Econometric theory and estimation methods of autoregressive spatial models

WORKSHOP3rd Regional Science Association of the Americas (RSAMERICAS) and the 5th SOCHER conference

Coro ChascoUniversidad Autónoma de Madrid

coro.chasco@uam.es

University of Tarapacá (Arica, Chile), Sept. 25th 2013

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Contents

1. Introduction to spatial econometrics 2. Spatial autocorrelation 3. Introduction to spatial dependence

models

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Main references

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CHASCO, C. (2003), Econometría espacial aplicada a la predicción-extrapolaciónde datos microterritoriales, Comunidad de Madrid, Chapters 1, 2, 3 & 4.

https://www.researchgate.net/profile/Coro_Chasco/?ev=hdr_xprf

CHASCO, C. & G. FERNÁNDEZ-AVILÉS (2009), Análisis de datos espacio-temporales para la economía y el geomarketing, NetBiblo, Chapters 1, 2, 3 & 4.

https://www.researchgate.net/profile/Coro_Chasco/?ev=hdr_xprf

Theoretical contents

Introduction to GeoDa

ANSELIN, L. (1988), Spatial econometrics: Methods and models, Kluwer Academic Publishers

http://books.google.co.uk/books/about/Spatial_Econometrics_Methods_and_Models.html?id=3dPIXClv4YYC&redir_esc=y

ANSELIN, L. (2005): Exploring spatial data with GeoDaTM: a workbook. Spatial Analysis, Laboratory y Center for Spatially Integrated Social Science, Urbana-

Champaign. https://geodacenter.asu.edu/og_tutorials

Theoretical contents

Introduction to GeoDa

IN ENGLISH:

IN SPANISH:

LESAGE, J. P. & R. K. PACE (2009), Introduction to Spatial Econometrics, Boca Raton, Taylor & Francis.

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1: Introduction to SpatialEconometrics

Fundamentals of spatial econometrics Spatial econometrics history Spatial data nature

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1.1. Fundamentals of spatial econometrics Subfield of classical econometrics Spatial interaction = autocorrelation Spatial heterogeneity: 1) heteroskedasticity and

2) spatial instability = discrete (regimes), continuous

Spatial effects: autocorrelation & heterogeneity Spatial effects do affect regression models Spatial econometrics spatial statistics

Regional & urban economics

Physical phenomena: biology & geology

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. ANSELIN, L. (2010), Thirty years of spatial econometrics, Papers in Regional Science 89-1, pp.3-25.

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1.2. Spatial econometrics history (I)

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0. SOME SPORADIC STATEMENTS A CENTURY AGO:

Stephan (1934): “Data of geographic units are tied together, likebunches of grapes, not separate, like balls in an urn. (…)

In dealing with social data, we know that by virtue of their verysocial character, persons, groups and their characteristics areinterrelated and not independent (…)

Sampling error formulas may yet be developed which are applicableto these data, but until then the older formulas must be used withgreat caution. Likewise, other statistical measures must be carefullyscrutinized when applied to these data…”

Student (Biometrika, 1914): “The problems arising fromcorrelation due to successive positions in space are exactly similar tothose due to successive occurrence in time, but as they are to someextent complicated by the second dimension, it is perhaps simpler toconsider correlation du to time.”

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1.2. Spatial econometrics history (II) Preconditions for growth: mid-70s until late 80s.The origin comes from 2 sources: 1) quantitative revolution in geography (e.g. Tobler, 1970) and 2) incorporation of spatial effects in regional and urban science (e.g. Arora and Brown 1977).

“Official” point of departure is 1979, with Paelinck et Klaassen, Spatial Econometrics.

Take off: 90s until 2006.Relevant advances: 1) « mainstream » econometrics shows interest in spatial econometrics; 2) development of the asymptotic theory and small sample propertiesof some methods; 3) generalization to panel data models and discrete variable models; 3) new estimation methods (IV, GMM, bayesian); 4) focus in spatial heterogeneity adopting more flexible models (e.g. semi-parametric methods).

Maturity: from 20062006: foundation of the World Spatial Econometrics Association in Rome.New spatial model specifications; development of spatial panel models, models for spatial latent variables and models for flows.

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1.2. Spatial econometrics history (III)Why spatial econometrics has become a prominent topic in the recent scientific literature?

