Nonlinear Spatial Dynamic Panel Data Models with Endogenous Dominant Units: An Application to Share Data

Abstract

This article develops a nonlinear spatial dynamic panel data model with one particularly interesting application to a structural interaction model for share data. To account for effects from dominant (popular) units, the spatial weights matrix in our model can allow for unbounded column sums. To account for heterogeneity, our model includes two-way fixed effects and heteroscedastic errors. We further consider the potential endogeneity of the spatial weight matrix constructed from socioeconomic distance. We investigate the quasi-maximum likelihood estimator (QMLE), generalized methods of moments estimator (GMME), and root estimator (RTE), and establish their consistency and asymptotic normality based on the near epoch dependence (NED) framework. The RTE can derive a relatively computationally simple and closed-form solution without evaluating the QMLE’s Jacobian matrix as well as the iterations by GMME. We consider both $n,T\to \infty$, and the strength of the dominant units is equal to 1 when $T\to \infty$. For the purpose of empirical analysis, we derive the marginal effects and their limiting distributions based on the proposed estimators. In an empirical application, we apply our model to China’s prefecture city-level data, revealing significant spillover effects of the tertiary industry share. These findings suggest that the development of the tertiary sector in large cities can foster its growth in small cities.

Keywords:

• Dominant units
• Endogenous weights
• Nonlinear spatial dynamic panel data model
• Share data

Acknowledgments

The authors thank to the editor, associate editor, and three referees for their constructive comments, and Yahui Chen and Xingbai Xu for their helpful discussions.

Disclosure Statement

The authors report there are no competing interests to declare.

Notes

1 The model can also handle the continuous data within a specific open interval, such as $(-1,1)$. Section S5.2 of the supplement gives more discussions.

2 Since the IV estimation is a special case of the GMM estimation, related discussions are in Section S2.1 of the supplement.

3 The RTE of the SAR model is discussed in Jin and Lee (2012) and Jin and Lee (2020). For the dynamic panel data model, the RTE is discussed in Jin, Lee, and Yu (2021).

4 As discussed in Footnote 5 in Qu, Lee, and Yu (2017), the spatial lag term $M_{\mathit{\text{nt}}}{Z}_{\mathit{\text{nt}}}$ could be specified in the Y_nt or Z_nt equations as long as M_nt is exogenous. But we cannot allow for endogenous M_nt . If M_nt is the endogenous W_nt , (2.1) and (2.3) become a highly nonlinear system and we have possible multiple solutions of Y_nt that is generated by the system. Additionally, there is no research on Z_nt being a nonlinear VAR process with spatial dependence currently. One closest study is a linear spatial VAR (SpVAR) model with time-invariant exogenous weights proposed by Yang and Lee (2021). In summary, the study of a nonlinear (SpVAR) model with time-varying exogenous/endogenous matrices is sufficient for another new study and is worth considering in the future. Due to these complicated issues, we do not cover the exogenous/endogenous weights matrix in (2.3).

5 In the simulation part, we illustrate the results of directly estimating the model without considering the “endogeneity” of spatial weights, especially for estimating λ₀ and ρ₀.

6 Some studies treat FEs as estimators, especially in the discrete nonlinear panel data model. For example, Fernández-Val and Vella (2011) and Fernández-Val and Lee (2013), while our approach is to eliminate FEs. Unlike the linear model of Lee and Yu (2010a), the model in this article is nonlinear and does not exist in a reduced form, and hence the transformation matrices $F_{n,n-1}$ and $F_{T,T-1}$ mentioned in Lee and Yu (2010a) cannot be applied. Therefore, we use the direct approach to consistently estimate the two-way FEs and concentrate them out from the log-likelihood function, which requires both n and T to be large. Details are in Footnote 8. When T is finite, the GMME based on moments after eliminating the two-way FEs can estimate parameters consistently when $0\le \delta <1$. However, when δ = 1 and T is finite, the probability of the related statistics converges to their expectation is $O(1/T)$, which will not satisfy the ST-NED condition, as $O(1/T)\to 0$ is required. To satisfy the ST-NED condition and $0\le \delta \le 1$, we need both n and T are large. Additionally, we may treat time FEs as additional regressors under finite T. From Table S25 of the supplement, we find that there is no significant difference between the two approaches. However, as T increases, especially when T = 20, the estimators obtained from the time dummy approach may have larger root mean squared errors. Due to the fact that T = 18 in our empirical application, compared with the time dummy approach, the method of eliminating FEs may obtain more robust estimates.

7 Thus, $Y_{n1}=F({\lambda }_0{W}_{n1}{Y}_{n1}+{\gamma }_0{Y}_{n0}+{\rho }_0{W}_{n0}{Y}_{n0}+X_{1n0}{\beta }_0+{\mathbf{c}}_{n0}+{\alpha }_{10}{l}_n+V_{n0})$ which can be correlated with two-way FEs and weighting matrix. The endogeneity is modeled in Assumption 1(i).

