space and growth: a survey of empirical evidence ... - ResearchGate

Région et Développement 31y = ρ Wy + α + βx+ ε ,(3)where x is an explanatory variable, and ρ is a parameter indicating the extent ofthe spatial interaction between observations with non-zero entries in W, thespatial weights matrix. The model can be rewritten in reduced form:where (1 −to:−y = ( I − ρW) α + βx+ ε .(4)1[ ]The marginal effect of an increase in x on y is:⎡⎤= = −∂x∂x⎣−1∂y∂⎣( I −ρW)βx⎦ −1⎡ ⎤( I ρW)β ,⎦−1ρ W ) is the spatial multiplier (see Anselin, 2002). This is equivalent∂y= (1 − ρW)∂x′− 1β.The correct interpretation of the spatial lag model and the conceptualdistinction between the various effects caused by the spatial multiplier can beseen by decomposing the spatial multiplier using the formula for a sum toinfinity (since |ρ| < 1):∂y2 2 3 3= ( I + ρW+ ρ W + ρ W + ...) β∂x′2 2 3 3= Iβ+ ρWβ+ ρ W β + ρ W β...The first term on the right hand side is a matrix with the direct effects onthe diagonal (the effects on y i of a marginal change in x i , where i refers to aspatial unit), and zeros in off-diagonal positions. The second term is a matrixwith zeros on the diagonal and indirect effects in the off-diagonal positions forthe regions j, defined as the neighbors of i in the spatial weights matrix. Theseindirect effects are spillovers of the direct effects, and both effects are local inthe sense that only the regions in which there has been an exogenous shock andtheir neighbors are affected. The third and higher-order terms refer to spatialspillovers induced by the direct and indirect changes in the first and secondterms, and can therefore be referred to as induced effects. Note that these inducedeffects comprise impacts on the higher-order neighbors (the neighbors of theneighbors of i) as well as feed-back effects on regions which are alreadyexperiencing direct and indirect effects. The latter has two importantimplications. First, this implies the spatial lag model links all the regions in the(5)(6)(7)