Spatiotemporal Difference-in-Differences: A Dynamic Mechanism of Socio-Economic Evaluation

Abstract

Advances in econometric modeling and analysis of spatial cross-sectional and spatial panel data assist in revealing the spatiotemporal characteristics behind socio-economic phenomena and improving prediction accuracy. Difference-in-differences (DID) is frequently used in causality inference and estimation of the treatment effect of the policy intervention in different time and space dimensions. Relying on flexible distributional hypotheses of treatment versus experiment groups on spillover, spatiotemporal DID provides space for innovation and alternatives, given the spatial heterogeneity, dependence, and proximity into consideration. This chapter gives a practical econometric evaluation of the dynamic mechanism in this spatiotemporal context as well as a toolkit for this fulfillment.

1. Introduction

Spatial panel data are used to investigate the spatial reliance in different regions, and some of the spillover effects are between regions, and ordinary least squares cannot reveal.

Spatial difference-in-differences (SDID) is used to investigate policy effect on the socio-economic variable, given the temporal-lagged variable into consideration. Usually, OLS fails to estimate the model with unbiased that Moran’s I reveals. The function of segregation of direct, indirect, and total effects in terms of the specification of the fixed and random effect model makes the SDID’s advantage over the traditional DID model.

2. Assumptions of SDID

2.1 DID model assumptions

To study the fixed effect of individual and time respectively, difference-in-differences (DID) specify the time and location designs in an experiment setting by an estimator of the fixed effect of panel data in average treatment effect on the treated [1, 2]. However, the spatial spillover effect complicates the estimation process. In the experiment group and treatment group, the impact of the treatment is measured before and after the treatment is applied.

Spatial Policy Effect:

$$Y_{\mathit{it}}={\mu }_t+C_i+\tau D_{\mathit{it}}+{\epsilon }_{\mathit{it}}$$E1

Individual fixed effect $C_i$ captures the difference between the treatment and control groups on the time-invariant characteristic. ${\mu }_t$, the temporal fixed effect of time-variant characteristics between control and treatment groups, is assumed to be of the same variance between the control and treatment groups on time-variant characteristics with respect to a specific time.

Time-variant control $D_{\mathit{it}}$ satisfies conditional independence assumption (CIA) to work as a core in the causality inference, while assuming time is exogenous, $\text{that is},E\left[{\epsilon }_{\mathit{it}}|\mathit{\text{treatment}},\mathit{\text{time}}\right]=0.$

The ordinary least square (OLS) is a consistent estimator for the effect of the causality inference.

$$\tau =E\left[Y_{i1}\left(1\right)-Y_{i1}\left(0\right)|D_i=1\right]$$E2

After the bias is removed, $E\left[\widehat{\tau }\right]=\tau +\mathit{\text{Treatment group Spillover}}+\mathit{\text{Control group Spillover}}+\mathit{\text{Spillover between different group type}}$

So far, only the spillover between treatment and control groups is not treated.

$$D_{\mathit{it}}=\left(D_{\mathit{it}}^{\left(1\right)}-{\overline{D}}^{\left(1\right)}\right)\times \left(D_{\mathit{it}}^{\left(2\right)}-{\overline{D}}^{\left(2\right)}\right)$$E3

$D_{\mathit{it}}$ is DID, the interaction term of temporal and spatial difference; that is, ${\widehat{\beta }}_{\mathit{DID}}$ in Figure 1, the observable time-dependent variable as control. $D_{\mathit{it}}^{\left(1\right)}$ is a dummy variable to denote the group being treated, valued at 1; or group not being treated, values at 0. $D_{\mathit{it}}^{\left(2\right)}$ is a dummy variable indicating a local policy or temporal element, values at 1 in the year of policy implementation and thereafter, or 0 for the years before. ${\overline{D}}^{\left(1\right)}{\overline{D}}^{\left(2\right)}$ are the means of two dummy variables.

Figure 1.

DID estimation illustration.