8 Since ${\nabla }_L={\text{diag}}_{t=1}^T({\nabla }_{\mathit{\text{nt}}})$ and the derivatives of Z_L do not include any parameters, we exclude them. From the first-order condition of (3.2), we have ${\widehat{\mathbf{c}}}_n=\frac{1}{T}{\sum }_{t=1}^T{s}_{\mathit{\text{nt}}}({\widehat{\theta }}_{\mathit{\text{qml}}})\prime s_{\mathit{\text{nt}}}({\widehat{\theta }}_{\mathit{\text{qml}}})$, ${\widehat{\alpha }}_t=\frac{1}{n}{l}_n^{\prime }((s_{\mathit{\text{nt}}}({\widehat{\theta }}_{\mathit{\text{qml}}})-\frac{1}{T}{\sum }_{h=1}^T{s}_{\mathit{\text{nh}}}({\widehat{\theta }}_{\mathit{\text{qml}}})))$, ${\widehat{\mathbf{c}}}_{2n}=\frac{1}{T}{\sum }_{t=1}^T{s}_{2\mathit{\text{nt}}}({\widehat{\theta }}_{\mathit{\text{qml}}})\prime s_{2\mathit{\text{nt}}}({\widehat{\theta }}_{\mathit{\text{qml}}})$ and ${\widehat{\alpha }}_{2t}=\frac{1}{n}{l}_n^{\prime }((s_{2\mathit{\text{nt}}}({\widehat{\theta }}_{\mathit{\text{qml}}})-\frac{1}{T}{\sum }_{h=1}^T{s}_{2\mathit{\text{nh}}}({\widehat{\theta }}_{\mathit{\text{qml}}})))$, where $s_{\mathit{\text{nt}}}(\theta )={\mathcal{Y}}_{\mathit{\text{nt}}}-T_{\mathit{\text{nt}}}\kappa$ and $s_{2\mathit{\text{nt}}}(\theta )={\mathcal{Z}}_{\mathit{\text{nt}}}-K_{\mathit{\text{nt}}}\varkappa$. Therefore, the consistent estimation of FEs based on the likelihood function requires large n and large T. Plugging the estimated FEs into (3.2), we have (3.3).

9 To derive ${\widehat{\kappa }}_{\mathit{\text{rt}}}$, we need to evaluate $G_{\mathit{\text{nt}}}(\lambda )$ based on an initial consistent estimator $\tilde{\lambda }$, which is also discussed in Jin and Lee (2020).

10 Note that ${\widehat{\kappa }}_{\mathit{\text{rt}}}$ is affected by the estimated ${\widetilde{ϵ}}_L={ϵ}_L(\tilde{\tau })$, which is evaluated by the initial consistent estimator $\tilde{\tau }$. The $C(\alpha )$-type moment proposed by Jin and Lee (2020) and Jin and Lee (2021) can eliminate the asymptotic impact on ${\widehat{\kappa }}_{\mathit{\text{rt}}}$, which is currently studied on another working paper by Chen et al. (2024).

11 Assumption 6(ii) assumes that the regressors in deviation form from cross section and time means are linearly independent so that there is no issue of multicollinearity. Of course, implicitly all the exogenous variables cannot be either time or cross-sectional invariant variables. If any of them is invariant in any dimension, the corresponding individual and/or time FEs would absorb them. Except for the invariant cases, we note that we allow variables in $X_{1\mathit{\text{nt}}}$ and $X_{2\mathit{\text{nt}}}$ to overlap so that there could be common variables. Theoretically and empirically, $X_{1\mathit{\text{nt}}}$ and $X_{2\mathit{\text{nt}}}$ could even be the same as identification is not an issue here. This is so, because the parameters in the equations Z_nt can be identified and consistently estimated solely from (2.3), and $X_{1\mathit{\text{nt}}}$ and the control variable is not linearly dependent, even though $X_{1\mathit{\text{nt}}}$ and $X_{2\mathit{\text{nt}}}$ might be so.

12 From Kuersteiner and Prucha (2020) and Jeong and Lee (2021), we can treat $({\xi }_l,{ϵ}_l)$ as the part of v_l such that ξ_l is the strict exogenous part while ϵ_l is the endogenous part.

13 Metropolitan areas generally consist of a densely populated urban core and its surrounding territories, sharing industry, infrastructure, and labor markets (Mills 1967).

14 In 2019, the added value of the tertiary industry accounted for 53.9% of GDP.

15 The classification is based on the “Notice on Adjusting the Criteria for the Classification of Urban Sizes” issued by the State Council of China in November 2014, using the urban permanent population as the statistical caliber.

16 An example is Wuhan in Hubei province affects Xinyang in Henan province, their distance is about 208km.

17 To further check the reasonableness of the specification of weight matrix, we assume that the unidirectional impact of these four types of cities is D →C →B →A and repeat the estimation. We find that λ is insignificant in all specifications, which verifies the weight setting before, that is, only large cities have spillover effects on small cities. Results are shown in Tables S32–S33 of the supplement.

18 Following Lee, Yang, and Yu (2022), ${\nu }_m(t)$ can be estimated by ${\widehat{\nu }}_m(t)=\frac{\hspace{0.17em}\mathrm{\text{ln}}\hspace{0.17em}m(t)}{\hspace{0.17em}\mathrm{\text{ln}}\hspace{0.17em}n}$. From Table 1, we find that $\widehat{\delta }=0.802,{\widehat{\nu }}_m=0.245$ and ${\widehat{\delta }}_g=0.905$. The estimation procedure of ${\widehat{\delta }}_g$ is in Claim S1 of the supplement. We also assume that types A and B are both dominant units, the results are in Table S28 of the supplement.

Additional information

Funding

Xi Qu gratefully acknowledges the National Natural Science Foundation of China (No. 72222007, No. 71973097, and No. 72031006).