fiscal decentralization matrix, the policy effect of reducing taxes and fees is shown as the negative externality of neighboring provinces, which tends to inhibit independent research and development process of neighboring provinces. That is to say, from the perspective of space, the positive incentive of local policy effect on enterprises’ independent research and development is only limited within the administrative planning area. Neighbors may be affected by local financial competition or economic development level differences, and can only rely on regions with strong independent research and development, thus forming the agglomeration phenomenon of a center and affiliated surrounding areas. The surrounding regions are highly dependent on each other, and they are tired of independent research and development in a short term. This phenomenon often proves that the neighboring provinces lag behind in certain innovation. They only adopt the introduction, dependence and reintroduction method through domestic purchase, and further lose the core competitiveness of independent innovation.

In the long run, no matter the adjacency matrix, geographical distance matrix, economic distance matrix or fiscal decentralization matrix, the policy effect of reducing taxes and fees will inhibit the independent research and development level of both local and neighboring provinces. This phenomenon also once again confirms that local enterprises have different behaviors in response to government incentives for tax and fee reduction policies. There are two possible reasons: on the one hand, local governments may completely abandon incentives for local independent research and development. The reason may be the contradiction between the long-term nature of independent research and development and the short tenure of local governments. In the short term, local governments are prone to pressure from political promotion incentives or economic performance evaluations, and usually implement fiscal and tax policies that stimulate independent research and development [29]. However, the term of office is short (generally 3–5 years), and the departure of officials means the separation of the relationship with the original administrative region.

As a result, local governments only pay attention to short-term interests and do not take long-term considerations when implementing the incentive policies of reducing taxes and fees. On the other hand, local enterprises may have strategic behaviors in response to tax and fee reduction policies, which confirms the research in the previous chapter of this study. That is, enterprises may choose different disposal methods based on different motivations for independent research and development. This creates the marginal effect of long term policy to drop. On the basis of measuring the material cost and time cost of technology introduction and independent research, enterprises are easy to choose to give up independent research and development. The related neighboring provinces, due to the lag of the overall regional technology level, also adopt the innovation path of foreign technology introduction→ dependence → reintroduction. This leads to the process of regional independent innovation to appear inhibitive. Meanwhile, it also further verifies the research hypothesis 3 of this study: when enterprises respond to tax and fee reduction policies, they will be motivated by different purposes and present strategic innovation behaviors, which will affect the choice of innovation path of enterprises.

It should be noted that no matter which matrix the above spatial weight is set to, the correlation between provinces is not limited by geographical, economic and institutional administrative barriers. At the same time, the externalities among provinces are not mainly manifested in the form of one-way spillovers within the region: according to calculation, tax and fee cuts have a two-way feedback effect on independent research and development in neighboring regions, and the size of the feedback effect is -8.42+3.44 = -4.98. Although feedback effect is small, but is a very important indicators. The feedback effects show that the policy spillover effect of neighboring provinces by the province to change outside of independent research and development process, but also by spillover effect to give effect to this province, which is verified in this study, the assumption of six: the assumption that the effect of tax and fee reduction policies of the independent research and development is not a single spillover in one-way, but has a two-way effect of spillover and feedback. The more developed areas not only serve as the starting point of technology spillover, but also receive feedback from the level of independent research and development in the less developed areas.

3.4 Robustness test

Considering the endogeneity treatment of spatial econometric models, we have constructed static and dynamic spatial models. The introduction of spatial and temporal parameters, as well as the spatial and time lag terms of the explained variables can overcome some endogeneity to a certain extent, and the decomposition of the coefficients of the dynamic Spatial Durbin Model can more purely determine the marginal utility of the parameters, which has good persuasiveness for the subsequent estimation results of this paper. Furthermore, different spatial weight types were constructed, such as the adjacency weight matrix, geographical weight matrix, economic weight matrix, and fiscal decentralization weight matrix, which can improve the explanatory power of the parameters.

To be explained variable selection of heterogeneity and sampling range selection bias, we intend to expand the government competence—using the prefecture level data and the DMSP/OLS night light data to characterize the local finance competition, and explain the local fiscal policy intervention to innovation, in particular, tax and fee reduction policies. In addition, explained variables are replaced in this section. Malmquist productivity index is used to decompose the index representing the level of independent innovation—technological progress index. Technological progress index can reflect the R&D input and output capacity as well as the independent innovation of enterprises [30, 31].