The strict set of assumptions of the DID is there is no spillover in treatment, control, or between treatment and control groups. When an assumption of same spillover effect within the control and treatment groups is added to the strict set, the assumption becomes relaxed restriction set 1. When there is no spillover between treatment and control groups, the assumption of relaxed restriction set 1 is changed to the spillover effect on control and treatment groups are same and the assumption becomes relaxed restriction set 2.

$$\widehat{\tau }=\widehat{E}\left[Y_{i1}-Y_{i0}|D_i=1\right]-\widehat{E}\left[Y_{i1}-Y_{i0}|D_i=0\right]$$E4

$\widehat{E}\left[Y_{i1}-Y_{i0}|D_i=1\right]$: Counterfactual trend + $\tau$ +treatment group spatial spillover

$\widehat{E}\left[Y_{i1}-Y_{i0}|D_i=0\right]$: Counterfactual trend + control group spatial spillover

The traditional DID

$$\tau =E\left[Y_{i1}\left(1\right)-Y_{i1}\left(0\right)|D_i=1\right]$$E5

2.2 Spatial model assumptions

The spatial spillover has the properties of spatial heterogeneity, spatial dependence (the butterfly effect and spatial association), and spatial proximity. Spatial heterogeneity denotes the different structures of spatial units in different locations [3, 4, 5]. Without the assumption on spatial homogeneity parameters, it is hard to estimate the model with the increase of observations. In the spatial model, most of the estimation assumes locations are regional homogeneous. The definition of spatial heterogeneity is the non-smoothness of a spatial random process, which comprises change of function form or parameters and heteroscedasticity of two categories.

Spatial dependence means the adjacent spatial locations have the propensity to be associated with each other and work in coordination synchronously.

Spatial proximity denotes that in spatial areas everything is related to everything else, but near things are more related than distant things (Waldo Tobler).

3. Model specifications-generalized spatial model

The panel data regression is a linear regression with the combination of three types of spatially lagged variables across time, which traces the same observation unit over different times. The observation unit’s characteristic over time and spatial location are the research interest. The classical panel data regression takes the following form based on Arbia [6], Cerulli [2], LeSage and Pace [3], and Wooldridge [5]:

$$y_{\mathit{it}}=X_{\mathit{it}}\beta +c_i+{\upsilon }_{\mathit{it}}$$E6

$i=1,2,\dots N$ index corresponding to i^th different observation units in the cross-sectional data.

$t=1,2,\dots T$ denotes time. $c_i$ is fixed effect w.r.t. observation unit, or spatial specific effect invariant to time. Alternatively, or simultaneously, $c_t$ a time-specific fixed effect w.r.t. different time, and invariant to observation unit can be embedded to the model. ${\upsilon }_{\mathit{it}}$ is an independent identical distributed error term, iid. (0, ${\sigma }^2$).

The generalized spatial model includes the following variations.

Spatial Autoregression Model (SAR-SDID)

$$y_{\mathit{it}}=\rho \sum _{j=1}^N{\omega }_{\mathit{ij}}{y}_{\mathit{jt}}+X_{\mathit{it}}\beta +\mathit{\tau D}\left(\right)+c_i+{\upsilon }_{\mathit{it}}$$E7

Before the spatial regression, Moran’s I test is used to measure and test spatial autoregression in general. It is also called a global spatial autoregression test; that is, it measures the degree of similarity to each other between the spatial observations in the sample.

$$I=\frac{n}{{\sum }_i{\sum }_j{\omega }_{\mathit{ij}}}\frac{{\sum }_i{\sum }_j{\omega }_{\mathit{ij}}\left(x_i-\overline{x}\right)\left(x_j-\overline{x}\right)}{{\sum }_i{\left(x_i-\overline{x}\right)}^2}$$E8

where n is the number of observations, and ${\omega }_{\mathit{ij}}$ is the element in i^th row and j^th column from the spatial weight matrix W. $x_i$, $x_j$ are i^th and j^th observation in the spatial unit, and $\overline{x}$ is the average of the observations.

