introduction to applied spatial econometrics attila varga dimetic pécs, july 3, 2009
TRANSCRIPT
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Introduction to Applied Spatial Econometrics
Attila Varga
DIMETIC Pécs, July 3, 2009
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Prerequisites
• Basic statistics (statistical testing)
• Basic econometrics (Ordinary Least Squares and Maximum Likelihood estimations, autocorrelation)
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EU Patent applications 2002
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Outline
• Introduction• The nature of spatial data• Modelling space• Exploratory spatial data analysis• Spatial Econometrics: the Spatial Lag and
Spatial Error models• Specification diagnostics• New developments in Spatial Econometrics• Software options
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Spatial Econometrics
„A collection of techniques that deal with the peculiarities caused by space in the statistical analysis of regional science models”
Luc Anselin (1988)
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Increasing attention towards Spatial Econometrics in Economics
• Growing interest in agglomeration economies/spillovers – (Geographical Economics)
• Diffusion of GIS technology and increased availability of geo-coded data
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The nature of spatial data
• Data representation: time series („time line”) vs. spatial data (map)
• Spatial effects:
spatial heterogeneity
spatial dependence
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Spatial heterogeneity
• Structural instability in the forms of:
– Non-constant error variances (spatial heteroscedasticity)
– Non-constant coefficients (variable coefficients, spatial regimes)
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Spatial dependence (spatial autocorrelation/spatial association)
• In spatial datasets „dependence is present in all directions and becomes weaker as data locations become more and more dispersed” (Cressie, 1993)
• Tobler’s ‘First Law of Geography’: „Everything is related to everything else, but near things are more related than distant things.” (Tobler, 1979)
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Spatial dependence (spatial autocorrelation/spatial association)
• Positive spatial autocorrelation: high or low values of a variable cluster in space
• Negative spatial autocorrelation: locations are surrounded by neighbors with very dissimilar values of the same variable
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EU Patent applications 2002
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Spatial dependence (spatial autocorrelation/spatial association)
• Dependence in time and dependence in space:– Time: one-directional between two
observations– Space: two-directional among several
observations
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Spatial dependence (spatial autocorrelation/spatial association)
• Two main reasons:
– Measurement error (data aggregation)– Spatial interaction between spatial units
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Modelling space
• Spatial heterogeneity: conventional non-spatial models (random coefficients, error compontent models etc.) are suitable
• Spatial dependence: need for a non-convential approach
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Modelling space
• Spatial dependence modelling requires an appropriate representation of spatial arrangement
• Solution: relative spatial positions are represented by spatial weights matrices (W)
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Modelling space
1. Binary contiguity weights matrices- spatial units as neighbors in different orders (first, second etc. neighborhood classes)
- neighbors:- having a common border, or- being situated within a given distance band
2. Inverse distance weights matrices
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Modelling space
• Binary contiguity matrices (rook, queen)
• wi,j = 1 if i and j are neighbors, 0 otherwise
• Neighborhood classes (first, second, etc)
W =
0100
1011
0101
0110
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Modelling space
• Inverse distance weights matrices
0)(
1
)(
1
)(
1)(
10
)(
1
)(
1)(
1
)(
10
)(
1)(
1
)(
1
)(
10
2
3,4
2
2,4
2
1,4
2
4,3
2
2,3
2
1,3
2
4,2
2
3,2
2
1,2
2
4,1
2
3,1
2
2,1
ddd
ddd
ddd
ddd
W =
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Modelling space
• Row-standardization:
• Row-standardized spatial weights matrices:
- easier interpretation of results (averageing of values)
- ML estimation (computation)
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Modelling space
• The spatial lag operator: Wy– is a spatially lagged value of the variable y– In case of a row-standardized W, Wy is the
average value of the variable: • in the neighborhood (contiguity weights)• in the whole sample with the weight decreasing
with increasing distance (inverse distance weights)
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Exploratory spatial data analysis
• Measuring global spatial association:
– The Moran’s I statistic:
a) I = N/S0 [i,j wij (xi -)(xj - ) / i(xi -)2]
normalizing factor: S0 =i,j wij
(w is not row standardized)
b) I* = i,j wij (xi -)(xj - ) / i(xi -)2
(w is row standardized)
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Global spatial association
• Basic principle behind all global measures:
- The Gamma index
= i,j wij cij
– Neighborhood patterns and value similarity patterns compared
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Global spatial association
• Significance of global clustering: test statistic compared with values under H0 of no spatial autocorrelation
- normality assumption
- permutation approach
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Local indicatiors of spatial association (LISA)
A. The Moran scatterplotidea: Moran’s I is a regression coefficient of a regression of Wz on z when w is row standardized:
I=z’Wz/z’z (where z is the variable in deviations from the
mean)- regression line: general pattern- points on the scatterplot: local tendencies- outliers: extreme to the central tendency (2 sigma rule)- leverage points: large influence on the central tendency (2 sigma rule)
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Moran scatterplot
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Local indicators of spatial association (LISA)
B. The Local Moran statistic
Ii = zijwijzj
– significance tests: randomization approach
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Spatial Econometrics
• The spatial lag model
• The spatial error model
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The spatial lag model
• Lagged values in time: yt-k
• Lagged values in space: problem (multi-oriented, two directional dependence)– Serious loss of degrees of freedom
• Solution: the spatial lag operator, Wy
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The spatial lag model
The general expression for the spatial lag model is
y = Wy + x +,
where y is an N by 1 vector of dependent observations, Wy is an N by 1 vector of lagged
dependent observations, is a spatial autoregressive parameter, x is an N by K matrix of
exogenous explanatory variables, is a K by 1 vector of respective coefficients, and is an N by
1 vector of independent disturbance terms.
