Trending Time-Varying Coefficient Spatial Panel Data Models

Abstract

This article investigates the estimation and inference of spatial panel data models in which the regression coefficient vector is a trending function. We use time differences to eliminate the individual effects and employ various GMM estimations for regression coefficients with both linear and quadratic moments. Time trend estimator based on these GMM estimations is also proposed. Monte Carlo experiments show that the finite sample performance of the estimators is satisfactory. As an empirical illustration, we investigate the trending pattern of the spillover effect of air pollution among Chinese cities from 2015 to 2021.

Keywords:

• Generalized method of moments
• Nonlinear time trend
• Spatial autoregressive models
• Time-varying coefficients

Supplementary Materials

The online Supplementary Materials contain several lemmas, proofs for the main theoretical results, discussions on estimating the time trend, simulation results related to Section 4, and additional empirical results related to Section 5.

Disclosure Statement

The authors report there are no competing interests to declare.

Notes

1 For a first-order spatial autoregressive (SAR) model where the spatial weights matrix is diagonalizable, the determinant of the Jacobian matrix in the ML estimation can be computed by its eigenvalues (see Ord, 2010). If the spatial weights matrix is not diagonalizable or we have higher-order spatial lags, the Ord device is not applicable for the ML estimation.

2 For recent developments, see Su and Jin (2010), Robinson (2012), Su (2012), Malikov and Sun (1989), Sun and Malikov (2018), and Hoshino (2007) among others.

3 The functional coefficients and trending coefficients models have the following differences. Firstly, as the coefficients in Sun and Malikov (2018) depend on z_it, a bivariate product kernel function is required in their first stage estimation due to the time difference transformation, because $\theta (z_1)-\theta (z_2)$ cannot be further simplified. In this article, we need only a univariate kernel function with t/T, because the time difference $\theta (z_1)-\theta (z_2)$ is asymptotically ignorable under large T. Next, the second stage estimation in Sun and Malikov (2018) is not needed in the time trend specification, which enhances computational burden. Finally, Sun and Malikov (2018) requires z_it to be strictly exogenous and have finite second moment, while this article has a special form of $z_{\mathit{\text{it}}}=t/T$. This implies that the functional coefficient setting in Sun and Malikov (2018) can employ existing technologies available for the conventional stationarity case, while the time–varying setting in this article can consider the nonstationarity of varying coefficients in the time–series dimension. We thank the associate editor and referees for pointing these out.

4 Recently, there is a growing research on the specification of spatial weights matrix, such as the endogeneity issue in Qu and Lee (2018) and Qu et al. (2021), nonparametric specification in Sun (2016), and matrix selection in Han and Lee (2022) and Zhang and Yu (2018). We will focus on the benchmark case with an exogenously given Wn in this article and those related issues can be extended in future research.

5 Instead of a functional specification of time-varying coefficients, another approach treats these coefficients as fixed parameters. See Hayakawa and Hou (2014) and Guo and Qu (2019).

6 In the panel data literature, both within transformation and the first difference can be applied to eliminate the fixed effects. Due to the trending feature, the first difference is convenient for asymptotic analysis. Although it causes serially correlated disturbances after the data transformation, the estimation efficiency can still be achieved as long as we use the optimal weighting matrix in the GMM setting.

7 To avoid using unnecessary specific notations, we abuse notations slightly, and a column vector with the structure $\frac{1}{T}(d_{\mathit{\text{nt}}}+A_n{ϵ}_{\mathit{\text{nt}}})$ is denoted loosely by $O_p\left(\frac{1}{T}\right)$, where d_nt is an $n\times 1$ vector whose elements are uniformly bounded, A_n is an n × n square matrix which is uniformly bounded in row sum and column sum, and ϵ_nt is an $n\times 1$ vector constructed from iid disturbances.

8 For the specification of disturbances, in principle, it can be relaxed to allow some degrees of serial correlation in the time dimension and spatial correlation in the cross-sectional dimension, which is considered in Liang et al. (2010), and even the time-varying spatial error. However, in this article, we choose to stick to the iid specification due to the following reasons. First, the spatial correlation specification for the disturbances might have the issue of whether it shall be time-invariant or time-varying. If it is time-varying similar to the trending regression coefficients, it might cause analytical complications. Second, the generalization of iid disturbances to the martingale difference array is feasible, but we focus on the estimation of time-varying features under the benchmark case. Therefore, we leave these generalizations for future research.

9 We note that when X_nt is stochastic and stationary, Bias ${}_1(\tau )$ would be zero and we can extend the bias to the second order such that the bias term will become $h_2·O(1)$. Bias terms under the stochastic and stationary X_nt can be found in Section S1.2 in the supplement file.

10 When V_nt is not normally distributed, the best quadratic matrix would be equal to $P_{\mathit{\text{nt}}}^{\ast }-\frac{{\eta }_4-3}{{\eta }_4-1}\text{diag}(P_{\mathit{\text{nt}}}^{\ast })$, where ${\eta }_4={\mu }_4/{\sigma }_0^4$. See Liu et al. (2017).

11 In addition, the usual leave-one-out cross-validation procedure to select bandwidth cannot be directly applied here. Due to the time-varying feature of the spatial autoregressive coefficient λ_t, we cannot leave one cross-sectional unit out as in Su (2012) and Liang et al. (2010) where the spatial effect coefficient is homogeneous, that is, ${\lambda }_t\equiv \lambda$ for some constant λ. Also, due to the first difference operation in order to eliminate the individual effects, leaving one time period out approach is not justified either, as an estimated $\delta (\tau )$ is associated with both $Y_{n\tau }$ and $Y_{n,\tau -1}$ in the estimation procedure.

12 The rook matrix represents a square tessellation with connectivity of four for the inner fields on the chessboard and two and three for the corner and border fields, respectively. Most empirically observed regional structures in spatial econometrics have connectivity close to the rook tessellation.

13 Dry deposition is the free fall of particulate matters directly from the atmosphere to the Earth, through sticking to other molecules during the suspension, whereas wet deposition is the process by mixing with suspended water in the atmosphere and then be washed away by rain, snow or fog.

14 We present the result using the $w_{\mathit{\text{ij}}}=\hspace{0.17em}\mathrm{\text{exp}}\hspace{0.17em}(-1.4d_{\mathit{\text{ij}}})$ specification following the suggestions from Zhang and Yu (2018). The empirical results from $w_{\mathit{\text{ij}}}=\hspace{0.17em}\mathrm{\text{exp}}\hspace{0.17em}(-d_{\mathit{\text{ij}}})$ is provided in the supplementary file, and the corresponding regression results are similar.

15 We use the standard normal density as our kernel function. In the supplemental file, we also report the empirical results with the Epanechnikov kernel and they have a similar pattern.

16 For robustness check, we also present the results with a different bandwidths (h = 0.1) in the online supplemental file. The trending patterns for the regression coefficients and time effects are similar, and as expected, the case of h = 0.1 yields a smoother curve.

Additional information

Funding

Chang gratefully acknowledges the financial support from the National Science and Technology Council (MOST-111-2118-M-170-001-MY2). Song gratefully acknowledges the financial support from the National Natural Science Foundation of China (No. 72373007). Yu gratefully acknowledges the financial support from the National Natural Science Foundation of China (No. 71925006). Song and Yu also gratefully acknowledge the financial support from the National Natural Science Foundation of China (No. 72333001) and support from the Center for Statistical Science of Peking University, China and Key Laboratory of Mathematical Economics and Quantitative Finance (Peking University) of the Ministry of Education, China.