The positive Moran’s I denotes a positive correlation, while a negative value means a negative correlation. The standardization of the spatial weight matrix simplifies the notation as follows.

$$I=\frac{X^{\prime }\mathit{WX}}{X^{\prime }X}$$E9

It is obvious that Moran’s I is the Pearson correlation coefficient between X and WX. It follows an asymptotic distribution, which simplifies the process of using Moran’s I to test spatial autocorrelation in the residual of the test. It is a statistical inference through the z-test.

A partial Moran’s test is

$$I_i=\frac{n\left(x_i-\overline{x}\right){\sum }_{i\ne j}{\omega }_{\mathit{ij}}\left(x_j-\overline{x}\right)}{{\sum }_i{\left(x_i-\overline{x}\right)}^2}$$E10

Partial Moran’s test $I_i$ averages over $i$ is the overall Moran’s I. Partial Moran’s test $I_i$ is often used. Overall Moran’s I is the slope of the scatter plot as in Figure 2, where the X-axis denotes X and Y-axis denotes lagged variable WX of X. The slope of the line reflects the relation between the observed variable and the spatial lagged observed variable. The line across the 1st and 3rd coordinates denotes a positive spatial correlation.

Figure 2.

Moran’s I scatter plot of Columbus crime data.

‘Columbus’ data are generated by geographical information systems (GIS) [7] based on US census data¹ on Columbus boundary systems, with the data type “.shp.” The file stores feature geometry such as coordinates of polygon centroids and their boundary. The W matrix is derived from the contiguity-based neighbors’ list [6].

The summary statistics of the “crime” variable in the “Columbus” data, totally 49 observations, are as follows (Table 1):

Minimum	1st quantile	Median	Mean	3rd quantile	Maximum
0.1783	20.0485	34.0008	35.1288	48.5855	68.8920

Table 1.

Summary statistics of crime variable.

The test of Moran’s I depends on the distribution of its input variable and is done via Monte Carlo simulation. Through the random multiple interchanging location of the spatial units, the model recalculates the weight matrix W and Moran’s I statistic to obtain Moran’s I after multiple replacements. In a frequency rectangle picture, Moran’s I empirical distribution is developed and compared with the statistic from the direct calculation.

The assumption of the method is when the spatial units are randomly distributed, the variable is not autocorrelated, and Moran’s I is close to 0. When Moran’s I is with a low probability of occurrence, the null hypothesis of no autocorrelation is rejected. When the input variable is the residual of classical linear regression, assumed to follow a normal distribution, Moran’s I follows a normal distribution as well, where the z-test is applicable to avoid the heavy calculation of Monte Carlo simulation.

3.1 Spatial error model (SEM-SDID)

$$y_{\mathit{it}}=X_{\mathit{it}}\beta +\mathit{\tau D}\left(\right)+c_i+u_{\mathit{it}}$$E11

$$u_{\mathit{it}}=\rho \sum _{j=1}^N{m}_{\mathit{ij}}{u}_{\mathit{jt}}+{\upsilon }_{\mathit{it}}$$E12

3.2 Spatial Durbin model (SDM-SDID)

$$y_{\mathit{it}}=\rho \sum _{j=1}^N{\omega }_{\mathit{ij}}{y}_{\mathit{jt}}+X_{\mathit{it}}\beta +\sum _{j=1}^N{\omega }_{\mathit{ij}}{X}_{\mathit{jt}}\theta +\mathit{\tau D}\left(\right)+\sum _{j=1}^N{\omega }_{\mathit{ij}}D\left(.\right)+\pi +c_i+{\upsilon }_{\mathit{it}}$$E13

3.3 Spatial lag of X model (SXL-SDID)

$$y_{\mathit{it}}=X_{\mathit{it}}\beta +\sum _{j=1}^N{\omega }_{\mathit{ij}}{X}_{\mathit{jt}}\theta +\mathit{\tau D}\left(\right)+c_i+\sum _{j=1}^N{\omega }_{\mathit{ij}}D\left(.\right)u_{\mathit{jt}}+{\upsilon }_{\mathit{it}}$$E14

