introduction to applied spatial econometrics attila varga dimetic pécs, july 3, 2009
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Introduction to Applied Spatial Econometrics
Attila Varga
DIMETIC Pécs, July 3, 2009
Prerequisites
• Basic statistics (statistical testing)
• Basic econometrics (Ordinary Least Squares and Maximum Likelihood estimations, autocorrelation)
EU Patent applications 2002
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
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)
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
The nature of spatial data
• Data representation: time series („time line”) vs. spatial data (map)
• Spatial effects:
spatial heterogeneity
spatial dependence
Spatial heterogeneity
• Structural instability in the forms of:
– Non-constant error variances (spatial heteroscedasticity)
– Non-constant coefficients (variable coefficients, spatial regimes)
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)
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
EU Patent applications 2002
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
Spatial dependence (spatial autocorrelation/spatial association)
• Two main reasons:
– Measurement error (data aggregation)– Spatial interaction between spatial units
Modelling space
• Spatial heterogeneity: conventional non-spatial models (random coefficients, error compontent models etc.) are suitable
• Spatial dependence: need for a non-convential approach
Modelling space
• Spatial dependence modelling requires an appropriate representation of spatial arrangement
• Solution: relative spatial positions are represented by spatial weights matrices (W)
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
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
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 =
Modelling space
• Row-standardization:
• Row-standardized spatial weights matrices:
- easier interpretation of results (averageing of values)
- ML estimation (computation)
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)
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)
Global spatial association
• Basic principle behind all global measures:
- The Gamma index
= i,j wij cij
– Neighborhood patterns and value similarity patterns compared
Global spatial association
• Significance of global clustering: test statistic compared with values under H0 of no spatial autocorrelation
- normality assumption
- permutation approach
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)
Moran scatterplot
Local indicators of spatial association (LISA)
B. The Local Moran statistic
Ii = zijwijzj
– significance tests: randomization approach
Spatial Econometrics
• The spatial lag model
• The spatial error model
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
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.
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
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
The Spatial Lag model
• IV estimation (2SLS)
– Suggested instruments: spatially lagged exogenous variables
The Spatial Error model
y = x +
with
= W + ,
where is the coefficient of spatially lagged autoregressive errors, W. Errors in are independently distributed.
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
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)
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)
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)
Spatial econometrics: New developments
• Estimation: GMM
• Spatial panel models
• Spatial Probit, Logit, Tobit
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
Software options
• GEODA – easiest to access and use
• SpaceStat
• R
• Matlab routines
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