Spatial Methods in Econometrics

Daniela GumprechtDepartment for Statistics and Mathematics,

University of Economics and Business Administration, Vienna

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Content• Spatial analysis – what for?• Spatial data• Spatial dependency and spatial

autocorrelation• Spatial models• Spatial filtering• Spatial estimation• R&D Spillovers

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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).

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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.

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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

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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.

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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): WyW nn spatial weights matrixy n1 vector of observations on the random variableElements W: non-stochastic and exogenous

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SAR and SMA processes• Simultaneous SAR process:

y = ρWy+ε = (I-ρW)-1ε• Spatial moving average process:

y = λWε+ε = (I+λW)εy centred variable I nn identity matrixε i.i.d. zero mean error terms with

common variance σ²ρ, λ autoregressive and moving

average parameters, in most cases |ρ|<1.

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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)’

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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.

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Moran‘s I • Measure of spatial autocorrelation:

I = e’(1/2)(W+W’)e / e’ee 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)]²M = I-X(X’X)-1X’ projection matrix

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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]

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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).

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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.

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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)

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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

*).

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Spatial estimation• S2SLS (from Kelejian and Prucha, 1995).

It consists of IV or GMM estimator of the auxiliary parameters:

(ρI,σI ²) = Arg min {[Γ(ρ,ρ²,σ²)-γ]’[Γ(ρ,ρ²,σ²)-γ]}with ΩI=Ω(ρI,σI ²) = σI ²[I-W(ρI)]-1[I-W(ρI)’]-1

where ρ[-a,a], σ²[0,b]FGLS estimator:

βIFGLS = [X’ΩI-1X]-1X’ΩI-1y

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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.

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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.

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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.

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R&D Spillovers• Model:

logFit = it0+it

dlogSitd+it

flogSitf

regional index i and temporal index t Fit total factor productivity (TFP)

Sitd domestic R&D expenditures

Sitf foreign R&D expenditures

it0 intercept (varies across countries)

itd 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)

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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.

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R&D Spillovers• The bilateral trade intensity between

country i and j:wI ij = (bij+bji)/2

wI ij = 0 for i = j• bij are the bilateral import shares of

country i from country j

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R&D Spillovers• Distance between two countries:

inverse connectivity 1 / wI ij • The higher the connectivity the smaller

the distance and vice versa. dij = wI ij-1 for all i and j

dii = 0• Distance matrix D: symmetric nn

matrix (231 distances for n = 22).

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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,y Σi(di-diP)2

xnx1, ynx1 coordinates of points

di original distances

diP distances in the projection space IR2

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R&D Spillovers

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R&D Spillovers

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R&D Spillovers• C&H results: using a standard fixed

effects panel regression they yielded logFit = it

0+0,097 logSitd+0,0924 logSit

f

(10,6836)*** (5,8673)***• Domestic and foreign R&D expenditures

have a positive effect on total factor productivity of a country.

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R&D Spillovers• Results using a dynamic random

coefficients model:logFit = it

0+0,3529 logSitd-0,085 logSit

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.

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R&D Spillovers• Spatial analysis: standard fixed effects

model with a spatial lagged exogenous variable:

• logFit = it0+0,0673 Sit

d+0,1787 bijtSitd

(4,1483)*** (8,2235)***• Domestic and foreign R&D expenditures

have a positive effect on total factor productivity of a country.

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R&D Spillovers• Spatial analysis: dynamic random

coefficients model with a spatially lagged exogenous variable:logFit = it

0+0,1252 Sitd+0,1663 bijtSit

d

(2,2895)** (2,1853)**• Domestic and foreign R&D expenditures

have a positive effect on total factor productivity of a country.

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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