The spatial weight matrix ${\omega }_{\mathit{ij}}$ and $m_{\mathit{ij}}$ are invariant to time changes. Spatial and temporal-specific effects can be treated as fixed or random effects. If treated as a fixed effect, the specific effect is taken as a parameter to estimate. Whereas a random effect, it is treated as a random variable following iid. (0, ${\sigma }^2$). The deterministic factor is to tell whether $c_i$ is correlated with $X_{\mathit{it}}$. The fixed effect is used to treat the correlation, while the random effect is for the uncorrelation. The random effect has the advantage of improving effectiveness with the observation number and distinguishes the factor invariant to time. Because $c_i$ is invariant to time, it is hard to separate observed information from individual effect. $\rho$ the spatial lag term is used to test the spatial spillover effects between neighboring regions [3]. The positively significant coefficient of $\rho$ indicates a positive spatial spillover effect.

To account for the direct impact and indirect impact, the transformation of Eq. 13 is taken as follows.

In matrix format, let the constant vector ${\mathbb{l}}_n$ and relevant parameters $\alpha$ to be embedded in Eq. 13.

$$y={\left(I_n-\mathit{\rho W}\right)}^{-1}{\mathbb{l}}_n\alpha +{\left(I_n-\mathit{\rho W}\right)}^{-1}\mathit{X\beta }+{\left(I_n-\mathit{\rho W}\right)}^{-1}\epsilon$$E15

$$y=\sum _{r=1}^k{S}_r\left(W\right)X_r+{\left(I_n-\mathit{\rho W}\right)}^{-1}{\mathbb{l}}_n\alpha +{\left(I_n-\mathit{\rho W}\right)}^{-1}\epsilon$$E16

The sum of the rows of $S_r\left(W\right)$ denotes the total impact of a region to an observation (ATITO); the sum of columns of $S_r\left(W\right)$ is the total impact of a region from an observation (ATIFO). The average of the sum of rows or columns is the average total impact (ATI). The average of elements on the main diagonal is the average direct impact (ADI), and average indirect impact (AII) is defined as the difference between average total impact (ATI) and average direct impact (ADI).

$$S_r\left(W\right)={\left(I_n-\mathit{\rho W}\right)}^{-1}{\beta }_r=\frac{\partial E\left(y\right)}{\partial x_r}=\left[\dots \begin{array}{cc}\frac{\partial E\left(y_1\right)}{\partial x_{1r}}\dots .& \frac{\partial E\left(y_1\right)}{\partial x_{\mathit{nr}}}\\ \frac{\partial E\left(y_n\right)}{\partial x_{1r}}\dots ..& \frac{\partial E\left(y_n\right)}{\partial x_{\mathit{nr}}}\end{array}\dots \right]$$E17

where ADI denotes the average impact of change of local explanatory variable $x_r$ on the specific local dependent variable y.

$$\mathit{ADI}=n^{-1}\sum _{i=1}^n\frac{\partial E\left(y_i\right)}{\partial x_{\mathit{ir}}}=n^{-1}\mathit{tr}\left[S_r\left(W\right)\right]$$E18

ATITO is the average impact on the specific local dependent variable y from the change in the explanatory variable $x_r$ of all regions.

$$\mathit{\text{ATITO}}=n^{-1}\sum _{i=1}^n\sum _{j=1}^n{S}_r{\left(W\right)}_{\mathit{ij}}=n^{-1}\sum _{i=1}^n\sum _{j=1}^n\frac{\partial E\left(y_i\right)}{\partial x_{\mathit{jr}}}$$E19

ATIFO is the average impact on all regional dependent variable y from the change in the explanatory variable $x_r$ of a specific region.

$$\mathit{\text{ATIFO}}=n^{-1}\sum _{j=1}^n\sum _{i=1}^n{S}_r{\left(W\right)}_{\mathit{ij}}=n^{-1}\sum _{j=1}^n\sum _{i=1}^n\frac{\partial E\left(y_i\right)}{\partial x_{\mathit{jr}}}$$E20

ATITO and ATIFO are equal values and are called by a joint name ATI.

$$\mathrm{AII}=\mathrm{ATI}-\mathrm{ADI}$$E21

4. Tests on model assumptions

A robust test is performed to study whether the estimation is sensitive to the change in the width of the event window. Wald test follows an asymptotic chi-squared distribution with N degree of freedom.

