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## Spatial Methods in Econometrics

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**Spatial Methods in Econometrics**Daniela Gumprecht Department for Statistics and Mathematics, University of Economics and Business Administration, Vienna**Content**• Spatial analysis – what for? • Spatial data • Spatial dependency and spatial autocorrelation • Spatial models • Spatial filtering • Spatial estimation • R&D Spillovers**Spatial data – what for?**• Exploitation of regional dependencies (information spillover) to improve statistical conclusions. • Techniques from geological and environmental sciences. • Growing number of applications in social and economic sciences (through the dispersion of GIS).**Spatial data**• Spatial data contain attribute and locational information (georeferenced data) . • Spatial relationships are modelled with spatial weight matrices. • Spatial weight matrices measure similarities (e.g. neighbourhood matrices) or dissimilarities (distance matrices) between spatial objects.**Spatial dependency**• “Spatial dependency is the extent to which the value of an attribute in one location depends on the values of the attribute in nearby locations.” (Fotheringham et al, 2002). • “Spatial autocorrelation (…) is the correlation among values of a single variable strictly attributable to the proximity of those values in geographic space (…).” (Griffith, 2003). • Spatial dependency is not necessarily restricted to geographic space**Spatial weight matrices**• W = [wij], spatial link matrix. • wij = 0 if i = j • wij > 0 if i and j are spatially connected • If w*ij = wij / Σj wij,W* is called row-standardized • W can measure similarity (e.g. connectivity) or dissimilarity (distances). • Similarity and dissimilarity matrices are inversely related – the higher the connectivity, the smaller the distance.**Spatial stochastic processes**• Spatial autoregressive (SAR) processes. • Spatial moving average (SMA) processes. • Spatial lag operator is a weighted average of random variables at neighbouring locations (spatial smoother): Wy W nn spatial weights matrix y n1 vector of observations on the random variable Elements W: non-stochastic and exogenous**SAR and SMA processes**• Simultaneous SAR process: y = ρWy+ε = (I-ρW)-1ε • Spatial moving average process: y = λWε+ε = (I+λW)ε y centred variable I nn identity matrix ε i.i.d. zero mean error terms with common variance σ² ρ, λ autoregressive and moving average parameters, in most cases |ρ|<1.**SAR and SMA processes**• Variance-covariance matrix for y is a function of two parameters, the noise variance σ² and the spatial coefficient, ρ or λ. • SAR structure: Ω(ρ) = Cov[y,y] = E[yy’] = σ²[(I-ρW)’(I-ρW)]-1 • SMA structure: Ω(λ) = Cov[y,y] = E[yy’] = σ²(I+ λW)(I+ λW)’**Spatial regression models**• Spatial lag model: Spatial dependency as an additional regressor (lagged dependent variable Wy) y = ρWy+Xβ+ε • Spatial error model: Spatial dependency in the error structure (E[uiuj] ≠ 0) y = Xβ+u and u = ρWu+ε y = ρWy+Xβ-ρWXβ+u Spatial lag model with an additional set of spatially lagged exogenous variables WX.**Moran‘s I**• Measure of spatial autocorrelation: I = e’(1/2)(W+W’)e / e’e e vector of OLS residuals • E[I] = tr(MW) / (n-k) • Var[I] = tr(MWMW’)+tr(MW)²+tr((MW))² / (n-k)(n-k+2)–[E(I)]² M = I-X(X’X)-1X’ projection matrix**Test for spatial autocorrelation**• One-sided parametric hypotheses about the spatial autocorrelation level ρ H0: ρ ≤ 0 against H1: ρ > 0 for positive spatial autocorrelation. H0: ρ ≥0 against H1: ρ < 0 for negative spatial autocorrelation. • Inference for Moran’s I is usually based on a normal approximation, using a standardized z-value obtained from expressions for the mean and variance of the statistic. z(I) = (I-E[I])/√Var[I]**Spatial filtering**• Idea: Separate regional interdependencies and use conventional statistical techniques that are based on the assumption of spatially uncorrelated errors for the filtered variables. • Spatial filtering method based on the local spatial autocorrelation statistic Gi by Getis and Ord (1992).**Spatial filtering**• Gi(δ) statistic, originally developed as a diagnostic to reveal local spatial dependencies that are not properly captured by global measures as the Moran’s I, is the defining element of the first filtering device • Distance-weighted and normalized average of observations (x1, ..., xn) from a relevant variable x. Gi(δ) = Σjwij(δ)xj / Σjxj, i ≠ j • Standardized to corresponding approximately Normal (0,1) distributed z-scores zGi, directly comparable with well-known critical values.