Research Article: Bayesian panel smooth transition model with spatial correlation

Date Published: March 4, 2019

Publisher: Public Library of Science

Author(s): Kunming Li, Liting Fang, Tao Lu, Lizhi Xing.


In this paper, we propose a spatial lag panel smoothing transition regression (SLPSTR) model ty considering spatial correlation of dependent variable in panel smooth transition regression model. This model combines advantages of both smooth transition model and spatial econometric model and can be used to deal with panel data with wide range of heterogeneity and cross-section correlation simultaneously. We also propose a Bayesian estimation approach in which the Metropolis-Hastings algorithm and the method of Gibbs are used for sampling design for SLPSTR model. A simulation study and a real data study are conducted to investigate the performance of the proposed model and the Bayesian estimation approach in practice. The results indicate that our theoretical method is applicable to spatial data with a wide range of spatial structures under finite sample.

Partial Text

In panel data regression models, cross sectional and time effects are usually introduced to represent individual heterogeneity. And the coefficients of explanatory variables are assumed to be constant for all section units and periods. In practice, this assumption is sometimes unreasonable. For example, many empirical studies have found that, the impact of exchange rate fluctuation on domestic inflation is not the same in countries with different degree of openness. At this point, if we use traditional panel data models to study the relationship between exchange rate and inflation based on data of different countries, this is equivalent to imposing the assumption that exchange rate has the same effect on inflation in all countries., which is somewhat far-fetched In order to overcome this drawback of traditional panel data models, economists propose random coefficients panel data models and varying coefficients panel data models in which the coefficients can vary with section units and times. The panel data threshold regression model (PTR) proposed by [1] is a widely used varying coefficients model in which the coefficients vary when threshold variables are in different threshold intervals that represent several regimes. Therefore, the coefficients are time-varying if threshold variables change over time. However, the changes in coefficients of PTR model are discontinuous in general since there are usually only a limited number of threshold intervals, which indicates that the transition between different regimes is abrupt. This limits the application scope of the model to a certain extent.. As an effective extension of the PTR model, the panel data smoothing transition regression model (PSTR) proposed by [2] and [3] allows coefficients to vary continuously with transition variables, which effectively ensures the continuity of regimes transition PSTR model actually allows coefficients to change with cross sections and times, which is a sufficient relaxation of heterogeneity assumption in panel data model, and we can easily find that the PTR model is a special case of the PSTR model. As individual heterogeneity can be fully portrayed in PSTR model, this model has been widely used in empirical research in many areas, such as [2], [4], and so on.

The spatial lag panel smooth transition regression (SLPSTR) model considered in this paper has the form as follows
where the subscript i,t indicates i-th cross-section and t-th period respectively, yit is dependent variable, Y = (y11,y21,⋯,yN1,y12,⋯,yNT)′ is NT × 1 vector of dependent variables and W is NT × NT spatial weight matrix, xit is k × 1 vector of independent variables, β0,β1 are k × 1 vectors of coefficients, μi represents the individual fixed effects, εit is random error term and εit ∼ N(0,σ2), g(qit;γ,c)=[1+exp(−γ(∏j=1m(qit−cj)))]−1 is transition function and evidently we have 0 < g(qit;γ,c) < 1, where c = (c1,c2,⋯,cm)′ is m × 1 vector of location parameters, γ > 0 is scale parameter. Without loss of generality, we set m = 1 to simplify mathematical deduction.

We first build the Bayesian analysis framework of model (3) before giving a specific estimation step.

In this section, we conduct a Monte Carlo simulation to investigate the performance of Bayesian approach under small sample. The data generating process (DGP) is given by
where g(qit; γ, c) = (1 + exp(-γ(qit—c)))-1, ρ = 0.75, γ = 1.5, c = 3.5, β0=(β01,β02)′=(2,5)′, β1 = (β11,β12)′ = (4,6)′ and εit ∼ N (0,0.25). In order to investigate the effect of spatial structure of data on the results of Bayesian, the weight matrix in [13] (Case matrix) and the Rook weight matrix in [14] are used respectively.. Other parameters and variables in DGP are set as follows.

In this section, we apply our SLPSTR model and Bayesian estimation method to explore the impact of economic structure on economic growth, which has been a hot topic in the field of economic development (see [15], [16], [17], [18], [19]). Different from the existing literature, our study is based on the perspective of return to factors. Note that economic growth depends on the level of economic productivity which will ultimately reflect in the productivity level of factor that can be measured by the output elasticity of factor. This indicates that economic structure will affect economic growth through output elasticity of factor, so we take the following Cobb-Douglas aggregate production function as the basic model to conduct our empirical test:
where Y, K and L are output, capital stock and labor force, α,β represent output elasticity of capital and labor respectively.

In this paper, we propose a spatial lag panel smoothing transition model by introducing cross-section correlation of dependent variable into panel smoothing transition model and develop a Bayesian estimation approach for this model. In Bayesian analysis, Metropolis-Hastings algorithm and Gibbs method are used for sample design based on prior setting of each parameter in the model. A simulation study and real data estimation are conducted to investigate the practical effect of the SLPSTR model.. The numerical simulation results show that the proposed Bayesian method can perform well for a wide range of spatial data under infinite sample. The real data estimation results demonstrate the application value of the theoretical methods proposed in this paper.




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