Spatial Econometrics - DIMETIC

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SpatialSpatial EconometricsEconometrics

Rosina Moreno

AQR Research Group-IREA

Universidad de Barcelona

OutlineOutline

Introduction

Spatial effects

Exploratory Spatial Data Analysis (ESDA)

Spatial Regressions

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I) Introduction to Spatial

Econometrics

IntroductionIntroduction toto SpatialSpatial EconometricsEconometrics

In the last decades a big effort has been made with the objective:

To increase the availability of databases for a

dissaggregated level:

regional and local databases

This has favoured a growing number of applied works in thescope of Regional and Urban Economics

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ATTENTION!!!! ... when cross-section data are used two types ofproblem may appear:

•• SpatialSpatial HeterogeneityHeterogeneity::

When data from very different spatial units are used to explain a phenomenon. Heteroskedasticity or structural instability.

•• SpatialSpatial AutocorrelationAutocorrelation::

Coincidence of value similarity with locational similarity.

Although spatial heterogeneity can be tackled by means of thestandard econometric toolbox, not the case for spatialautocorrelation because of the multidirectionality of theinterdependence of the relations

SPATIAL ECONOMETRICSSPATIAL ECONOMETRICS

Anselin (1988).. “A collection of techniques that deal with the

peculiarities caused by space in the statistical analysis of regional

science models”

Historically, although the origine could be settled in 1948, Spatial Econometrics originated as an identifiable field in Europe in the early 1970s to deal with sub-country data in regional econometric models (Hordijk and Paelinck, 1976):

• the role of spatial dependence in spatial models

• the asymmetry in spatial relations

• the importance of explanatory factors located in other spaces

• explicit modeling of space (inclusion of topological variables)

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Has recently gained a more central place

in applied and theoretical economics

• In the past : models that explicitly incorporated space (urbanand regional economics, real state economics, economicgeography, transportation economics)

• More recently : spatial econometric methods have increasinglybeen applied in more traditional fields of economics(international economics, labor economics, public economics, agricultural and environmental economics).

WHY?

SpatialSpatial InteractionInteraction

Some major factors to this new attention to spatial interaction:

• A growing interest within theoretical economics in models thatmove away from the atomistic agent acting in isolation to anexplicit accounting for the interaction of that agent with others.

• Spatial aspects of Marshall externalities, agglomeration economies and other spillovers : New Economic Geography (Krugman, Glaeser et al, Durlauf, Ciccone)

• The explosive diffusion of geographic information systems(GIS) technology and the associated availability of geo-codedsocio-economic datasets for empirical analysis.

theoretical interest + GIS + spatial data + spatial econometrics techniques

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In spite of the introduction of spatial interaction ...

econometric theory and practice have been dominated

by a focus in the time dimension

No reference to the concept of spatial autocorrelation in any ofthe commonly cited econometrics texts

(Judge et al, 1995, Greene, 1993, Davidson and MacKinnon, 1993). A rare exception is Johnston, 1984

Anselin and Florax (1995): “...In spite of these important methodological developments, it would be an overstatement to suggest that spatial economometrics

has become accepted practice in current empirical research in regional science and regional economics”

H. Kelejian and D. Robinson, E. Can, L. Anselin, A. Varga, S. Rey,

B. Fingleton, R. Paci, E. Lopez-Bazo ...

II) Spatial Effects

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SPATIAL HETEROGENEITY

Structural instability in the form of:

non-constant error variances (spatial heteroskedasticity)

model coefficients (variable coefficients, spatial regimes).

Can be tackled by means of the standard econometric tools, but...

3 reasons why it is important to consider spatial heterogeneity explicitly:

• The structure behind the instability is spatial (or geographic) in the sensethat the location of the observations is crucial in determining the form of theinstability.

• Because the structure is spatial, heterogeneity often occurs jointly withspatial autocorrelation (tests for heteros can be misleading)

• In a single cross-section, spatial autocorrelation and spatial heterogeneitymay be observationally equivalent.

SPATIAL SPATIAL

AUTOCORRELATION/DEPENDENCEAUTOCORRELATION/DEPENDENCE

The coincidence of value similarity with locational similarity.