The test on correlation coefficient $\rho$ is positive significance.

The estimation is done with spatial inverse-distance contiguity weight matrix.

A parallel test is used to test that the change of temporally dependent variable as control will or will not impact the direction of policy influence. A good control ensures the conditional independent assumption (CIA) holds. The different outcomes between the treatment group subject to intervention and the control group in the absence of intervention produce reliable results if both groups are similar in their characteristics and have parallel trends before the intervention, that is, the parallel trends assumption. If the assumption holds, the different outcomes between groups attributed to the intervention ([1, 8, 9, 10, 11]). It differs from the Granger test in that a parallel test is performed in the periods before policy intervention to reveal the significant parameters $H0:{\tau }_0={\tau }_{-1}=\dots ={\tau }_{-k}=0$ that spans over more than two time periods, while the Granger test requires only a minimum of two periods and is much simpler [12]. If the assumption ${\mu }_t$ of the same fixed time effect of both groups is the same holds, incorporating new control variable $X_{\mathit{it}}$ will not change the estimation of parameters except their variance.

$$Y_{\mathit{it}}={\mu }_t+C_i+\tau D_{\mathit{it}}+{\epsilon }_{\mathit{it}}$$E22

$$Y_{\mathit{it}}={\mu }_t+C_i+\tau D_{\mathit{it}}+\sum _{k=0}^m{\tau }_{-k}{D}_{i,t-k}+\sum _{k=1}^q{\tau }_{+k}{D}_{i,t+k}+X_{\mathit{it}}\beta +{\epsilon }_{\mathit{it}}$$E23

Granger test is focused on the after-event periods, to investigate whether the parameters of DID after the policy intervention, ${\tau }_1\dots {\tau }_k$, are significant. Using lags and leads provides a test to determine whether past treatments affect the current outcome or for the presence of anticipatory effects, that is, to estimate ${\tau }_{-k}$ and ${\tau }_k$, thus challenging the conventional idea that causality works only “from the past to the present” [13].

To be specific, Buerger et al. [12] use Granger equations to test the parallel trends assumption, the most important DID framework assumption to improve evidence on causal claims.

When relaxing the parallel assumption, the placebo test answers the question that does the policy matter if one period before or behind implementation? It is tested on the significance of ${\tau }_1,{\tau }_{-1}$. As a “fake” treatment effect in the pre-period, which is another way to observe parallel trends [14] while requiring three or more time periods prior to the treatment implementation [15].

Hausman test is the test on the choice of fixed effect model or random effect model.

If Corr($c_i$, $X_{\mathit{it}}$) = 0, parameters of FE or RE models are consistent estimators. Although the estimators are almost the same, the RE model estimation is more effective.

If Corr($c_i$, $X_{\mathit{it}}$)$\ne$ 0, the estimators follow different asymptotic distributions, and the estimators are significantly different. Only the FE estimator is consistent.

Under the normality assumption, the maximum-likelihood estimator $\widehat{{\theta }_r}$ of the random effect, model is consistent and asymptotic effective, while $\widehat{{\theta }_f}$ is consistent and asymptotic effective only in the existence of a correlation between individual effect and exogenous variable.

Hausman test compares the difference of two estimators to infer the existence of correlation via the statistic:

$$n{\left(\widehat{{\theta }_r}-\widehat{{\theta }_f}\right)}^{\prime }{\mathrm{\Omega }}_n^{+}\left(\widehat{{\theta }_r}-\widehat{{\theta }_f}\right)$$E24

${\mathrm{\Omega }}_n.$ is the covariance matrix of $\sqrt{n}\left(\widehat{{\theta }_r}-\widehat{{\theta }_f}\right)$ under the null hypothesis. ${\mathrm{\Omega }}_n^{+}$ is a generalized inverse matrix of ${\mathrm{\Omega }}_n$. This statistic follows a ${\chi }^2\left(\mathit{rank}\left({\mathrm{\Omega }}_n\right)\right)$

The Lagrangian multiplier is used to test the possible spatial autocorrelation in the residual of the model, which is like Moran’s I test on the potentially existing spatial autocorrelation. The difference lies in the individual effect as the spillover effect in the dependent variable lagged spatial model.