From a theoretical point of view:There is a need to better understand the fundamental processes behind the complex dynamics that result in the existence of spatial and space-time interactions: not only having evidence of, but how and why these interactions occurred.

From a practical point of view:. Increasing availability of spatial data: systems of sensors (even in social science) provide ever larger streams of extremely fine grained data (on a time, geographical and individual scale)=spatial data-mining. The usual paradigms (sample-population, spatial stochastic process) are insufficient to meaningfully manage such massive data sets.. The development of GIS and software packages (SpaceStat, Geoda, PySpace, libraries for Matlab, R, Stata…).

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. ARBIA, G. (2011), A lustrum of SEA: Recent research trends following the creation of the Spatial Econometrics Association (2007–

2011), Spatial Economic Analysis 6(4), 377-395.

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1.3. Spatial data nature Time: continous and unidimensional

Past (t-1)

Future (t+1)

Space: continuous and bidimensional

i

North

South

EastWest

NortheastNorthwest

SoutheastSouthwest

Present (t)

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FISCHER, M. (2006), Spatial Analysis in GeoComputation, Springer, Berlin Heidelberg; Chapter 2.

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1.3. Spatial data nature (II)

Year

Year

Time series:Trim.

Month

Country Region

Province

Spatial data:

FRECUENCIES

SCALES

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1.3. Spatial data nature (III)

Year 0

B.C.

A.C.

2 spatial references:

Axis X:

Longitude

Axis Y: Latitude

Spain: coordinates(-300,4200) Equator:

latitude 0

Greenwich: longitude 0

(0,0)

1 temporal reference:

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1.3. Spatial data nature (IV)

FUENTE DEL BERROFUENTE DEL BERROFUENTE DEL BERROFUENTE DEL BERROFUENTE DEL BERROFUENTE DEL BERROFUENTE DEL BERROFUENTE DEL BERROFUENTE DEL BERROFUENTE DEL BERROFUENTE DEL BERROFUENTE DEL BERROFUENTE DEL BERROFUENTE DEL BERROFUENTE DEL BERROFUENTE DEL BERROFUENTE DEL BERROFUENTE DEL BERROFUENTE DEL BERROFUENTE DEL BERROFUENTE DEL BERROFUENTE DEL BERROFUENTE DEL BERROFUENTE DEL BERROFUENTE DEL BERROFUENTE DEL BERROFUENTE DEL BERROFUENTE DEL BERROFUENTE DEL BERROFUENTE DEL BERROFUENTE DEL BERROFUENTE DEL BERROFUENTE DEL BERROFUENTE DEL BERROFUENTE DEL BERROFUENTE DEL BERROFUENTE DEL BERROFUENTE DEL BERROFUENTE DEL BERROFUENTE DEL BERROFUENTE DEL BERROFUENTE DEL BERROFUENTE DEL BERROFUENTE DEL BERROFUENTE DEL BERROFUENTE DEL BERROFUENTE DEL BERROFUENTE DEL BERROFUENTE DEL BERROFUENTE DEL BERROGOYAGOYAGOYAGOYAGOYAGOYAGOYAGOYAGOYAGOYAGOYAGOYAGOYAGOYAGOYAGOYAGOYAGOYAGOYAGOYAGOYAGOYAGOYAGOYAGOYAGOYAGOYAGOYAGOYAGOYAGOYAGOYAGOYAGOYAGOYAGOYAGOYAGOYAGOYAGOYAGOYAGOYAGOYAGOYAGOYAGOYAGOYAGOYAGOYAGOYA

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A

M-30

Type of time series data:

Type of spatial data:

Polygons (e.g. districts)

Points (e.g. outlets)

Lines (e.g. roads)

Points (e.g.months)

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2. Spatialautocorrelation

Existence of a functional relationship between what happens at one point in space and what happens elsewhere.

Two causes: measurements errors and spatial interactions.

Space as explicative of human behaviour: distance matters.

What is observed at one point is determined by what happens elsewhere in the system.

ANSELIN, L. (1988), “Spatial Econometrics: Methods and Models”.

Kluwer Academic Publishers; pp. 11-31

Rta. disp. por hab. (1997)(miles ptas.)

