# **2.1 A factor-augmented dynamic spatial model of growth**

Let _git_ denote the real output growth in country _i_ = 1 _, . . . , n_ at time _t_ = 1 _, . . . , T_ , and let _gt_ = ( _g_ 1 _t, ..., gnt_ )<sup>_′_</sup> . We assume that _gt_ follows a spatial dynamic panel data (SDPD) model of the form: Thus, each country’s real output growth is related to current real output growth in (economically) neighboring countries, _Wgt_ , where _W_ is an _n × n_ spatial weight matrix; lagged real output growth in the own country, _gt−_ 1, as well as in neighboring countries _Wgt−_ 1; a set of _r_ time-varying unobserved factors _ft_ common to all countries; and a stochastic disturbance _Vt_ .

This general specification allows for both spatial dependence, unobserved common factors and growth persistence. Spatial dependence, embedded in the spatially-lagged dependent variable _Wgt_ as well as its time-lagged value _Wgt−_ 1, reflects the effects of current and lagged real output growth of nearby countries on the real output growth of a particular country, see e.g Ho, Wang and Yu (2013) and Ertur and Koch (2007). The extent of spatial dependence is measured by the contemporaneous spatial autoregressive parameter _ρ_ and the space-time lag coefficient _λ_ .<sup>2</sup> The relative contribution of each country to the overall spatial effect is measured by the spatial weight matrix _W,_ which can be understood as providing a measure of economic proximity between countries.

In turn, the unobserved common factors _ft_ capture systemic shocks that affect real output growth across all countries. The _n × r_ matrix of factor loadings Ψ measures the (possibly heterogeneous) effect of the factors on each country’s growth. Finally, growth persistence is captured by the parameter _β_ on the lagged endogenous variable.

Equation (1) nests various models as special cases. For example, in the absence of spatial dependence ( _ρ_ = 0 and _λ_ = 0), equation (1) simplifies to a factor-augmented model relating real output growth to observable lagged growth plus latent common factors, see e.g. Kose, Otrok and Whiteman (2003), Jorg and Sandra (2016) or Moench, Ng and Potter (2013).<sup>3</sup> In these specifications, the spatial dependence between countries is parameterized by the _n×n_ spatial weight matrix _W_ . The matrix is assumed to be non-stochastic, with the properties (i) _Wij ≥_ 0 for _i_ = _j_ , and (ii) _Wij_ = 0 for _i_ = _j_ . The first property indicates that the elements of _W_ are non-negative known constants. The second property states

> 2The parameter _λ_ , termed ’diffusion parameter’ by Shi and Lee (2017), captures spatio-temporal correlations in output growth that may result from partial adjustment or inter-temporal decision making by economic agents, see e.g., Tao and Yu (2012).

3Equation (1) can also be seen as a variant of the GVAR model of Chudik and Pesaran (2016) imposing constant _β, λ_ and _ρ_ across. In turn, the common factor framework in (1) encompasses individual and time period fixed effects as a particular case (Shi and Lee (2017)). To see this, consider the specification where _ζ_ = ( _ζ_ 1 _ζ_ 2 _· · ζ_ )<sup>_′_</sup> are individual effects, and _ξt_ are time effects with _ι_ = (1 1 _· ·_ 1)<sup>_′_</sup> , where 5