Under the setting of the individual and temporal dual fixed effect model in Eq.13, the two restrictive constraint tests on the coefficients are as follows.

$${\mathrm{H}}_0^{\mathrm{a}}:\mathrm{\theta }=0$$

$${\mathrm{H}}_0^{\mathrm{b}}:\mathrm{\theta }+\mathrm{\rho \beta }=0$$

If $H_0^a$ holds, Eq. 13 becomes Eq. 7. Under $H_0^b$, it becomes Eq. 11.

If both hypotheses are rejected, Eq. 13 is selected.

If $H_0^a$ is accepted robustly and the RLM test indicates a spatial autoregression model, Eq. 7 is selected.

If $H_0^b$ is accepted robustly and the RLM test indicates a spatial error model, Eqs. 11 and 12 are selected.

If the two restrictive constraint tests yield a different result from that of RLM, Eq. 13 is selected.

5. Application: an example

In Gu [16] policy evaluation research, DID estimator is renewed as development in academic patent activities following a spatial autoregressive process with respect to the dependent variable. The DID is proposed as a spatial DID estimator to account for spatial spillover effects. The empirical analysis of 31 Chinese provinces indicates that an incentive patent policy plays a positive role in the output and commercialization of academic patents during the period from 2010 to 2019. Incentive patent policies are found to play as a placebo in academic patent activities.

The traditional DID method ignores the geographical proximity and spatial spillover effects of academic patent activities. Gu [16] shows the spatial DID model is used to find out three treatment effects, that is, treatment effects based on patent incentive policies and spillover effects within the treatment and control groups. Spatial DID models, including the spatial dependence between adjacent provinces, effectively investigate the spatial spillover effects of policies.

The number of academic patents granted (NGP) in each province is a common indicator of the output of academic patents, and the commercialization rate of academic patents (CAP) and the number of academic patents sold divided by the number of patents granted to the university are used as two explanatory variables in the research.

GDP per capita (PGDP), the number of universities (NCU) in a province, the teacher-to-student ratio (TSR), and the number of enterprises above the designated size (NIE) as indicators of the scale of large industrial enterprises in a region are four explanatory variables in the model.

$${\mathit{NGP}}_{\mathit{it}}=C+\mathit{\rho W}{\mathit{NGP}}_{\mathit{it}}+{\beta }_1{\mathit{\text{PGDP}}}_{\mathit{it}}+{\beta }_2{\mathit{NCU}}_{\mathit{it}}+{\beta }_3{\mathit{TSR}}_{\mathit{it}}+{\beta }_4{\mathit{NIE}}_{\mathit{it}}+{\beta }_5{\mathit{DID}}_{\mathit{it}}+{\epsilon }_{\mathit{it}}$$E25

$${\mathit{CAP}}_{\mathit{it}}=C+\mathit{\rho W}{\mathit{CAP}}_{\mathit{it}}+{\beta }_1{\mathit{\text{PGDP}}}_{\mathit{it}}+{\beta }_2{\mathit{NCU}}_{\mathit{it}}+{\beta }_3{\mathit{TSR}}_{\mathit{it}}+{\beta }_4{\mathit{NIE}}_{\mathit{it}}+{\beta }_5{\mathit{DID}}_{\mathit{it}}+{\epsilon }_{\mathit{it}}$$E26

$${\mathrm{\epsilon }}_{\mathrm{it}}\sim \mathrm{N}\left(0,{\mathrm{\sigma }}^2{\mathrm{I}}_{\mathrm{n}}\right),\mathrm{i}=1,\mathrm{..31}$$