1.400 a 1.8001.125 a 1.400

900 a 1.125

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2.1. Formal expression of spatialautocorrelation

Spatial autocorrelation: functional relationship between what happens at one point in space and what happens elsewhere.

Session 2

This expression is not very useful since it would result in an unidentifiable system: (N2 – N)parameters and N observations.

It is necessary to impose some restrictions on this function.

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2.2. Spatial weight matrix The definition of a spatial lag

variable is largely arbitrary which other units have an

influence on the particular one?

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Spatial weight matrix allows to relate a variable at one point in space to observations for that variable in other spatial units in the system

2.3. Spatial weight matrix (II)

Lag operator Time series:

Space:

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=

Spatial lag operator: W

W is often standardised such that the row elements sum to one, as it facilitates the interpretation of the model coefficients

2.3. Spatial weight matrix (III)

W

W*

Row-standardised matrixes not always lead to meaningful economic interpretation, so we must think carefully about pros and cons.

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Common specifications of the spatial weight matrix:

1. Contiguity-based matrices: having a common border (binary 0-1)2. Distance-based matrices (Euclidean or Arc Distance).

a) Distance band weights: wij=1 if i,j are at a critical distance cut-off point; wij=0, otherwise. Symmetric

b) K-nearest neighbors weights: wij=1 if j is one of the kth i’s nearest neighbors; wij=0, otherwise. Non-symmetric.

c) Inverse distance weights: . Symmetric.3. Kernel matrices: fixed and adaptive.4. Distance by relative frontier longitude: . Non-

symmetric.5. Accessibility matrices: networks.6. Economic or technological-distance matrices (e.g. trade flows).

2.3. Spatial weight matrix (IV)

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21ij ijw d

( ) ( ) a bij ij ijw d b

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2.4. Global spatial autocorrelation: Moran’s I test

. GETIS, A. (2010), Spatial autocorrelation. In Fischer M. and A. Getis (eds.), Handbook of Applied Spatial Analysis. Sobtware Tools,

Methods and Applications, Springer, Berlin Heilderberg.

W*

Possitive aut.

Negativ aut.

Moran’I theoretical mean: E(I) =

N: sample size

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Permutation: randomisation with unknowndistribution function

1) A reference distribution for I is generated empirically.2) Randomly permuting observations & computing Moran’s for a set of n! new samples

3) E[I] & SD[I] are computed directly from the generated distribution of Moran’s Is

4) Significance of z(I): in a standard normal table.

2.4. Global spatial autocorrelationMoran’s I (II)

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2.4. Global spatial autocorrelationMoran’s I (III)

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3. Specification of spatial dependence modelsEmpirical applications with spatial models must coverthree steps:

Specification of the nature of spatial dependence: whichlocations interact (the global sample/local subsamples)?

Testing the presence of spatial dependence: which type of spatial dependence (substantive/residual)?

Estimation of the spatial dependence model: which estimation method (ML, spatial TSLS, GMM, semi-parametric, bayesian)?

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3. Specification of spatial dependence models (II)

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. ANSELIN, L. (2003), Spatial externalities, spatial multipliers, and spatial econometrics, International

Regional Science Review 26(2), pp. 153–166.

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3. Specification of spatial dependence models (III)

Basic spatial regression model:y = N + X +

Spatial lag model (SAR):y = Wy + N + X +

Spatial error model (SEM):y = N + X + u ; u = Wu +

Spatial Durbin model (SDM)y = Wy + N + X + WX +

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3. Specification of spatialdependence models (IV)

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3.1. Basic spatial model3.2. Spatial lag model (SAR)3.3. Spatial error model (SEM)3.4. Spatial Durbin model (SDM)

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3.1. Basic spatial process The general purpose of linear regression analysis is to find

out a (linear) relationship between a dependent variable and a set of explanatory variables.

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XyxyK

kikiki

1

1

Depending on the nature of spatial dependence, there are threespatially lagged variables that can be added to this model:

. Substantive: Wy, WX (spatial interactions in variables).

. Residual spatial dependence: W (measurement errors).

