SPATIAL ECONOMETRICS
Tobler’s first law of geography: everything is related to everything else, but near things are
more related than distance things. There is strong concentration over space. Many economic
processes are spatially correlated. Poverty, GDP, and unemployment often display spatially
correlated patterns, or are clustered in space. (Cor)relation over space are rather common.
Spatial modelling is becoming increasingly important: new and georeferenced data, advanced
software, and new methods and regression techniques.
Why not use OLS? (assumptions)
- Random sampling
- Linearity in parameters
- Mean independence of error term
- No perfect multicollinearity
- Large outliers are unlikely
- Omitted variable bias
Spatial data consist out of a time component and a spatial component. Examples are: house
price data, income data, education performance of pupils at different schools, emissions,
population density.
Spatial heterogeneity: socio-economic variables are unequally distributed over space. Spatial
dependence: the variable is clustered over space. Spatial relations are multi-directional. One
spatial unit can affect several other spatial units directly. Spatial relations are reciprocal
(=wederzijds). Spatial unit A affects B and B affects A at the same time.
There is a different between time (panel data, temporal) and linear space (e.g. beach, spatial):
- Time has a natural origin, linear space has not: difficult to tell where something starts.
- Time do not have reciprocity (=interactie) (present data does not influence the data in
the past), linear space does have (location 2 influences 1 and 3 but not directly 4): can
go either way.
- Time and linear space are unidirectional
Two-dimensional space is even multidirectional: one element can influence several other
elements at the same time. Locations can influence each other.
The spatial structure is defined through a spatial weights matrix W: which consist out of nxn
elements with discrete or continuous elements. How does it relate to another location? It can
be measured with: Euclidian distance, network distance (road network), spatial interactions,
and social networks. Definition:
- Contiguity matrix: adjacent = 1st order contiguous, neighbor of neighbor = 2nd order
contiguous. If locations are adjacent, directly related, the value is 1, otherwise it is 0. If
the weight of neighbor of neighbors’ is 1 à 2nd order. The 1st order contiguity-based
relation is when two areas have a relation if and only if they share a border. Otherwise
the relation is zero. Relations between direct adjacent neighbors are called 1st order
contiguity. Between areas that share a common neighbor are 2nd order contiguity. If
regions or countries are isolated (e.g. islands) this might pose a problem.
, - Distance matrix: k-nearest neighbors. Weight is 1 for the k nearest neighbors,
otherwise 0. Inverse distance weights (1/distance): weight is lower when the location
is further away and cut-off distance, where the weight 1 within 5 kilometers and 0
otherwise. The relation between two areas is measured by some notion of distance of
travel time.
Equation: y=rho*W*y+beta1*X1+beta2*X2+error term. Rho determines the amount of
interaction and W determines the interaction/spatial weights structure.
Matrices can be standardized. The most common principle is row-standardization: all entries
on a given row of W are divided by the sum of all those entries (row-sum).
& &(#)*+
𝑤 ∗ #$ = ∑$ !"& = ∑ ,-. 01 &(#)*+ 20&
#%& !#
k=other locations, ∑4356 𝑤#3 =sum of neighbors’ connections, w*ij=share of neighbors’
connection. Row-standardization is only possible for averages, the size of the area does not
matter. It cannot be used for urban and rural areas because this will give incorrect values. It is
impossible to compare both areas. A disadvantage is that all entries on a role sum up to 1 and
that spatial weight matrices that initially were symmetrical lose their symmetry.
Neighborhoods surrounded by many offices are treated similarly as neighborhoods
surrounded by few offices.
The remarks regarding distance weight matrices are:
- Check for exogeneity of matrix. Euclidean distance is not something that can be
changed = exogeneous. Number of friends can change = endogenous. Reversed
causality: higher income and more friends.
- Connectivity
- Symmetry
- Standardization
- Distance decay. The choice is arbitrary: sometimes theory may help which may to find
the optimal distance decay parameter empirically. Alternative: different x-variables
capturing concentric rings.
Spatial autocorrelation (dependence, clustering): is the value of the variable is correlated over
8 ;<=>;<
space. It is measured with Moran’s I: 𝐼 = 9 × ;<=;<
It is the oldest and most common general
'
misspecification test. The advantage of this test-statistic is that is has power against all kinds
of spatial dependence processes. The disadvantage is that it is not known against which kind
of spatial process.
R=number of spatial units, S0=sum of all elements of the spatial weight matrix, W=spatial
weight matrix, 𝑥< = 𝑥 − 𝑥̅ is a vector with the variable of interest (subtract the average). H0:
there is spatial randomness. Ha: there is spatial dependence.
Moran’s I is a correlation coefficient. Use row-standardized spatial weight matrix to interpret.
X’ = transpose of the matrix. Assume that the distribution of I is normal to investigate if it is
statistically significance. Bootstrapping: take samples of the samples and calculate Moran’s I
each time.
How to determine spatial autocorrelation:
- Determine distance between all neighborhoods using centroids
6
- Use inverse distance function 𝑤#$ = B( to determine spatial weights in weight matrix.
!"