Except for the policy variable ${\mathit{DID}}_{\mathit{it}}$, $\mathit{\rho W}{\mathit{NGP}}_{\mathit{it}}$, and $\mathit{\rho W}{\mathit{CAP}}_{\mathit{it}}$, the rest variables are controls. If the second right-hand side variables in Eqs. 25 and 26, $\mathit{\rho W}{\mathit{NGP}}_{\mathit{it}}$ and $\mathit{\rho W}{\mathit{CAP}}_{\mathit{it}}$, are omitted, two equations are the traditional DID. ${\mathit{DID}}_{\mathit{it}}$ is the multiplication of two dummy variables, denoting whether and when the policy is implemented.

The data obtained from the China Statistical Yearbook span from 2011 to 2020, 10 years in 31 provinces, which makes 310 observations in total. The data on the commercialization of academic patents are obtained from the Compilation of Science and Technology Statistics in Universities, compiled by the Science and Technology Department of the Ministry of Education of China.

The population is divided into an experimental group comprised of 17 provinces, and a control group including 14 provinces. Two sets of models are consisted of fixed effect or random effect factorization and applied to Eqs. 25 and 26, totally four models (Table 2).

	NGP		CAP
	Model 1	Model 2	Model 3	Model 4
	Fixed effects	Random effects	Fixed effects	Random effects
PGDP	82.174***; (5.91)	77.775***; (5.93)	0.048*; (1.65)	0.049**; (2.31)
NCU	200.218***; (8.46)	82.571***; (5.16)	0.195***; (3.92)	0.018; (1.17)
TSR	109274.4***; (3.53)	72722.79**; (2.55)	230.979***; (3.61)	18.035; (0.38)
NIE	−79.33***; (−2.74)	−52.884*; (−1.83)	−0.135**; (−2.24)	−0.091**; (−2.2)
DID	1592.831**; (2.28)	1788.006**; (2.48)	4.394***; (3.02)	3.251**; (2.24)
Year; 2011	−961.286***; (−2.85)	−621.956*; (−1.75)	−2.409***; (−3.19)	0.692; (0.88)
….. Table	Omitted	intentionally
$\rho$	0.652***; (7.89)	0.616***; (−7.2)	0.419**; (2.58)	1.864*; (1.77)
….. Table	Omitted	intentionally
Wald test of spatial terms	62.25***	52.83***	6.66**	100.93*

Table 2.

Results of estimation [16].

*

is significant at 0.1.

**

is significant at 0.05.

***

is significant at 0.001.

Positively significant $\rho$ and Wald test of spatial terms indicate the spatial spillover effects are not ignorable. Significant coefficients of DID indicate the incentive patent policy promotes the output and commercialization of patents. Hausman test is ignored due to the insignificant difference between FE and RE models. The SDID is applicable (Tables 2 and 4).

SDID spillover effect develops indirect effects in adjacent areas, outperforming the DID model; that is, the indirect effect in models 3 and 4 of dependent variable CAP are insignificant. In this way, the policy effect in the neighborhood provinces is further segregated (Table 3).

	NGP		CAP
	Model 1	Model 2	Model 3	Model 4
	Fixed effects	Random effects	Fixed effects	Random effects
Direct effect	1662.371**; (2.28)	1851.767**; (2.48)	4.444***; (3.02)	3.392**; (2.36)
Indirect effect	2719.377*; (1.78)	2617.57*; (1.88)	2.914; (1.38)	1.305; (1.13)
Total effect	4381.748**; (2.04)	4469.337**; (2.21)	7.358**; (2.4)	4.697**; (2.11)

Table 3.

Results of policy effect tests [16].

*

is significant at 0.1.

**

is significant at 0.05.

***

is significant at 0.001.

The placebo tests in Table 4 show that there is no change of significance in the DID if the policy is implemented in the year before or after the actual year of implementation. The result is problematic to convince that the incentive patent policy promotes the outcome or commercialization of a patent. The DID is rather a placebo without an effect on the patent on its own, whereas a proxy of province systemic difference makes the diversity.