3.1. Basic spatial process (II)

OLS are BLUE if:

Assumptions about u:1. Mean zero2. Constant variance (homoskedasticity)3. No spatial autocorrelation4. Normality

Basic assumptions:1. Linearity2. Full rank (no multicollinearity)2. Good specification3. Nonstochastic regressors

OLS estimators: minimizes the sum of squared

residuals of the regression

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• Normality of the errors

S: asymmetry

K: kurtosis

Jarque-Bera is an asymptotic discrepancy test:

22 1 36 4

N kJB S k Consequences: Most hypothesis tests and a large number of regression diagnostics are based on the assumption of a normal error distribution: maximum-likelihood estimations, t-Student, F-tests, Lagrange Multiplier tests (Breusch-Pagan, spatial autocorrelation)!!!!!!

Solution: Logs or any other Box-Cox variable transformations, other estimation methods (IV, GMM), other tests (Koenker-Basset, Kelejian-Robinson).

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• Heteroskedasticity

Definition: Regression disturbance do not have a constant variance over all observations (i.e., is not homoskedastic).Causes in spatial data analysis:

1) When using data for irregular spatial units (with different area).

2) When there are systematic regional differences in the relationshipsyou model (i.e., spatial regimes).

3) When there is a continuous spatial drift in the parameters of the model (i.e., spatial expansion).Consequences:

1) A standard regression model that ignores this will be misspecified.

2) OLS estimates are unbiased, but they will no longer be most efficient.

3) Inference based on the usual t and F statistics will be misleading.

4) R2 measure of goodness-of-fit will be wrong (based on e’e, instead e’e-1e).

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• Spatial autocorrelation

Concept: Spatial autocorrelation, or more generally, spatial dependence, is the situation where the dependent variable or error term at each location is correlated with observations on the dependent variable or values for the error term at other locations. The general case is formally:

1) “y” in location “i” is correlated with “y” in location “j”:

W*2) “e” in location “i” is correlatedwith “i” in location “j”:

Spatial-lag

Spatial-error

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• Spatial autocorrelation (II)

Consequencies The consequences of ignoring spatial autocorrelation in a regression model, when it is in fact present, depend on the form for the alternative hypothesis, H1:

H0:==0 If H1:� � 0 (spatial-lag) If H1:� � 0 (spatial-error)

b: Biased and inefficient

Specification error: omitting a

significant explanatory variable

b: Unbiased but inefficient

t-Student, F tests are misleading

R2 invalid

y Wy X u y X uu Wu

y X uu W

Sustantive spatial dependence Error spatial dependenceNo spatial dependence

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• Spatial autocorrelation (III)

There are 7 spatial autocorrelation tests: 4 (error) + 2 (lag) + 1 (SARMA)

1) Moran’s I:

I. H1:� � 0 (spatial-error)

2

1 1 1

N N N

ij i j ii j i

e WeI w e e ee e

Inference is based on a standardized z-value ~ N(0,1) (asymptotically).

Require normality for the error terms

The theoretical E(I), SD(I) are more complex.

The interpretation of the statistic is the same as for the general case.

It is by far the most familiar test, it is fairly unreliable:

- This statistic picks up a range of misspecification errors: from non-normality and heteroskedasticity to spatial lag dependence.

- It does not provide any guidance for H1 in terms of which of the substantive or errordependence is the most likely alternative.

u = +W

. CLIFF, A. y J. ORD (1981), “Spatial processes, models and applications”. London: Pion

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• Spatial autocorrelation (IV)

2) Burridge’s Lagrange Multiplier test (1980)

I. H1: 0 (spatial-error)

Asymptotic test, which follows a �2 distribution with 1 degree offreedom. It requires normality in the errors. The test is the same for both H1 of SAR and SMA in the errors.

u = +W2

2

2'

'ERR

ML

e We

LMtr W W W

3) Anselin & Florax’s Lagrange Multipliertest on errors, robust to ignored spatial lag (1995)

Asymptotic test � �2, 1 d.f. Requires normality in the errors

. ANSELIN, L & R FLORAX (1995), ““Small sample properties of tests for spatial dependence in regression models: Some further results”. In Anselin A and R. Florax, New directions in spatial econometrics, pp. 21–74. Berlin: Springer-Verlag.