Positive spatial autocorrelation: High or low values for a variable tendto cluster in space.

A sample contains less information than an uncorrelated counterpart

Negative spatial autocorrelation: Locations tend to be surrounded by neighbors with very dissimilar values.

Implies a checkerboard pattern of values and does not always have a meaningful interpretation.

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Two main reasons:

1º Measurement Errors

“scarse correspondence between the spatial extension of the economic

phenomenon and the units of observation. SPUREOUS AUTOCORRELATION”

For instance: Urban Systems (joint labour markets) y municipalities

2º Spatial Interaction between spatial units

“Spatial Externalities espaciales (spillovers): Spill-Over effects from infrastructures

or technological diffusion among economies”

A B C

1 2

Firms in space- heterogeneous

Example

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Firms in space:- heterogeneous- linked to each other

A B C

D E F

Firms in space:- heterogeneous- linked to each other

- found in different regions

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A B C

D E F

Firms in space:- heterogeneous- linked to each other

- located in different regions

Give place to regions- heterogeneous- linked

Spatial autocorrelation can be formally expressed by the moment condition:

This covariance becomes meaningful whenthe particular configuration of nonzero i,j pairs has an interpretation

in terms of spatial structure or spatial interaction.

In a cross-sectional setting with N observations, there is insufficientinformation to estimate the N by N covariance matrix.

Not even with asymptotics.

* SIMILARITIES BETWEEN TEMPORAL

AND SPATIAL AUTOCORRELATION

jiforYYCov ji ≠≠ 0][

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In the space, the relations are multidirectional. This requires a specific notion:

W : Spatial weights matrix

N by N positive and symmetric matrix:

wij = 1 when i and j are neighbors

wij = 0 otherwise

Row-standardized weights matrix:

• all weights between 0 and 1

• the spatial parameters are comparable between models (smoothing of theneighboring values)

A SPATIAL LAG OPERATORA SPATIAL LAG OPERATOR : a weighted average ofthe values at neighbouring regions : Wy

∑=

j

ij

ijs

ij w

ww

NotionsNotions ofof neighborhoodneighborhood::

• traditional approach: geography or spatial arrangement ofobservations

• physical contiguity (rook, queen)

• inverse function of the distance (gravity models):

• sociometrics: whether two individuals belong to the same social network based on economic distance

• trade relations

HigherHigher orderorder ofof spatialspatial laglag operatorsoperators:: WW22y, Wy, W33y, ...y, ...

ji

s

ijxx

w−

= 1

w dij ija

ijb= − β

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Spatial

dependence

At a univariate level: ESDA

In a regression model: Spatial Regression

III) Exploratory SpatialData Analysis

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IsIs itit necessarynecessary toto carrycarry out out anan ESDA in regional ESDA in regional studiesstudies??

We generate 3 series:

Series 1: random distribution in space

Series 2: 11 region-centres randomly distributed

Series 3: certain jerarchy with a clear centre

SERIE 1

SERIE 2 SERIE 3

Índice Gini= 0.26

Índice Theil=0.05

Índice Gini= 0.16

Índice Theil=0.02

Índice Gini= 0.16

Índice Theil=0.02

10000<=y<13000

y<10000

13000<=y<16000

19000<=y<22000

16000<=y<19000

22000<=y

10000<=y<13000

5000<=y<10000

13000<=y<16000

19000<=y<22000

16000<=y<19000

22000<=y

10000<=y<13000

5000<=y<10000

13000<=y<16000

19000<=y<22000

16000<=y<19000

22000<=y

IsIs itit necessarynecessary toto carrycarry out out anan ESDA in regional ESDA in regional studiesstudies??

Descriptive analysis of the distribution in the space of the variables:

map with spatial distribution + indexes of concentration

NON-SPATIAL (each region isisolated from the rest, a randomdistribution in the space is

considered)

It is a subjetive informationand highly dependent of, forexample, the number ofintervals chosen.

SOLUTION?