**Spatial filtering**• Expected value of Gi(δ) (over all random permutations of the remaining n-1 observations) E[Gi(δ)] = Σjwij(δ) / (n-1) represents the realization at location i when no autocorrelation occurs. • Its ratio to the observed value indicates the local magnitude of spatial dependence. • Filter the observations by: xi* = xi[Σjwij(δ) / (n-1)] / Gi(δ)**Spatial filtering**• (xi-xi*) purely spatial component of the observation. • xi* filtered or “spaceless” component of the observation. • If δ is chosen properly the zGi corresponding to the filtered values xi* will be insignificant. • Applying this filter to all variables in a regression model isolates the spatial correlation into (xi-xi*).**Spatial estimation**• S2SLS (from Kelejian and Prucha, 1995). It consists of IV or GMM estimator of the auxiliary parameters: (ρ̃,σ̃²) = Arg min {[Γ(ρ,ρ²,σ²)-γ]’[Γ(ρ,ρ²,σ²)-γ]} with Ω̃=Ω(ρ̃,σ̃²) = σ̃²[I-W(ρ̃)]-1[I-W(ρ̃)’]-1 where ρ[-a,a], σ²[0,b] FGLS estimator: β̃FGLS = [X’Ω̃-1X]-1X’Ω̃-1y**R&D Spillovers**• Theories of economic growth that treat commercially oriented innovation efforts as a major engine of technological progress and productivity growth (Romer 1990; Grossman and Helpman, 1991). • Coe and Helpman (1995): productivity of an economy depends on its own stock of knowledge as well as the stock of knowledge of its trade partners.**R&D Spillovers**• Coe and Helpman (1995) used a panel dataset to study the extent to which a country’s productivity level depends on domestic and foreign stock of knowledge. • Cumulative spending for R&D of a country to measure the domestic stock of knowledge of this country. • Foreign stock of knowledge: import-weighted sum of cumulated R&D expenditures of the trade partners of the country.**R&D Spillovers**• Panel dataset with 22 countries (21 OECD countries plus Israel) during the period from 1971 to 1990. • Variables total factor productivity (TFP), domestic R&D capital stock (DRD) and foreign R&D capital stock (FRD) are constructed as indices with basis 1985 (1985=1). • Panel data model with fixed effects.**R&D Spillovers**• Model: logFit = it0+itdlogSitd+itflogSitf regional index i and temporal index t Fit total factor productivity (TFP) Sitd domestic R&D expenditures Sitf foreign R&D expenditures it0intercept (varies across countries) itd coefficient, corresponds to elasticity of TFP with respect to domestic R&D itf coefficient, corresponds to elasticity of TFP with respect to foreign R&D (itf)**R&D Spillovers**• Assumption: variables R&D spending are spatially autocorrelated => no need to use separate variables for domestic and foreign R&D spendings. • Trade intensity: average of bilateral import shares between two countries = connectivity- or distance measure.**R&D Spillovers**• The bilateral trade intensity between country i and j: w̃ij = (bij+bji)/2 w̃ij = 0 for i = j • bij are the bilateral import shares of country i from country j**R&D Spillovers**• Distance between two countries: inverse connectivity 1 / w̃ij • The higher the connectivity the smaller the distance and vice versa. dij = w̃ij-1 for all i and j dii = 0 • Distance matrix D: symmetric nn matrix (231 distances for n = 22).**R&D Spillovers**• Plot the distances between all countries. • Project all 231 distances from IR21 to IR2. • Minimize the sum of squared distances between the original points and the projected points: minx,yΣi(di-diP)2 xnx1, ynx1 coordinates of points di original distances diP distances in the projection space IR2**R&D Spillovers**• C&H results: using a standard fixed effects panel regression they yielded logFit = it0+0,097 logSitd+0,0924 logSitf (10,6836)*** (5,8673)*** • Domestic and foreign R&D expenditures have a positive effect on total factor productivity of a country.**R&D Spillovers**• Results using a dynamic random coefficients model: logFit = it0+0,3529 logSitd-0,085 logSitf (7,7946)*** (-1,1866) • Domestic R&D expenditures have a positive effect on total factor productivity of a country, foreign R&D spending have no effect.**R&D Spillovers**• Spatial analysis: standard fixed effects model with a spatial lagged exogenous variable: • logFit = it0+0,0673 Sitd+0,1787 bijtSitd (4,1483)*** (8,2235)*** • Domestic and foreign R&D expenditures have a positive effect on total factor productivity of a country.**R&D Spillovers**• Spatial analysis: dynamic random coefficients model with a spatially lagged exogenous variable: logFit = it0+0,1252 Sitd+0,1663 bijtSitd (2,2895)** (2,1853)** • Domestic and foreign R&D expenditures have a positive effect on total factor productivity of a country.**R&D Spillovers**• Conclusion: • Different estimation techniques lead to different results • Still not clear whether foreign R&D spending have an influence on total factor productivity. • Further research needed