21

1 112 2

21 1

' 'ML ML

e W e e W yT RJLM EL

T T RJ

11 1 1

1 2

'

ML

W Xb M W XbRJ T

211

'11 WWWtrT 2

ML e e N

33

• Spatial autocorrelation (IV)

4) Kelejian-Robinson test (1992)

I. H1: 0 (spatial-error)

Asymptotic test, which follows a �2 distribution with P degrees of freedom. It does not require normality for the error terms. It also is applicable to both linear and nonlinear regressions Only for large sample test (may not have much power for small data sets). It’s not known its performance with linearity and normality in the errors

u = +W

. KELEJIAN H & D ROBINSON (1992), “Spatial autocorrelation: a new computationally simple test with an application to per capita county policy

expenditures”. Regional Science and Urban Economics 22; pp. 317-31.

KR is obtained from an auxiliary regression of cross products of residuals and cross products of the explanatory variables (collected in a matrix Z with P columns). The cross products are for all pairs of observations for which a nonzero correlation is postulated (but each pair is only entered once), for a total of hN pairs.

� is the coefficient vector in this auxiliary regression.� is the resulting residual vector.

nh

ZZKR

'''

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• Spatial autocorrelation (V)

5) Anselin’s Lagrange Multipliertest for spatial lag (1988)

Asymptotic test, which follows a �2 distribution with 1 degree of freedom. It is only valid under the assumption of normality in the error terms. Only for large sample test (may not have much power for small data sets).

II. H1: 0 (spatial lag)

. ANSELIN, L (1988), Lagrange Multiplier test diagnostics for spatial dependence and spatial heterogeneity, Geographical Analysis 20, 1-17.

2

2

2

2

'

''

LAGML

ML

e Wy

LMWXb MWXb

tr W W W

M = I – X[X’X]-1 X’

6) Anselin & Florax’s Lagrange Multipliertest for spatial lags, robust to ignored spatial error (1995)

2

1 12 2

1

' '

ML ML

e W y e W e

LM LERJ T

Asymptotic test � �2, 1 d.f. Requires normality in the errors

11 1 1

1 2

'

ML

W Xb M W XbRJ T

211

'11 WWWtrT 2

ML e e N

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• Spatial autocorrelation (VI)

7) Anselin’s Lagrange MultiplierSARMA test

Asymptotic test, which follows a 2 distribution with 2 degrees of freedom. Only valid in linear models under the assumption of normality in the errors

. ANSELIN, L (1988), Lagrange Multiplier test diagnostics for spatial dependence and spatial heterogeneity, Geographical Analysis 20, 1-17.

III. H1: 0 ; 0 (SARMA)

2 2

2 2 2

1 1

' ' 'ML ML ML

e Wy e We e We

SARMARJ T T

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• Spatial autocorrelation (VII)

Consider 7 spatial autocorrelation tests: 4 (error) + 2 (lag) + 1 (SARMA)

1) Moran’s I2) LM-ERR

4) Kelejian-Robinson3) Robust LM-ERR

5) LM-LAG6) Robust LM-LAG

7) Lagrange Multiplier

II. H1: 0 (spatial-lag)

III. H1: 0 ; 0 (SARMA)

I. H1: 0 (spatial-error)unreliable

SAR/SMA

omitted lag

non-norm.

omitted err.

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Spatial-lag model or mixed regressive spatial autoregressive model orsimultaneous spatial autoregressive model, SAR model.

2.2. Spatial lag model (SAR)

Spatial-lag model allows to asses:1. The degree of spatial dependence while controlling for the effect of these other variables.2. The significance of the other (non-spatial) variables, after the spatial dependence is controlled for.

: spatial autoregressive parameter Wy: spatial lag variable; when W is row-standardized: [-1,+1]

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XWyy

1 1 1̂ L L L L L L L L L Ly y y y y y y y u y y y u

1. Biased:

2. Inconsistent:

/1 1 If 0 0 LE y u y I W u E W I W u u

1ˆ L L LE E y y y u

1

1 0

: 0

L L LL L LN N

y y y uPLim y y y u PLimN N

If Q

Only if = 0

2.2.1. Estimation of the SAR modelOLS are not only inefficient but biassed: ignoringWy is an omitted variables problem.

Appropriate estimation methods for the SAR model:1. ML: based on an underlying normal distribution of the errors.2. Spatial TSLS, GMM, boostrap, bayesian.

