Spatial Methods in Econometrics Daniela Gumprecht Department for Statistics and Mathematics, University of Economics and Business Administration, Vienna

2 Content Spatial analysis – what for? Spatial data Spatial dependency and spatial autocorrelation Spatial models Spatial filtering Spatial estimation R&D Spillovers

3 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).

4 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.

5 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

6 Spatial weight matrices W = [w ij ], spatial link matrix. w ij = 0 if i = j w ij > 0 if i and j are spatially connected If w * ij = w ij / Σ j w ij, 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.

7 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 n  n spatial weights matrix y n  1 vector of observations on the random variable Elements W: non-stochastic and exogenous

8 SAR and SMA processes Simultaneous SAR process: y = ρ Wy+ ε = (I- ρ W) -1 ε Spatial moving average process: y = λ W ε + ε = (I+ λ W) ε ycentred variable I n  n identity matrix ε i.i.d. zero mean error terms with common variance σ ² ρ, λ autoregressive and moving average parameters, in most cases | ρ |<1.

9 SAR and SMA processes Variance-covariance matrix for y is a function of two parameters, the noise variance σ ² and the spatial coefficient, ρ or λ. SAR structure: Ω ( ρ ) = Cov[y,y] = E[yy’] = σ ²[(I- ρ W)’(I- ρ W)] -1 SMA structure: Ω ( λ ) = Cov[y,y] = E[yy’] = σ ²(I+ λ W)(I+ λ W)’

10 Spatial regression models Spatial lag model: Spatial dependency as an additional regressor (lagged dependent variable Wy) y = ρ Wy+X β + ε Spatial error model: Spatial dependency in the error structure (E[u i u j ] ≠ 0) y = X β +u and u = ρ Wu+ ε y = ρ Wy+X β - ρ WX β +u Spatial lag model with an additional set of spatially lagged exogenous variables WX.

11 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) -1 X’ projection matrix

12 Test for spatial autocorrelation One-sided parametric hypotheses about the spatial autocorrelation level ρ H 0 : ρ ≤ 0 against H 1 : ρ > 0 for positive spatial autocorrelation. H 0 : ρ ≥ 0 against H 1 : 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]

13 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 G i by Getis and Ord (1992).

14 Spatial filtering G i ( 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 (x 1,..., x n ) from a relevant variable x. G i ( δ ) = Σ j w ij ( δ )x j / Σ j x j, i ≠ j Standardized to corresponding approximately Normal (0,1) distributed z-scores z Gi, directly comparable with well-known critical values.

15 Spatial filtering Expected value of G i ( δ ) (over all random permutations of the remaining n-1 observations) E[G i ( δ )] = Σ j w ij ( 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: x i * = x i [ Σ j w ij ( δ ) / (n-1)] / G i ( δ )

16 Spatial filtering (x i -x i * ) purely spatial component of the observation. x i * filtered or “spaceless” component of the observation. If δ is chosen properly the z Gi corresponding to the filtered values x i * will be insignificant. Applying this filter to all variables in a regression model isolates the spatial correlation into (x i -x i * ).

17 Spatial estimation S2SLS (from Kelejian and Prucha, 1995). It consists of IV or GMM estimator of the auxiliary parameters: ( ρ̃, σ̃ ²) = Arg min {[ Γ ( ρ, ρ ², σ ²)- γ ]’[ Γ ( ρ, ρ ², σ ²)- γ ]} with Ω̃ = Ω ( ρ̃, σ̃ ²) = σ̃ ²[I-W( ρ̃ )] -1 [I-W( ρ̃ )’] -1 where ρ  [-a,a], σ ²  [0,b] FGLS estimator: β̃ FGLS = [X’ Ω̃ -1 X] -1 X’ Ω̃ -1 y

18 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.

19 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.

20 R&D Spillovers Panel dataset with 22 countries (21 OECD countries plus Israel) during the period from 1971 to 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.

21 R&D Spillovers Model: logF it =  it 0 +  it d logS it d +  it f logS it f regional index i and temporal index t F it total factor productivity (TFP) S it d domestic R&D expenditures S it f foreign R&D expenditures  it 0 intercept (varies across countries)  it d coefficient, corresponds to elasticity of TFP with respect to domestic R&D  it f coefficient, corresponds to elasticity of TFP with respect to foreign R&D (  it f )

22 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.

23 R&D Spillovers The bilateral trade intensity between country i and j: w ̃ ij = (b ij +b ji )/2 w ̃ ij = 0 for i = j b ij are the bilateral import shares of country i from country j

24 R&D Spillovers Distance between two countries: inverse connectivity 1 / w ̃ ij The higher the connectivity the smaller the distance and vice versa. d ij = w ̃ ij -1 for all i and j d ii = 0 Distance matrix D: symmetric n  n matrix (231 distances for n = 22).

25 R&D Spillovers Plot the distances between all countries. Project all 231 distances from IR 21 to IR 2. Minimize the sum of squared distances between the original points and the projected points: min x,y Σ i (d i -d i P ) 2 x nx1, y nx1 coordinates of points d i original distances d i P distances in the projection space IR 2

26 R&D Spillovers

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28 R&D Spillovers C&H results: using a standard fixed effects panel regression they yielded logF it =  it 0 +0,097 logS it d +0,0924 logS it f (10,6836)*** (5,8673)*** Domestic and foreign R&D expenditures have a positive effect on total factor productivity of a country.

29 R&D Spillovers Results using a dynamic random coefficients model: logF it =  it 0 +0,3529 logS it d -0,085 logS it f (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.

30 R&D Spillovers Spatial analysis: standard fixed effects model with a spatial lagged exogenous variable: logF it =  it 0 +0,0673 S it d +0,1787 b ijt S it d (4,1483)*** (8,2235)*** Domestic and foreign R&D expenditures have a positive effect on total factor productivity of a country.

31 R&D Spillovers Spatial analysis: dynamic random coefficients model with a spatially lagged exogenous variable: logF it =  it 0 +0,1252 S it d +0,1663 b ijt S it d (2,2895)** (2,1853)** Domestic and foreign R&D expenditures have a positive effect on total factor productivity of a country.

32 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