Download Spatial Statistics in Econometrics

Spatial Methods in
Econometrics
Daniela Gumprecht
Department for Statistics and Mathematics,
University of Economics and Business Administration,
Vienna
Content
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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
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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): Wy
W nn spatial weights matrix
y n1 vector of observations on the random
variable
Elements 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)ε
y
centred variable
I
nn 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.
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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’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
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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 zvalue 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:
(ρ̃ ,σ̃ ²) = Arg min {[Γ(ρ,ρ²,σ²)-γ]’[Γ(ρ,ρ²,σ²)γ]}
with Ω̃=Ω(ρ̃ ,σ̃ ²) = σ̃ ²[I-W(ρ̃ )]-1[I-W(ρ̃ )’]-1
where ρ[-a,a], σ²[0,b]
FGLS estimator:
β̃ FGLS = [X’Ω̃-1X]-1X’Ω̃-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+itdlogSitd+itflogSitf
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
itf
coefficient, corresponds to elasticity of
TFP with respect to foreign R&D (itf)
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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:
w̃ij = (bij+bji)/2
w̃ij = 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 / w̃ij
• The higher the connectivity the smaller the
distance and vice versa.
dij = wĩ j-1 for all i and j
dii = 0
• Distance matrix D: symmetric nn 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 = 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.
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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.
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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.
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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.
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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
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