A complementary analysis, taking special account of the space

Objective statistical testing of the presence of a spatial dependence process

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ESDA

A methodology to generate insight into patterns and associations in data, without prior assumptions. Focusing explicictly on spatial effects.

techniques to describe spatial distributions

• spatial trends

discover patterns of spatial association

• spatial clustering

indentify atypical locations

• spatial outliers

suggest different spatial regimes or other forms of spatial instability

• spatial non-stationarity

ESDA : spatialspatial associationassociation

Draw a distinction between:

methods appropriate for the geostatistical or distance-based

perspective

• locations are considered to form a sample of an underlyingcontinuous distribution

and those geared to the lattice or neighborhood view of spatial data

• the data are conceptualized as a single realization of a spatialstochastic process

• spatial interaction is conceptualized as a step function

• the overall interaction (covariance) in the observed data is obtainedby assuming a particular form for the spatial stochastic process : W

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ESDA ESDA toolstools

A. Description and visualization of spatial distributions

B. Visualization of global spatial association and detection ofspatial non-stationarity

C. Local indicators of spatial association (LISA)

Description of the tool

Empirical example with real data

Software

Most of those ESDA tools use the capacity of data visualization andmanipulation given by GIS : GIS ArcView (ESRI, 1995) or GEODA

allows for dynamically linked windows of the data

different views of data:

• histogram, box plot, scatterplot

views dynamically linked:

• click on one, corresponding points (areas) on others highlight

geographic brushing:

• map as a view of data

Before: Vinculation ArcView with other software. For example: SpaceStatSpaceStat. .

AnselinAnselin, 1999. , 1999. ExtensionExtension ofof SpaceStatSpaceStat forfor ArcViewArcView

NowadaysNowadays: GEODA: GEODA

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• 108 EU12 regions (NUTS1 and NUTS2)

• spatial exploratory analysis of the variable GDPpw (in logs and relative to the European average), 1975-1992

• weights matrix: first order contiguity (standardized)

• GEODA

EmpiricalEmpirical ExerciseExercise

lny)e1(ag Ty

β−−−=

Growth equation (beta-convergence equation)

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A) A) VisualizationVisualization techniquestechniques

Spatial distribution of ln GDPpw, 1975

Spatial distribution of ∆ln GDPpw, 1975-1992

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Box map of ∆ln GDPpw, 1975-1992

B/ Global B/ Global spatialspatial autocorrelationautocorrelation

Null hypothesis: No spatial association

• values observed at a location do not depend on values observed at neighboring locations

• observed spatial patterns of values is equally likely as any otherspatial pattern

• the location of values may be altered without affecting theinformation content of the data

Alternative hypothesis: spatial association

• Positive spatial association

• similar values tend to cluster in space

• neighbors are similar

• Negative spatial association

• neighbors are dissimilar

• chekerboard pattern

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B.1. Spatial Autocorrelation Statistics

jixx

xxxxw

S

NI

N

1i2

i

N

ij jiij

0≠

−

−−=

∑

∑

= )(

))((

ji

xx

xxdw

dGN

1i

N

1jji

N

1i

N

1jjiij

≠=

∑∑

∑∑

= =

= =)(

)(

Z(I) not sig.: Absence of autocorrelation

(random distribution)

Z(I) >0 sig.: Positive Autocorrelation

(spatial clustering of similar values)

Z(I) <0 sig.: Negative Autocorrelation

(checkerboard pattern, “competition”)

Z(G) not sig.: Absence of spatial autocorrelation

Z(G) >0 sig.: Dominance of concentration of highvalues of the variable

Z(G) <0 sig.: Dominance of concentration of lowvalues of the variable

Moran’s I (Moran, 1948) G of Getis and Ord (Getis and Ord, 92)

B.2. B.2. MoranMoran ScatterplotScatterplot

Linear Spatial Association

• linear association between value at i and weighted average of neighbors:

four quadrants:

• high-high, low-low = spatial clusters

• high-low, low-high = spatial outliers

y.vsWyory.vsyw ij jij∑

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-3 -2 -1 0 1 2 3

Y

WY

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-3 -2 -1 0 1 2 3

Y

WY

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-3 -2 -1 0 1 2 3

Y

WY

No spatial dependence Positive spatial dependence Negative spatial dependence

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Relationship with Moran’s I

• this is the slope coefficient in a regression of Wy on y

• the statistic can be visualized as the slope of a straight line in a scatterplot

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Variable Moran’s I Geary’s C G, Getis and Ord

Ln GDP pw, 1975 11.605* -10.406* 2.543*

∆ ln GDP pw, 1975-1992 9.975* -8.749* -

Local Local IndicatorsIndicators of of SpatialSpatial AssociationAssociation (LISA)(LISA)

C/ Local Spatial Autocorrelation

Identify Hot Spots (in the absence of global association)

• significant local clusters (association around an individual location)

• significant local outliers

• high surrounded by low and vice versa

Indicate Local Instability

• local deviations from global pattern of spatial association

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LISA satisfies two requirements

• indicate significant spatial clustering for each location(association around an individual location)

• sum of LISA proportional to a global indicator of spatialassociation

LISA Forms of Global Statistics

• local Moran, local Getis and Ord

jJj

ij

i

2i

ii zw

Nz

zI

i

∑∑ ∈

=/

Local Moran (Ii) (Anselin, 1995) New Gi (Ord and Getis, 1995)

Z(Ii) >0 sig.: Cluster of similar values around i

Z(Ii) <0 sig.: Cluster of dissimilarvalues around i

New-Gi >0 sig.: Cluster of highsimilar values around i

New-Gi <0 sig.: Cluster of lowsimilar values around i

G dw d x

xj ii

ij jj

N

jj

N( )

( )= ≠=

=

∑

∑

1

1

1N)])xx((S[I =I =I /i2

ioi

i

i

i

−−γ ∑∑∑

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Significance of Ii, ∆Ln GDPpw 1975-1992

Spatial distribution of Ii, ∆Ln GDPpw 1975-1992

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CONCLUSIONS:CONCLUSIONS:

∆GDPpw presents a clear positive spatial dependence process:

• Similar values in neighboring regions

• Persistence of the process in the whole period (interdependence)

Important!!

• To wonder which factors can be explaining these results(externalities, common factors, ...)

• This spatial process may appear when the variable is used in theregression model (severe consequences)

NECESSITY OF A SPATIAL REGRESSION ANALYSISNECESSITY OF A SPATIAL REGRESSION ANALYSIS

IV) Spatial Regression

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FOUR ELEMENTS OF SPATIAL ECONOMETRICSFOUR ELEMENTS OF SPATIAL ECONOMETRICSFOUR ELEMENTS OF SPATIAL ECONOMETRICS

Specifying the structure of spatial dependence

• which locations interact (previous ESDA)

Testing for the presence of spatial dependence

• what type of dependence, which alternative

Estimating models with spatial dependence

• spatial lag, spatial error

Spatial prediction

ConsequencesConsequences ofof ignoringignoring spatialspatial autocorrelationautocorrelation

Ignoring a Spatial Lag:

Ignoring Spatial Error:

• biased and inconsistent estimation of the parameters (even withan error term spatially non-correlated)

• inefficient estimation of the parameters

• biased residual variance inefficient OLS

• biased inference (t-tests and R2)

• wrong specification test are still valid(structural instability, heteroskedasticity tests)

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• As an additional regressor in the form of a spatially lagged dependent variable:

)I,0(N~u

uXWyy2σ

+β+ρ=

• In the error structure:

)IN(0,~u

uW

Xy

2σ

+ελ=εε+β=

)I,0(Nu

uuW

Xy

2

1

1

σ∼

+θ=εε+β=

ψ+ν=εε+β=

W

Xy

• Autorregresive structure:

• Moving Average structure(Cliff y Ord, 1981):

• Error Components Model(KR, 1993):

SPATIAL DEPENDENCE AS

SPATIAL DEPENDENCE AS

A NUISANCE. SPATIAL

A NUISANCE. SPATIAL

ERROR MODELS

ERROR MODELS

SUBSTANTIVE SPATIAL

DEPENDENCE. SPATIAL

LAG MODELS

A/ SPATIAL REGRESSION SPECIFICATIONSA/ SPATIAL REGRESSION SPECIFICATIONSA/ SPATIAL REGRESSION SPECIFICATIONS

• Higher order spatial processes:

• Modelo SARMA(p,q): Huang (1984)

• Other spatial processes:

• Mixed regressive-spatialregressive:

)I,0(Nu

uuW...uWuWuW

XyW...yWyWyWy

2

qq332211

1pp332211

σ∼

+θ++θ+θ+θ=ε

ε+β+ρ++ρ+ρ+ρ=

• Mixed Regressive- SpatialAutorregressive Model :

y W y X

W u

u N I

= + += +

∼

ρ β εε λ ε

σ

1 1

2

20( , )

y W y X W R u

u N I

= + + +

∼

ρ β β

σ1 1 2 2

20( , )

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B/ TESTING SPATIAL DEPENDENCEB/ TESTING SPATIAL DEPENDENCE

• Ad-hoc tests valid to test the null of absence of error spatial dependence(the alternative hypothesis is not a specific spatial process model)

• Maximum Likelihood Based Tests well structured H0 and HA

• Moran’s I for regression residuals

• Kelejian and Robinson’s test

• Wald test

• Likelihood Ratio Tests

• Lagrange Multiplier/Score tests

(LM-LAG, LM-ERR, LM-EL, LM-LE)

Based on ML estimation ofthe spatial model

Based on OLS residuals

Moran’s I :

(1948)

IN

S

e We

e e= '

'

LM-ERR :

(1980, 1992)T1=tr(W’W+W

2)

[ ]21

1

2

2

~'

χT

sWee

ERRLM =−

LM-LAG :

(1988, 1992)

( )[ ]

2

1211

2112

RJTT

s/Wy'e)RJ(Ts

We'e

ELLM−

β−ρ

−β−ρ

−

−

=−

RJ=[T1+(WXβ)’M(WXβ)/s2]

LM LAG

e Wys

RJ− =

−

'2

2

ρ βLM LE

e Wys

e We s

RJ T− =

−

−−

' ' /22

1

2

ρ β

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III/ ESTIMATION OF SPATIAL ECONOMETRICS MODELSIII/ ESTIMATION OF SPATIAL ECONOMETRICS MODELS

• Maximum Likelihood Estimation most widely used alternative to OLS in thepresence of a spatial process

• The resulting system for first derivatives is highly non-linear, with full matrices for A and B. Use of numerical methods.

• The complication is reduced when obtaining a concentrated LF:

νν−++Ω−π−=φ '2

1AlnBlnln

2

1ln

2

N)(Lln

A=(I-ρW1), B=(I-λW2) )XAy(Bu 2/12/1 β−Ω=Ω=ν −−

( ) ( )

ρ−ρ−−ρ−+=

N

eeeeln

2

NWIlncteLLn L0

'L0

c

λ−λ−−λ−+=N

e)WI()'WI('eln

2

NWIlncteLLn c

• Other estimation methods:

• IV

• GMM

• Robust methods of estimation based in resampling techniques (bootstrap)

Robust to non-normality but not very efficient

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EMPIRICAL EXERCISE: CONVERGENCE EQUATIONEMPIRICAL EXERCISE: CONVERGENCE EQUATION

• Exploratory Spatial Analysis high degree of spatial dependence in European regions productivity per worker is not randomly distributed in the space

• GrowthGrowth equationequation: Beta : Beta convergenceconvergence equationequation

• Not considering any external effect in growth process

Motivated by:

• spatial autocorrelation in the factors affecting these variables

• potential interaction in the levels of technology of the regions due to technology spillovers

This process of spatial autocorrelation may cause a problem of spatial autocorrelation in the residuals of growth models applied to cross-sectional data

This process invalidates OLS Depending on the models, different conclusions

elny)e1(ayg T +−−= β−

Estimate equation growth by OLS, obtaining the statistics for spatial dependence

Spatial Specification Strategy ((underunder thethe assumptionassumption ofof non non theoreticaltheoreticalmodelmodel includingincluding a a spatialspatial dependencedependence processprocess withwith economiceconomic justificationjustification):):