	NGP		CAP
	Model 5	Model 6	Model 7	Model 8
	1 year earlier	1 year later	1 year earlier	1 year later
PGDP	82.595***; (5.92)	82.703***; (5.96)	0.049*; (1.68)	0.049*; (1.7)
NCU	201.749***; (8.5)	199.52***; (8.44)	0.199***; (3.98)	0.195***; (3.91)
TSR	106525.2***; (3.43)	109694.8***; (3.55)	223.24***; (3.48)	230.945***; (3.6)
NIE	−81.847***; (−2.82)	−76.23***; (−2.63)	−0.141**; (−2.33)	−0.131**; (−2.15)
DID	1480.638*; (1.82)	1624.062**; (2.5)	4.687***; (2.77)	3.724***; (2.74)
Year; 2011	−1040.216***; (−3.06)	−926.287***; (−2.75)	−2.669***; (−3.49)	−2.31***; (−3.06)
Table	Omitted	intentionally
$\rho$	0.654***; (7.91)	0.65***; (−7.86)	0.41**; (2.5)	0.432***; (2.71)
Table	Omitted	intentionally
Wald test of spatial terms	62.63***	61.76***	6.23**	7.33*

Table 4.

Results of placebo tests [16].

*

is significant at 0.1.

**

is significant at 0.05.

***

is significant at 0.001.

6. Conclusion

This chapter outlines the methodology and application of DID in spatial analysis. The impact of incentive policy on economic activities is controversial. The empirical evidence results from a correlation test rather than causality analysis. SDID as a tool to segregate the direct effect, indirect effect, and total effect in the fixed-effect and random-effect models finds a causal relationship between the policy and relevant economic activities under the influence while dealing with the spillover effects in quasi-natural experiments. The placebo effect of policy can expand the horizon of policy evaluation, which helps consolidate the scientific foundation of policy evaluation. Regional policies are proxies for other variables that characterize the systemic differences in policies between regions.

In the policy evaluation, the SDID reveals the spatial spillover effect on the neighborhood regions, causing them to imitate the policies and promote economic activities. It is not appropriate to study the policy effect independently, but a comprehensive evaluation from a local perspective is preferred.

Acknowledgments

I thank Jiafeng Gu, Tongying Liang, and Feifei Liang for their generous comments.

Conflict of interest

The author declare no conflict of interest.

Stata Code

/*Create time fixed effect and individual fixed effects.*/

sort time id

by time: gen ind = _n

sort id time

by id: gen T = _n

/*Generate treat and post two dummy variables, with 2010 set as time spot of policy intervention and observations from 17 to 31 are treatment group, and rest is control group.*/

gen treat = 0

replace treat = 1 if id >17

gen after = 0

replace after = 1 if time > = 2010

/* Create Weight matrix:*/

spmatrix create idistance M /*spatial inversed distance matrix*/

spmatrix dir

spmatrix create contiguity W/*spatial distance matrix*/

spmatrix dir

estat moran, errorlag (W)

estat moran, errorlag (M)

gen treatafter = treat*after

spmatrix create contiguity W if year == 2010

spxtregress NGP treatafter PGDP NCU TSR NIE i.time,re dvarlag (W)

gen treatafter = treat*after

spmatrix create contiguity W if year == 2010

spxtregress CAP treatafter PGDP NCU TSR NIE i.time,re dvarlag (W)

/*General application*/

1)SAR-SDID

> spmatrix create contiguity W if year == 2010

>spxtregress NGP treatafter PGDP NCU TSR NIE i.time, re dvarlag (W)

2)SEM-SDID

> spmatrix create contiguity W if year == 2010

>spxtregress NGP treatafter PGDP NCU TSR NIE i.time,re errorlag (W)

3)SDM-SDID

> spmatrix create contiguity W if year == 2010

>spxtregress NGP treatafter PGDP NCU TSR NIE i.time, re dvarlag (W) ivarlag (W: X)

4)SXL-SDID

> spmatrix create contiguity W if year == 2010

>spxtregress NGP treatafter PGDP NCU TSR NIE i.time,re ivarlag (W: DX)