The dynamic general nesting spatial econometric model for spatial panels with common factors: Further raising the bar

Rev Reg Res https://doi.org/10.1007/s10037-021-00163-w ORIGINAL PAPER The dynamic general nesting spatial econometric model for spatial panels with common factors: Further raising the bar J. Paul Elhorst1 Accepted: 3 November 2021 © Springer-Verlag GmbH Germany, part of Springer Nature 2021 Abstract The dynamic general nesting spatial econometric model for spatial panels with common factors is the most advanced model currently available. It accounts for local spatial dependence by means of an endogenous spatial lag, exogenous spatial lags, and a spatial lag in the error term. It accounts for dynamic effects by means of the dependent variable lagged in time, and the dependent variable lagged in both space and time. Finally, it accounts for global cross-sectional dependence by means of cross-sectional averages or principal components with heterogeneous coefficients, which generalizes the traditional controls for time-invariant and spatialinvariant variables by unit-specific and time-specific effects. This paper provides an overview of the main arguments in favor of each of these model components, as well as some potential pitfalls. Keywords Spatial panels · Dynamic effects · Spatial spillovers · Common factors · Estimation JEL Classification C21 · C23 · C51 Research highlights Presents most advanced spatial econometric model currently available. Accounts for dynamic effects, local spatial dependence and global cross-sectional dependence. Provides overview of arguments in favor and against different model components.  J. Paul Elhorst 1 Department of Economics, Econometrics and Finance, University of Groningen, P.O.Box 800, 9700AV Groningen, The Netherlands K J. P. Elhorst 1 Introduction Spatial econometrics is a subfield of econometrics dealing with spatial lags among geographical units. The early literature in this field started with contributions of Moran (1948), Whittle (1954), and Ord (1975), followed by the seminal contribution of Anselin (1988),1 and a series of textbooks by LeSage and Pace (2009), Elhorst (2014), Kelejian and Piras (2017), and Beenstock and Felsenstein (2019). According to Elhorst (2014), three generations of spatial econometric models could be distinguished about halfway through the decade 2010–2020. The first generation consists of models based on cross-sectional data. The second generation comprises non-dynamic models based on spatial panel data. These models might just pool time-series cross-sectional data, but more often they also control for fixed or random spatial and/or time-period specific effects. The third generation of spatial econometric models encompasses dynamic spatial panel data models. Today (read: 2021), a fourth generation of spatial econometric models has developed: the general nesting spatial (GNS) econometric model for spatial panels with common factors (CF). This model accounts for local spatial dependence by means of an endogenous spatial lag, exogenous spatial lags, and a spatial lag in the error term. It accounts for dynamic effects by means of the dependent variable lagged in time, and the dependent variable lagged in both space and time. Finally, it accounts for global cross-sectional dependence by means of cross-sectional averages or principal components with heterogeneous coefficients, which generalizes the traditional controls for time-invariant and spatial-invariant variables by unit-specific and time-specific effects. With these properties it is the most general spatial econometric model currently available. The aim of this paper is threefold. First, the full model is set out mathematically. Second, the rationale behind each term that is part of the model is explained. Third, potential objections or pitfalls of including certain terms are discussed from a statistical or an economic viewpoint. Finally, different kinds of data are discussed: regional or macroeconomic data, microeconomic data, and economichistorical data. According to Elhorst (2010), the year 2007 marks a sea change in the spatial econometricians’ way of thinking. Prior to 2007 they were interested mainly in models containing one spatial lag, while after 2007 the interest in models containing more than one spatial lag increased. For this reason he added the words “Raising the Bar” to the title of his paper. The interest for common factors and the distinction between weak and strong cross-sectional dependence, which occurred about halfway through the decade 2010–2020, is another sea change in the spatial econometricians’ way of thinking, explaining the title of this paper. 1 See references in this book for a more comprehensive review, and Anselin (2010) for a recent update. K The dynamic general nesting spatial econometric model for spatial panels with common... 2 Model The general nesting spatial econometric model for spatial panels with common factors reads as P Yt D Yt 1 C W Yt C W Yt 1 C Xt ˇ C W Xt  C r rT frt C ut (1) ut D W ut C "t where Yt D .y1t ; :::; yN t /T denotes an N × 1 vector consisting of one observation on the dependent variable yit for every unit i (i = 1, ..., N) in the sample at time t (t = 1, ..., T). Yt 1 and WYt represent, respectively, the temporal and spatial lag of Yt, and W Yt 1 the spatiotemporal lag of Yt, while τ, ρ, and η are the response parameters of these variables, better known as, respectively, the serial, spatial and spatiotemporal autoregressive coefficients. The N × N matrix W is a nonnegative matrix of known constants describing the spatial arrangement of the units in the sample. Its diagonal elements are set to zero to prevent units from explaining themselves. Xt is an N × K matrix of explanatory variables and WXt an N × K matrix of contemporeous spatial lags of these explanatory variables. The impacts of these variables are measured by, respectively, the K × 1 vectors β and θ. The N × 1 vectors ut and εt denote the error terms of the model. It is assumed that ut follows a firstorder spatial autoregressive process with spatial autocorrelation coefficient λ, which may be labeled as a spatial lag in the error term, and that "t D ."1t ; :::; "N t /T is a vector of disturbance terms, where εit are independently and identically distributed error terms for all i with zero mean and variance σ2. Since the spatial econometric model in Equation (1) contains spatial lags in the dependent variable, in each of the explanatory variables, and in the error term, it is also known as a general nesting spatial model (Elhorst 2014). The determinants of the model described so far capture potential local spatial dependence (weak cross-sectional dependence) among the observations. The common factors frt (r = 1, ..., R) capturing potential global cross-sectional dependence can take three forms. First, if two factors are considered, f1t D .1; :::; 1/T and f2t D .1 ; :::; T /T , and the parameter restrictions 1T D .v1 ; :::; vN / and 2T D .1; :::; 1/ are imposed, the model boils down to a dynamic GNS mo (...truncated)