Non Non rejectionrejection ofof thethe nullnull::

We accept the traditionalequation, concluding there isno interdependence among

EU regions

Technology and productivityin one region only depend onthe factors in this region

RejectionRejection ofof thethe nullnull::

1) Moran’s I and LM-ERR are significant:• Spatial error model (ML)• λ measures the intensity of the interdependence among residuals• Consequence of the existence of dependence between some ofthe variables not included in the model or due to the existence ofmeasurement errors due to lack of correspondence between theadministrative and natural borders• Test the consistence of this model with the COMFAC test

2) Moran’s I and LM-LAG are significant:• Spatial Lag Model (ML)• γ measures the intensity of the interdependence between the endogenous in one location and in its neighbours• Introduce a spatial lag of the explanatory variable (Wlny) andanalize its significance with a LR test.

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Estimation of the growth equation. 108 EU regions

• Estimated rate of convergence slightly over an annual 2%

• R2 =0.60, little reliability because of a spatial dependence process

• Moran’s I and LM reject the null, LM-ERR > LM-LAG

lny

β implícita

-0.3652(0.029)

0.0267

R2lnL

0.600261.62

I-MoranLM-ERRLM-EL LM-LAGLM-LE

7.42*48.85*17.26*31.63*0.05

Data set : europa Dependent Variable : TASA Number of Observ ations: 108 Mean dependent var : 0.0803599 Number of Variab les : 2 S.D. dependent var : 0.216294 Degrees of Freed om : 106 R-squared : 0.600175 F- statistic : 159.116 Adjusted R-squared : 0.596403 Prob(F-statistic ) :7.86316e- 023 Sum squared residual: 2.02014 Log likeli hood : 61.6186 Sigma-square : 0.0190579 Akaike info crit erion : - 119.237 S.E. of regression : 0.138051 Schwarz criterio n : - 113.873 Sigma-square ML : 0.018705 S.E of regression ML: 0.136766 --------------------------------------------------- ------------------- Variable Coefficient Std.Error t- Statistic Probability --------------------------------------------------- ------------------ CONSTANT 1.681615 0.1276347 13. 17522 0.0000000 LPW75 -0.3652242 0.02895361 -12. 61412 0.0000000 --------------------------------------------------- ------------------- REGRESSION DIAGNOSTICS MULTICOLLINEARITY CONDITION NUMBER 19.16424 TEST ON NORMALITY OF ERRORS TEST DF VALUE PROB Jarque-Bera 2 6.883302 0.0320118 DIAGNOSTICS FOR HETEROSKEDASTICITY RANDOM COEFFICIENTS TEST DF VALUE PROB Breusch-Pagan test 1 0.1083306 0.7420528 Koenker-Bassett test 1 0.08396926 0.7719888 SPECIFICATION ROBUST TEST TEST DF VALUE PROB White 2 5.107975 0.0777709 DIAGNOSTICS FOR SPATIAL DEPENDENCE FOR WEIGHT MATRIX : europe.GAL (row-standardized w eights) TEST MI/DF VALUE PROB Moran's I (error) 0.502301 7.4282274 0.0000000 Lagrange Multiplier (lag) 1 31.6383627 0.0000000 Robust LM (lag) 1 0.0455570 0.8309832 Lagrange Multiplier (error) 1 48.8572063 0.0000000 Robust LM (error) 1 17.2644005 0.0000325 Lagrange Multiplier (SARMA) 2 48.9027632 0.0000000

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Similar values are close in space

-0.417 - -0.219

-0.219 - -0.054

-0.054 - 0.043

0.043 - 0.142

0.142 - 0.363

S

N

EW

Spatial distribution of residuals (OLS)

Residual Map + Results from spatial dependence tests

Possibility of the existence of interdependences among regions

Re-estimate growth equation including a SAR error term

Spatial Error Model

Highly significant autorregressive parameter

LR test, appropriate model

Lny

β implícita

-0.308

(0.046)