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SAR(1) model:

Ly Wy u y u

(I – W)-1= I + W + 2W2 + . . .:

Multiplier effect (on the explanatory variables): the value of y in location i is affected by the explanatory variables of this location i and by these same variables in all others locations j = 1,…, N, j i through the inverse matrix (I – W)-1. This effect reduces with distance, since1 and wij1 (in most cases).

Diffusion efect (on the error term): a random shock affecting one localtion i propagates to all other locations in the sample through the spatial transformation (I – W)-1. This effect also reduces with distance.

If were known, could be estimated by OLS:

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Under invertibility conditions of the spatial multiplier [I - W], a reduced form is derived:

2.2.2. Properties of the SAR modela) Multiplier and diffusion effects

1 1 1( ) ( ) ( ) ( )y I W X I W E y X I W X Constraint: ρ and W must be the same for the spatial multiplier for both X and .

2 2 ... y X WX W X v

. ANSELIN, L. (2003), Spatial externalities, spatial multipliers, and spatial econometrics, International Regional Science Review 26(2), pp. 153–166.

y Wy X

GLOBAL spatial autocorrelation

The elements of the main diagonal of Cov(u) are not equal.

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2.2.2. Properties of the SAR model

b) Variance covariance matrix Full matrix: each location is correlated with the rest of the locations

(correlation reduces with distance)=global spatial autocorrelation

1 2 1 1( ) ( ) ( ') u I W E u u I W I W

c) Induced heteroskedasticity

2 1 1( ) ( ) ( ') yCov y I W I W( , )i jCov y y

XWyy

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2.2.2. Properties of the SAR model

d) Interpretation of the coefficients A change in location i in any given explanatory variable xk will affect

the value of y in location i itself (a direct impact) and potentially will affect all other locations indirectly (an indirect impact).

The interpretation of is straightforward (as a marginal effect). However, the cannot be interpreted as marginal effects:

. LESAGE, J.P. and R.K. PACE (2009), Introduction to spatial econometrics,

Chapman & Hall/CRC, Chapter 2.

1 1

k

N

N N k k N

S W

I W y X

y I W I x I W

y X Basic model: ˆ , ,      and      0,i ik

ik jk

y yi r j i

x x

SAR model:

(I – W)-1= I + W + 2W2 + . . .

1 ˆ ˆN N k N k

k

yI W I I

x

0,ik ij

jk

yS W j i

x

Since (I – W)-1= full matrix:

2 2 ... y X WX W X v XWyy

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2.2.2. Properties of the SAR modeld) Interpretation of the coefficients (II)

Direct impact:Includes the effect of changes in the value of an xk in location i on the values of the y variable in location i. For each location, there is a different direct effect, creating a kind of “interactive heterogeneity”.

Indirect impact:Includes the effect of “feedback loops” where observation i affects observation j and observation j also affects i, as well as longer paths, where the effect might go from observation i to j and to N, and back to i.

^ ^ ^

^ ^

diagonal

non-diagonal

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2.2.2. Properties of the SAR modeld) Interpretation of the coefficients (III)

Example of matrix Sk(W):

Principal diagonal: direct impact; non-diagonal: indirect impacts

^ ^

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2.2.2. Properties of the SAR modeld) Interpretation of the coefficients (IV)

Average direct impact:The impact of changes in the ith observation of xk (which we denote xik) on yi . Similar to typical regression coefficient interpretations that represent average response of the dependent to independent variables changes over the sample.

Average total impact “to” an observation:The sum across the ith row of Sr(W) = total impact on individual observation yi resulting from changing the kth explanatory variable by the same amount across all Nobservations: xK + δ.N (where δ is the scalar change). There are N of these sums given by the column vector cK = Sr(W)N.

ktr S W N

N k N k Nc N S W N

Average total impact “from” an observation:The sum down the jth column of Sr(W) would yield the total impact over all yi from changing the kth explanatory variable by an amount in the jth observation: xjk + δ. There are N of these sums given by the row vector rK = ’N Sr(W).

k N N k Nr N S W N

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2.2.2. Properties of the SAR modeld) Interpretation of the coefficients (V)

The “to” an observation (column ck): considers how changes in all observations influence a single observation j versus other observations.

The “from” an observation (row rk): relates how changes in a single observation j influences own observation j versus other observations.