In order to ensure the accuracy of sample selection, we intercept data of the periods, integrate the data structure of prefecture-level cities, integrate the time range that can be captured by DMSP/OLS night light data, and the relevant variables that can be obtained by each prefecture-level city, eliminate the indicators that are seriously missing in the data, and perform logarithmic processing on the data with large fluctuations. Data samples in this section are selected from 243 panel data of prefecture-level cities from 2001 to 2013, with a total of 3159 samples. Furthermore, the dynamic Spatial Durbin Model was used for parameter estimation, and the parameter estimates were similar to those of the model above, and all of them passed the effect decomposition test. The estimation results of Durbin model in dynamic space are shown in Table 6, and the partial differential decomposition results of each parameter effect are shown in Table 7.

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Table 6. Spatial interactive dynamic Durbin model test of night light data.

https://doi.org/10.1371/journal.pone.0288535.t006

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Table 7. Dynamic spatial effect decomposition of night light data.

https://doi.org/10.1371/journal.pone.0288535.t007

As can be seen from the estimation results in the tables above, after replacing the explained variables and sample range, and introducing the objective indicator DMSP/OLS night light data for variable substitution, the significance of the estimated results is less different from the direction of the policy effect, which basically conforms to the estimation results of the spatial model mentioned above. In conclusion, the direct short-term effect is significant and indirect effect in both the short and long term are characterized by negative significantly, which verified the above estimates—tax and fee reduction policies tend to encourage enterprises to promote the level of local independent research and development in the short term, while its impact on surrounding areas presents a negative externality. In addition, the estimated value of local technological progress index in response to tax and fee reduction policies is significant, which also reflects that there is a time lag in the incentive effect of policy on independent research and development, which is relatively easy to understand, and technological progress has a certain inertia. When an entity attempts to conduct core technology research and development, it often increases research and development expenses due to considerations of cost investment, future returns, and market position [32]. This further accelerates the process of independent research and development.

3.5 Further study

The dynamic Spatial Durbin Model was adopted above to empirically test the relationship between the effect of tax and fee reduction policies and the process of independent innovation. However, fiscal policy has the characteristics of wide coverage and long duration, and its influence on independent innovation is complex. Based on the above considerations, this study discusses the conduction between policy effect and independent research and development by selecting two main factors of Cochrane’s production function that affect technological progress, namely labor and capital. The estimated evidence can provide a reference for the incentive process of independent research and development while further testing the empirical results above. This section continues to use the sample selection criteria of the above studies. On the premise of continuing the existing explanatory variables and control variables, the panel data of 243 prefecture-level cities from 2001 to 2019 are constructed, with a total sample size of 5260. In addition, since the estimated values in this section are mainly economic indicators, the spatial weight matrix of economic distance is introduced and the estimates of other spatial weight matrices are similar. Among them, the specific parameter estimation results are shown in Table 8, and the partial differential of each coefficient is decomposed into Table 9.

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Table 8. Test of spatial interaction between labor and capital factors.

https://doi.org/10.1371/journal.pone.0288535.t008

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Table 9. Dynamic spatial effect decomposition of labor and capital factors.

https://doi.org/10.1371/journal.pone.0288535.t009

According to the estimates in Tables 8 and 9 above, after partial differential decomposition of the partial effects of labor factors and capital factors, it can be seen that all the estimated parameters are basically significant. Tax and fee reduction policies can significantly affect the ratio of labor to capital, their respective price levels and input-output rates. Among them, there is a negative correlation between direct effect and indirect effect on capital price, while there is a positive correlation between labor price, capital input output rate and labor input output rate. This suggests that tax and fee reduction policies can affect the technology selection bias and improve the productivity of two factors by adjusting the price of capital and labor. It is testified the hypothesis four, which is that the tax and fee reduction policies closely associated with area of technology selection bias, and effect on the surrounding areas by spillover. In addition, the effect of the policy of local can affect the surrounding area through the external overflow and this is reflected in the convergence of labor and capital factors in neighboring areas (both positive and negative prices and productivity). In the long run, the demand of the dominant factors in this region will lead the independent research and development bias of the whole region by means of the factor supply and technology imitation in the neighboring regions. This tests Hypothesis 5 of this study, which is that tax and fee reduction policies will affect the selection of labor and capital factors in the region’s preference for independent research and development. At the same time, this dominant factor will lead the technological research and development direction of the whole region by being imitated by neighboring regions.