0.025

λ

2R

lnL

0.8426

(0.040)

0.776

86.86

LR-ERR 50.49*

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REGRESSION SUMMARY OF OUTPUT: SPATIAL ERROR MODEL - MAXIMUM LIKELIHOOD ESTIMATION Data set : europa Spatial Weight : europe.GAL Dependent Variable : TASA Number of Observations: 108 Mean dependent var : 0.080360 Number of Variab les : 2 S.D. dependent var : 0.216294 Degree of Freedo m : 106 Lag coeff. (Lambda) : 0.842671 R-squared : 0.776029 R-squared (BUSE) : - Sq. Correlation : - Log likelihood : 86.867871 Sigma-square : 0.010478 Akaike info crit erion : -169.736 S.E of regression : 0.102362 Schwarz criterio n : -164.371479 --------------------------------------------------- -------------------- Variable Coefficient Std.Error z-value Probability --------------------------------------------------- -------------------- CONSTANT 1.484249 0.1932838 7.67 9119 0.0000000 LPW75 -0.3081088 0.04659497 -6.61 2491 0.0000000 LAMBDA 0.8426711 0.0403697 20.8 7385 0.0000000 --------------------------------------------------- -------------------- REGRESSION DIAGNOSTICS DIAGNOSTICS FOR HETEROSKEDASTICITY RANDOM COEFFICIENTS TEST DF VALUE PROB Breusch-Pagan test 1 12 .68615 0.0003684 DIAGNOSTICS FOR SPATIAL DEPENDENCE SPATIAL ERROR DEPENDENCE FOR WEIGHT MATRIX : europe.GAL TEST DF VALUE PROB Likelihood Ratio Test 1 50 .49845 0.0000000 ========================= END OF REPORT =========== ===================

Estimate the spatial lag model, to confirm our election

Highly significant

Appropriate model

Higher for the SEM

Lny

β implícita

γ

-0.2183

(0.034)

0.0120

0.5354

(0.070)

2R

lnL

0.713

77.41

LR-LAG 31.58*

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REGRESSION SUMMARY OF OUTPUT: SPATIAL LAG MODEL - MAXIMUM LIKELIHOOD ESTIMATION Data set : europa Spatial Weight : europe.GAL Dependent Variable : TASA Number of Observations: 108 Mean dependent var : 0.0803599 Number of Variab les : 3 S.D. dependent var : 0.216294 Degrees of Freed om : 105 Lag coeff. (Rho) : 0.535429 R-squared : 0.713377 Log likelihood : 77.4121 Sq. Correlation : - Akaike info crit erion : -148.824 Sigma-square : 0.0134091 Schwarz criterio n : -140.778 S.E of regression : 0.115798 --------------------------------------------------- -------------------- Variable Coefficient Std.Error z-value Probability --------------------------------------------------- -------------------- W_TASA 0.5354294 0.07074347 7.5 68605 0.0000000 CONSTANT 1.005509 0.1540436 6.5 27433 0.0000000 LPW75 -0.2183492 0.03410261 -6.4 02713 0.0000000 --------------------------------------------------- -------------------- REGRESSION DIAGNOSTICS DIAGNOSTICS FOR HETEROSKEDASTICITY RANDOM COEFFICIENTS TEST DF VALUE PROB Breusch-Pagan test 1 5. 465147 0.0193994 DIAGNOSTICS FOR SPATIAL DEPENDENCE SPATIAL LAG DEPENDENCE FOR WEIGHT MATRIX : europe.GAL TEST DF VALUE PROB Likelihood Ratio Test 1 31 .58684 0.0000000

WEAK POINTS OF SPATIAL ECONOMETRICSWEAK POINTS OF SPATIAL ECONOMETRICS

• ML ESTIMATION:

Some problems due to the necessity of using numerical methods to solve the non-linear optimization

• Development of new methods of estimation

High computational cost when N is high (determinants, inverses, ... of NxN matrices)

•Using techniques for sparse matrices (application of algorithms)

• Space-Time Models:

Simply adapting testing and estimation methods from cross-section dimension presents the disadvantage of not considering the nature of panel data.