Spatial Econometrics with R

Spatial econometrics involves the study and modeling of spatial relationships and dependencies in econometric data. It is an essential field for understanding spatial phenomena in economics, geography, and various other disciplines. This guide covers the theoretical foundation of spatial econometrics and practical examples using R Programming Language.

Introduction to Spatial Econometrics

Spatial Econometrics is a subfield of econometrics that deals with spatial interdependence and spatial heterogeneity in regression models. It is used to analyze spatial data where observations are not independent but influenced by their spatial location.

1. Spatial Autocorrelation: Measures the degree to which similar values occur near each other in space.
2. Spatial Lag Models (SLM): Incorporate spatially lagged dependent variables to account for spatial autocorrelation.
3. Spatial Error Models (SEM): Incorporate spatially autocorrelated error terms to account for spatial dependency in residuals.
4. Spatial Durbin Models (SDM): Extend spatial lag models by including spatial lags of both dependent and independent variables.

Spatial Weight Matrices

A critical component in spatial econometrics is the spatial weight matrix (W), which defines the spatial structure of the data. It quantifies the spatial relationships between observations.

• Contiguity-Based Matrices: Define neighbors based on shared boundaries.
• Distance-Based Matrices: Define neighbors based on distance thresholds.

Practical Examples of Spatial Econometrics Using R

First Install and load the necessary packages for spatial econometrics:

install.packages(c("spdep", "sf", "spatialreg"))
library(spdep)
library(sf)
library(spatialreg)

Example 1: Spatial Autocorrelation

Load a spatial dataset and prepare the spatial weight matrix:

# Load an example dataset
nc <- st_read(system.file("shape/nc.shp", package="sf"))

# Create a spatial weight matrix based on contiguity
nb <- poly2nb(nc)
W <- nb2listw(nb, style="W")
head(nc)

Output:

Simple feature collection with 6 features and 14 fields
Geometry type: MULTIPOLYGON
Dimension:     XY
Bounding box:  xmin: -81.74107 ymin: 36.07282 xmax: -75.77316 ymax: 36.58965
Geodetic CRS:  NAD27
   AREA PERIMETER CNTY_ CNTY_ID        NAME  FIPS FIPSNO CRESS_ID BIR74 SID74 NWBIR74
1 0.114     1.442  1825    1825        Ashe 37009  37009        5  1091     1      10
2 0.061     1.231  1827    1827   Alleghany 37005  37005        3   487     0      10
3 0.143     1.630  1828    1828       Surry 37171  37171       86  3188     5     208
4 0.070     2.968  1831    1831   Currituck 37053  37053       27   508     1     123
5 0.153     2.206  1832    1832 Northampton 37131  37131       66  1421     9    1066
6 0.097     1.670  1833    1833    Hertford 37091  37091       46  1452     7     954
  BIR79 SID79 NWBIR79                       geometry
1  1364     0      19 MULTIPOLYGON (((-81.47276 3...
2   542     3      12 MULTIPOLYGON (((-81.23989 3...
3  3616     6     260 MULTIPOLYGON (((-80.45634 3...
4   830     2     145 MULTIPOLYGON (((-76.00897 3...
5  1606     3    1197 MULTIPOLYGON (((-77.21767 3...
6  1838     5    1237 MULTIPOLYGON (((-76.74506 3...

Calculate Moran's I for the dataset

Moran's I is a measure of spatial autocorrelation:

# Calculate Moran's I for a variable
moran.test(nc$BIR74, listw = W)

Output:

	Moran I test under randomisation

data:  nc$BIR74  
weights: W    

Moran I statistic standard deviate = 2.4055, p-value = 0.008074
alternative hypothesis: greater
sample estimates:
Moran I statistic       Expectation          Variance 
      0.139319332      -0.010101010       0.003858258 

• The Moran I statistic is 0.1393, indicating a slight positive spatial autocorrelation.
• The Z-score is 2.4055, suggesting the observed Moran I is 2.4 standard deviations above the expected value under the null hypothesis.
• The p-value is 0.008074, which is less than the common significance level of 0.05, indicating that the spatial autocorrelation is statistically significant.

In conclusion, there is significant evidence to reject the null hypothesis of no spatial autocorrelation in the BIR74 variable, suggesting that areas with high values of BIR74 tend to be near other areas with high values, and areas with low values of BIR74 tend to be near other areas with low values.

Example 2: Spatial Lag Model (SLM)

Fit a spatial lag model using the lagsarlm function from the spatialreg package:

# Load necessary libraries
library(sf)
library(spdep)
library(spatialreg)

# Load the example dataset
nc <- st_read(system.file("shape/nc.shp", package="sf"))

# Create a spatial weight matrix based on contiguity
nb <- poly2nb(nc)
W <- nb2listw(nb, style="W")

# Fit a spatial lag model with the correct variable names
slm <- lagsarlm(SID74 ~ NWBIR74 + BIR74, data = nc, listw = W)
summary(slm)

Output:

Call:lagsarlm(formula = SID74 ~ NWBIR74 + BIR74, data = nc, listw = W)

Residuals:
      Min        1Q    Median        3Q       Max 
-11.42866  -1.56581  -0.54971   1.13618  14.20567 

Type: lag 
Coefficients: (asymptotic standard errors) 
              Estimate Std. Error z value  Pr(>|z|)
(Intercept) 0.89337654 0.63911040  1.3978  0.162160
NWBIR74     0.00357200 0.00054522  6.5515 5.696e-11
BIR74       0.00052249 0.00020037  2.6077  0.009116

Rho: 0.043388, LR test value: 0.33608, p-value: 0.5621
Asymptotic standard error: 0.0764
    z-value: 0.5679, p-value: 0.5701
Wald statistic: 0.32251, p-value: 0.5701

Log likelihood: -261.2544 for lag model
ML residual variance (sigma squared): 10.879, (sigma: 3.2983)
Number of observations: 100 
Number of parameters estimated: 5 
AIC: 532.51, (AIC for lm: 530.84)
LM test for residual autocorrelation
test value: 0.021662, p-value: 0.88299

The spatial lag model for SID74 indicates that NWBIR74 and BIR74 have significant positive effects on SID74, with p-values of 5.696e-11 and 0.009116, respectively. The spatial autoregressive coefficient (Rho) is 0.043388, but it is not statistically significant (p-value: 0.5701). The model's residuals show no significant spatial autocorrelation (p-value: 0.88299). The model's log-likelihood is -261.2544, and the AIC is 532.51, indicating a slightly worse fit than a standard linear model (AIC for lm: 530.84).

Conclusion

Spatial econometrics provides powerful tools for analyzing spatial dependencies in econometric data. Using R, you can easily perform spatial econometric analysis, including testing for spatial autocorrelation, fitting spatial regression models, and visualizing spatial data. This guide covered the fundamental theory and practical examples to get you started with spatial econometrics in R.