1 ˆ ˆˆ ˆ1N k k N N k N N N k N kc N r N S W N I W N

Summary measure of total impacts:

kdirect

N k Ntotal

indirect total direct

M k tr S W N

M k S W N

M k M k M k

Summary measures of impacts - direct, indirect, total:

47

2.3. Spatial-Error Model (SEM) Spatial-error model or spatial

autoregressive error term modelThe spatial error model is a special case of a so called non-spherical error model. Thespatial dependence in the error term can takeon a number of different forms. In mostsoftware, only a spatial autoregressive processfor the error term can be estimated.

1

1....................

y X u

SAR u Wu

SMA u W

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Betas: are marginal effects

:

y X WuSince u y Xy X W y X

y Wy X WX

LOCAL: no multiplier effect of WX on y

1

y X uSAR u Wu Betas are marginal effects:

1) No multiplier (global) effects of WX on y.

2) Only a diffusion effect on the error term.

1( ) ( )y X I W u E y X X

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2.3. Spatial-Error Model (SEM), II Spatial-error model or spatial autoregressive error term model

OLS are unbiased but inefficient. ML ispreferred, though the estimation of the nuisance parameter can be biased.

For non-normal errors: spatial TSLS (spatial Durbin), GMM and bayesian.

Non-diagonal error variance matrix:

1

y X uSAR u Wu If were known, the could be

estimated by means of OLS in a model with spatially filtered variables:

For a known , this method is called GLS. In most cases,

must be estimated jointlywith . In spatial models, FGLS is not applicable.

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2.4. Spatial Durbin model (SDM)

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LeSage et Pace (2009): useful model in empiricalapplications. When there are omitting variables, which are spatially

autocorrelated, the SDM limits the bias effect.

It combines a SAR and SEM model: global and local spillovers.

y X uy X Wu y X W y X

u Wu

Constrained:

Unconstrained (SDM):

y Wy X WXy Wy X WX

. = -

SEM model:

2.4. Spatial Durbin model (SDM), II

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Since SDM is a SAR model, we can compute the direct, indirect and total impacts:

SAR

y Wy X WX

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3. Specification strategies. ELHORST, P. (2010) Applied spatial

econometrics: raising the bar, Spatial Economic Analysis, 5(1), 9-28.

SDM: produces unbiased coefficient estimates also if the true data-generation process is any of the

other spatial regression specifications, except for the Manski model.

LESAGE, J. P. & R. K. PACE (2009), Introduction to Spatial Econometrics, Boca

Raton, Taylor & Francis.

BIAS & INCONSISTENCY:relevant variable omission =

Wy & WX.

INEFFICIENCY: Wu omission

. BIVAND, R. (2012) After "Raising the Bar'':

applied maximum likelihood estimation of families of models in spatial econometrics, Estadística Española,

54(177), 71-88.

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. CLASSIC strategy:

Anselin, L. (1988), Spatial econometrics: Methods and models, Kluwer Academic Publishers.

.. HENDRY-BASED strategy:

Florax, R.J.G.M, H. Folmer and S.J. Rey (2003), Specificationsearches in spatial econometrics: the relevance of Hendry’s

methodology, Regional Science and Urban Economics 33, pp. 557–579.

. LESAGE & PACE-BASED strategy:

Elhorst, P. (2010) Applied spatial econometrics: raising the bar, Spatial Economic Analysis, 5(1), 9-28.

3: Spatial dependence models: Specification (III)

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3: Spatial dependence models: Specification (IV)

Elhorst’s specification strategy (model comparison):

1. OLS model: y = N + X + 2. Use the LM-tests to test H0(no-spatial-autocorr.) against (each/both):

H1(0: spatial lag=SAR): y = Wy + N + X + H1(0: spatial error=SEM): y = N + X + u ; u = Wu +

3. If the OLS model is rejected in favour of each/both SAR/SEM: the spatial Durbin model (SDM): y = Wy + N + X + WX + 4. For ML estimation of spatial models, an LR test can be used to test:

H1( 0) & H1( ) SDM: y = Wy + N + X + WX + H0( = 0) + LM H1(0) SAR: y = Wy + N + X + H0( = = comfac) + LM H1(0) SEM: y = N + X + u ; u = Wu + If no agreement LR-LM SDM: y = Wy + N + X + WX +

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