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The spatial lag model
• Estimation
– Problem: endogeneity of wy (correlated with the error term)
– OLS is biased and inconsistent– Maximum Likelihood (ML) – Instrumental Variables (IV) estimation
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The spatial lag model
• ML estimation: The Log-Likelihood function
L = ln I - W - N/2 ln (2) - N/2 ln (2) - (y - Wy - x)’( y - Wy - x)/2 2
Maximizing the log likelihood with respect to , , and 2 gives the values of parameters that provide the highest likelihood of the joint occurrence of the sample of dependent variables
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The Spatial Lag model
• IV estimation (2SLS)
– Suggested instruments: spatially lagged exogenous variables
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The Spatial Error model
y = x +
with
= W + ,
where is the coefficient of spatially lagged autoregressive errors, W. Errors in are independently distributed.
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The Spatial Error model
• OLS: unbiased but inefficient
• ML estimation
The likelihood function for the regression with spatially autocorrelated error term is
L = ln I -W - N/2 ln (2) - N/2 ln (2) - (y - x)’(I - W)’(y - x)(I - W)/2 2
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Specification tests
Test
Formulation Distribution Source
MORAN
e’We/e’e
N(0,1)
Cliff and Ord (1981)
LM-ERR
(e’We/s2)2/T
2 (1)
Burridge(1980)
LM-ERRLAG
(e’We/s2)2 / [tr(W’W + W2) - tr(W’W + W2)A-1var()]
2 (1)
Anselin (1988/B)
LM-LAG
(e’Wy/s2) 2/ (RJ
2 (1)
Anselin (1988/B)
LM-LAGERR
(e'B'BWy) 2/(H - HVar()H'
2 (1)
Anselin et al. (1996)
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Steps in estimation
• Estimate OLS
• Study the LM Error and LM Lag statistics with ideally more than one spatial weights matrices
• The most significant statistic guides you to the right model
• Run the right model (S-Err or S-Lag)
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Table 6.2. OLS Regression Results for Log (Innovations) at the MSA Level (N=125, 1982) Model Jaffe ML -Spatial Lag Spatial Extended Jaffe Constant W_Log(INN) Log(RD) Log(RD75) Log(URD) Log(URD50) Log(LQ) Log(BUS) Log(LARGE) RANK
-1.045 (0.146)
0.540 (0.054)
0.112 (0.036)
-1.098 (0.143) 0.125
(0.055) 0.515
(0.053)
0.125 (0.035)
-1.134 (0.172)
0.504 (0.055) 0.001
(0.041) 0.132
(0.036) 0.037
(0.018)
-1.407 (0.212)
0.277 (0.057) -0.027 (0.037) 0.093
(0.034) 0.032
(0.015) 0.652
(0.163) 0.332
(0.057) -0.337 (0.094) 0.202
(0.101) R2 - adj Log-Likelihood
0.599 -65.336
-62.708
0.611 -62.402
0.725 -36.683
Kiefer-Salmon White B-P LM-Err D50 D75 IDIS2 LM-Lag D50 D75 IDIS2 LR-Lag D50
1.899
1.183
1.465 2.688 1.691
5.620 2.968 2.039
0.243
0.000 0.737 0.008
5.256
9.024
0.936 2.178 1.102
1.026 1.485 0.659
37.847
0.102 0.060 0.045
0.450 1.593 0.625
Notes: Estimated standard errors are in parentheses; critical values for the White statistic with respectively 5, 20, and 35 degrees of freedom are 11.07, 31.41, and 49.52 (p=0.05); critical value for the Kiefer-Salmon test on normality and the Breusch-Pagan (B-P) test for heteroskedasticity is 5.99 (p=0.05); critical values for LM-Err, LM-Lag and LR-Lag statistics are 3.84 (p=0.05) and 2.71 (p=0.10); spatial weights matrices are row-standardized: D50 is distance-based contiguity for 50 miles; D75 is distance-based contiguity for 75 miles; and IDIS2 is inverse distance squared.
Example: Varga (1998)
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Spatial econometrics: New developments
• Estimation: GMM
• Spatial panel models
• Spatial Probit, Logit, Tobit
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Study materials
• Introductory:– Anselin: Spacestat tutorial (included in the
course material)– Anselin: Geoda user’s guide (included in the
course material)
• Advanced:– Anselin: Spatial Econometrics, Kluwer 1988
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Software options
• GEODA – easiest to access and use
• SpaceStat
• R
• Matlab routines