Assessing Regional Economic Performance: Regional Competition in Spain Under a Spatial Vector Autoregressive Approach

Abstract

After a wave of empirical literature on regional competition focused on issues related to regional convergence, work developed recently has tried to address some of the shortcomings of the previous literature, developing a series of alternative approaches that center their attention on the assessment of regional economic performance. These alternative methodologies embrace a complex set of space-time interactions that take into account that a single region’s economic performance affects and is affected by other regions. In this context, and as an original contribution of this chapter, a spatial vector autoregressive (SpVAR) model for the Spanish regions during the period 1955 –2009 is presented. The SpVAR model considers spatial as well temporal lags of the variables. The estimated SpVAR is used to calculate impulse responses that provide insights about the effects of shocks to relative regional productive capacity on different regions. The empirical results suggest that the existence of trade linkages have had different significant impacts on the production shares of the 17 Spanish regions, but competition between regional economies prevails.

JEL codes: C31, C33, J24, O18, R11

Appendix

This appendix includes a brief discussion about multivariate spatial vector autoregressive –SpVAR- models, containing both the general formulation of such models and an explanation on the main properties of this type of econometric specifications.

Spatial VARs are a special type of vector autoregressions (Sims 1980), which include spatial as well as temporal lags of the state variables. Contrary to standard VARs, that do not allows the joint modeling of dynamic spatio-temporal interdependencies within a group of connected local economies (regions or states, metropolitan areas or local districts), SpVAR models permit that endogenous variables can exhibit co-movements over time and also over space.

Let ${{Y}_{{it}}} = ({{y}_{{1,it}}},{{y}_{{2,it}}}, \ldots, {{y}_{{G,it}}})$ and ${{X}_{{it}}} = ({{x}_{{1,it}}},{{x}_{{2,it}}}, \ldots, {{x}_{{K,it}}})$ denote two $G \times 1$ and $K \times 1$ vectors of stationary endogenous and exogenous variables, respectively, recorded on the ith region (i = 1,2,…,N) at time t (t = 1,2,…,T ). A reduced-form SpVAR specification of order p for these data can be written as

$$ {{Z}_{{it}}} = {{\Gamma}_{{0i}}} + {{\Gamma}_{{1i}}}{{Z}_{{i,t - 1}}} + \ldots + {{\Gamma}_{{pi}}}{{Z}_{{i,t - p}}} + {{\Phi}_{{0i}}}t + {{\Phi}_{{1i}}}{{X}_{{i,t}}} + {{U}_{{it}}} $$

where ${{Z}_{{it}}} = ({{Y}_{{it}}},Y_{{it}}^{*})$; ${{\Gamma}_{{ji}}}$ ( j = 0,1, …, p) and ${{\Phi}_{{ji}}}$ ( j = 0,1) are matrices of coefficients to be estimated; ${{U}_{{it}}}$ is a $2G \times 1$ vector of non-autocorrelated reduced-form disturbances with mean zero and a nonsingular covariance matrix, ${{\Sigma}_i}$; and $Y_{{it}}^{*} = ({{I}_G} \otimes W){{Y}_t}$, with ${{Y}_t} = ({{Y}_{{1t}}},{{Y}_{{2t}}}, \ldots, {{Y}_{{Nt}}})$ and W being a row-standardized $N \times N$ connectivity matrix with elements ${{w}_{{ij}}}$ fixed over time satisfying ${{w}_{{ii}}} = 0$ and $\sum\limits_{{j = 1}}^N {{{w}_{{ij}}}} = 1$. The components of the spatial lagged dependent vector $Y_{{it}}^{*}$ can be written as $y_{{g,it}}^{*} = \sum\limits_{{j = 1}}^N {{{w}_{{ij}}}{{y}_{{g,jt}}}}$ and would be the weighted average of $y_g$ in region i for regions s where ${{w}_{{is}}} \ne 0$.