Spatial structure of dataset. Increase gamma, more closed neighborhoods are relevant
Tobler’s first law of geography: everything is related to everything else, but near things are
more related than distance things. There is strong concentration over space. Many economic
processes are spatially correlated. Poverty, GDP, and unemployment often display spatially
correlated patterns, or are clustered in space. (Cor)relation over space are rather common.
Spatial modelling is becoming increasingly important: new and georeferenced data, advanced
software, and new methods and regression techniques.
Why not use OLS? (assumptions)
- Random sampling
- Linearity in parameters
- Mean independence of error term
- No perfect multicollinearity
- Large outliers are unlikely
- Omitted variable bias
Spatial data consist out of a time component and a spatial component. Examples are: house
price data, income data, education performance of pupils at different schools, emissions,
population density.
Spatial heterogeneity: socio-economic variables are unequally distributed over space. Spatial
dependence: the variable is clustered over space. Spatial relations are multi-directional. One
spatial unit can affect several other spatial units directly. Spatial relations are reciprocal
(=wederzijds). Spatial unit A affects B and B affects A at the same time.
There is a different between time (panel data, temporal) and linear space (e.g. beach, spatial):
- Time has a natural origin, linear space has not: difficult to tell where something starts.
- Time do not have reciprocity (=interactie) (present data does not influence the data in
the past), linear space does have (location 2 influences 1 and 3 but not directly 4): can
go either way.
- Time and linear space are unidirectional
Two-dimensional space is even multidirectional: one element can influence several other
elements at the same time. Locations can influence each other.
The spatial structure is defined through a spatial weights matrix W: which consist out of nxn
elements with discrete or continuous elements. How does it relate to another location? It can
be measured with: Euclidian distance, network distance (road network), spatial interactions,
and social networks. Definition:
- Contiguity matrix: adjacent = 1st order contiguous, neighbor of neighbor = 2nd order
contiguous. If locations are adjacent, directly related, the value is 1, otherwise it is 0. If
the weight of neighbor of neighbors’ is 1 à 2nd order. The 1st order contiguity-based
relation is when two areas have a relation if and only if they share a border. Otherwise
the relation is zero. Relations between direct adjacent neighbors are called 1st order
contiguity. Between areas that share a common neighbor are 2nd order contiguity. If
regions or countries are isolated (e.g. islands) this might pose a problem.
, - Distance matrix: k-nearest neighbors. Weight is 1 for the k nearest neighbors,
otherwise 0. Inverse distance weights (1/distance): weight is lower when the location
is further away and cut-off distance, where the weight 1 within 5 kilometers and 0
otherwise. The relation between two areas is measured by some notion of distance of
travel time.
Equation: y=rho*W*y+beta1*X1+beta2*X2+error term. Rho determines the amount of
interaction and W determines the interaction/spatial weights structure.
Matrices can be standardized. The most common principle is row-standardization: all entries
on a given row of W are divided by the sum of all those entries (row-sum).
& &(#)*+
𝑤 ∗ #$ = ∑$ !"& = ∑ ,-. 01 &(#)*+ 20&
#%& !#
k=other locations, ∑4356 𝑤#3 =sum of neighbors’ connections, w*ij=share of neighbors’
connection. Row-standardization is only possible for averages, the size of the area does not
matter. It cannot be used for urban and rural areas because this will give incorrect values. It is
impossible to compare both areas. A disadvantage is that all entries on a role sum up to 1 and
that spatial weight matrices that initially were symmetrical lose their symmetry.
Neighborhoods surrounded by many offices are treated similarly as neighborhoods
surrounded by few offices.
The remarks regarding distance weight matrices are:
- Check for exogeneity of matrix. Euclidean distance is not something that can be
changed = exogeneous. Number of friends can change = endogenous. Reversed
causality: higher income and more friends.
- Connectivity
- Symmetry
- Standardization
- Distance decay. The choice is arbitrary: sometimes theory may help which may to find
the optimal distance decay parameter empirically. Alternative: different x-variables
capturing concentric rings.
Spatial autocorrelation (dependence, clustering): is the value of the variable is correlated over
8 ;<=>;<
space. It is measured with Moran’s I: 𝐼 = 9 × ;<=;<
It is the oldest and most common general
'
misspecification test. The advantage of this test-statistic is that is has power against all kinds
of spatial dependence processes. The disadvantage is that it is not known against which kind
of spatial process.
R=number of spatial units, S0=sum of all elements of the spatial weight matrix, W=spatial
weight matrix, 𝑥< = 𝑥 − 𝑥̅ is a vector with the variable of interest (subtract the average). H0:
there is spatial randomness. Ha: there is spatial dependence.
Moran’s I is a correlation coefficient. Use row-standardized spatial weight matrix to interpret.
X’ = transpose of the matrix. Assume that the distribution of I is normal to investigate if it is
statistically significance. Bootstrapping: take samples of the samples and calculate Moran’s I
each time.
How to determine spatial autocorrelation:
- Determine distance between all neighborhoods using centroids
6
- Use inverse distance function 𝑤#$ = B( to determine spatial weights in weight matrix.
!"
Spatial structure of dataset. Increase gamma, more closed neighborhoods are relevant
