空间计量经济学课件(虞义华,中国人民大学)

7
Modelling Space
Spatial dependence modelling requires an appropriate representation of spatial arrangement Solution: relative spatial positions are represented by spatial weights matrices (W)
Hence, to estimate the model with a spatially lagged dependent variable, we apply the maximum likelihood estimation (MLE) method The log-likelihood function of the model is,
Spatial Weight Matrix(es)
Most common is using binary connectivity based on contiguity: wij = 1 if regions i and j are contiguous, wij = 0 otherwise
Spatial Weight Matrix(es)
w ij pij d ij
2
/(
pi 1 pi 2 ... pij d ij
2
)
Travel time of freight vehicles between the centers of regions (Tiiu Paas, Friso Schlitte ‘Spatial effects of regional income disparities and growth in the EU countries and regions ’). The matrix W is calculated as follows:
Results might be sensitive to the functional form of the weight matrix, But we do not have a priori information about nature of spatial dependence. The choice of matrix must be appropriate for the research problem in question Economic distance (Greenbaum , 2002; Case, Rosen and Hines, 1993) Economic potential (p represents economic importance, typically population or some other proxy for size) (Biles, 2003)
L = lnI - 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
wij w ji 1 1 2 (timeij time ji )
Trade flows (Aten 1997)
Spatial Model 1 (Spatial Lag Model, SAR)
y = αιn + λWy + Xβ + є, є ~ N(0, σ2In), or yi= α + λΣjwijyj + Xiβ + єi where W is the non-stochastic n × n spatial weights matrix in which the element mij is equal to 1/dij with dij being the distance between two cities i and j (i ≠ j) (Dubin, 1988; Biles, 2003; Hernandez, 2003; Garrett et al., 2007)
Various Spatial Models
Spatial Dependence Test --Moran (1950) Test
Moran’s (1950) I statistics is originally developed to examine the spatial correlation in random variables (Moran, 1950), which was later adapted to regression residuals (Cliff and Ord, 1981) Measure covariance in errors between joining districts relative to the variance in errors in a given district. The idea is similar to autocorrelation for time series data. The Moran’s I statistic is given by, I=ê’Wê/ ê’ê where êis the vector of regression residuals from the OLS model y = αιn + Xβ + є, є ~ N(0, σ2In). Under the null hypothesis of no spatial autocorrelation, the expected value of I is given by: E ( I ) [1/ (n 1)] ( If I > the expected value, …positive autocorrelation)
In making such specification, we assume that as the distance between cities i and j increases (decreases), Wij decreases (increases), which poses less (more) spatial weight to the city pair (i, j)
Spatial Model 2 (Spatial Error Model, SEM)
y = αιn + Xβ + μ, μ = ρWμ + є, є ~ N(0, σ2In)
Despite OLS estimation in the presence of spatial correlation among model disturbances yields unbiased coefficient estimator, estimates of standard errors will be inconsistent, which implies that t-statistics and F-statistics will be incorrect, and inferences based on them will be misleading Therefore, the spatial error model is also estimated using MLE. The log-likelihood function has the form,
How to estimate? --MLE
As opposed to the time series case (where GLS is appropriate), spatial GLS is also biased and inconsistent. For
formal proof see Anselin and Bera spatially lagged dependence, traditional OLS does not yield unbiased and consistent estimates as the autoregressive component of the model is correlated with residuals
Textbooks
James Lesage (得克萨斯州立大学) ()
Matlab Codes (esp. for spatial panel) can be downloaded from their websites
Paul Elhorst（格林宁根大学） (http://www.regroningen.nl/elhorst/software.shtml)
1. 2. 3. 4. 5. 6. 公共基础设施建设 公共卫生支出 建开发区 环保支出 电厂投资 。。。
Spatial Dependence
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
Potential Mechanisms
1. 2. 3. 4. 5. Yardstick competition Mimicking Expenditure externality/Free-riding Fiscal competition Common shock
5
Real Examples（Numerous）
(/wiki/Great-circle_distance)
Spatial Weight Matrix(es)
3. Distance based neighbors (k-nearest neighbors)
Other Matrix Specifications
o Distance (distance band, K-nearest neighbors)
How many “neighbors” to include, what distance do we use?
o General weights (social distance, distance decay)
2. Spatial Weight Matrix Based on Inverse Distance
dij Wij 0, otherwise
Great circle distance: d(longi,lati; longj,latj)=6378*acos{sin(lati*π/180)*sin(latj*π/180)+ cos(lati*π/180) *cos(latj*π/180)*cos[(longi-longj)*π/180]}
空间计量经济学入门
（Introduction to Spatial Econometrics）
中国人民大学经济学院 虞义华 yihua.yu@
Spatial Econometricians
1. 2. 3. 4. 5. 6. 7. 8. 9. Luc Anselin/Anil Bera/Sergio Rey James LeSage Paul Elhorst Raymond Florax Gianfranco Piras; Ingmar Prucha; Harry Kelejian Badi Baltagi/邓颖 Giuseppe Arbia Lung-fei Lee/虞吉海/金飞/瞿茜/…. ….
8
Spatial Weight Matrix(es)
Choose a neighborhood criterion Neighborhoods can be defined in a number of ways o Contiguity (common boundary)
What is a “shared” boundary?
Spatial Dependence
地理学第一定律（First Law of Geography） (Tobler, 1979) “Everything is related to everything else, but near things are more related than distant things” “任何事物都相关，只是相近的事物关联更紧密”