It can be seen that spatially heterogeneous model dynamics is allowed because parameters are assumed to vary unrestrictedly at the level of the individual regions. Also, it can be observed that contemporaneous relations across the states variables are not modeled explicitly but captured by the elements of the covariance matrix ${{\Sigma}_i}$.

Written in disaggregated form, the SpVAR model for region i takes the following form:

$$ \left\{ {\begin{array}{lll} {{{Y}_{{it}}} = \Gamma_{{0i}}^1 + \Gamma_{{1i}}^1{{Y}_{{i,t - 1}}} + \Gamma_{{2i}}^1Y_{{i,t - 1}}^{*} + \ldots + \Phi_{{0i}}^1t + \Phi_{{1i}}^1{{X}_{{it}}} + U_{{it}}^1} \\{Y_{{it}}^{*} = \Gamma_{{0i}}^2 + \Gamma_{{1i}}^2{{Y}_{{i,t - 1}}} + \Gamma_{{2i}}^2Y_{{i,t - 1}}^{*} + \ldots + \Phi_{{0i}}^2t + \Phi_{{1i}}^2{{X}_{{it}}} + U_{{it}}^2} \\\end{array} } \right. $$

This expression implies that a spatial VAR can be seen as a spatial-extended VAR model for the vector ${{Y}_{{it}}}$. For each endogenous variable $y_m$ of this vector (m = 1,2,…,G), the SpVAR with N regions takes the form of the following system of equations (similar specification exists for the external variables $y_m^{*}$):

$$ \left\{ \begin{array}{lll} \begin{array}{lll} {y_{{m,1t}} = \varphi_{{01,m}}^1 + \sum\limits_{{g = 1}}^G{\varphi_{{11,m,g}}^1{{y}_{{g,1,t - 1}}}} + \sum\limits_{{g = 1}}^G{\varphi_{{21,m,g}}^1y_{{g,1,t - 1}}^{*}} + \ldots +\phi_{{01,m}}^1t}\cr \quad\quad\quad\quad+ \sum\limits_{{k = 1}}^K{\phi_{{11,m,k}}^1{{x}_{{k,1t}}}} + u_{{m,1t}}^1\end{array}\\ \begin{array}{lll} {y_{{m,2t}} = \varphi_{{02,m}}^1 + \sum\limits_{{g = 1}}^G {\varphi_{{12,m,g}}^1{{y}_{{g,2,t - 1}}}} + \sum\limits_{{g = 1}}^G {\varphi_{{22,m,g}}^1y_{{g,2,t - 1}}^{*}} + \ldots + \phi_{{02,m}}^1t} \cr \quad\quad\quad\quad+ \sum\limits_{{k = 1}}^K {\phi_{{12,m,k}}^1{{x}_{{k,2t}}}} + u_{{m,2t}}^1 \\ \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \vdots\end{array} \\ \begin{array}{lll} {y_{{m,Nt}} = \varphi_{{0N,m}}^1 + \sum\limits_{{g = 1}}^G {\varphi_{{1N,m,g}}^1{{y}_{{g,N,t - 1}}}} + \sum\limits_{{g = 1}}^G {\varphi_{{2N,m,g}}^1y_{{g,N,t - 1}}^{*}} + \ldots + \phi_{{0N,m}}^1t}\cr \quad\quad\quad\quad+ \sum\limits_{{k = 1}}^K {\phi_{{1N,m,k}}^1{{x}_{{k,Nt}}}} + u_{{m,Nt}}^1\\\end{array}\end{array}\right.$$

These reduced-form equations include the deterministic and exogenous variables, a set of temporally lagged variables (as in the traditional VARs), and a set of new temporally lagged spatial-lag variables.

SpVARs can be used to simulate the spatio-temporal dynamic effects of exogenous shocks within the system. Now, the impulse response analysis is more general than in the traditional VARs: an exogenous shock that occurs in a given region (or a group of them) at a point time can affect the economic conditions of other regions in the next periods. Therefore, shocks can propagate over time as well as across space, permitting the existence of spatial spillover effects: an exogenous shock in a region can spill over to the locations considered as neighbours (${{w}_{{is}}} \ne 0$).