Spatial Data, Analysis, and Regression - III

A mini-course

Dani Arribas-Bel

Second block

Spatial Regression


  • Motivation
  • Specification

    • Spatial dependence
    • Spatial heterogeneity


  • Diagnostics
  • Estimation
  • Software implementation


Explicit introduction of spatial effects in an econometric framework


Space (by itself or as a proxy of something else) is relevant in many conceptual frameworks:

  • Spatial externalities/spill-overs
  • Spatial competition (e.g. spatial reaction functions, policy competition)


Even if models do not, real world occurs in space, and this (sometimes) creates problems:

  • Modifiable Areal Unit Problem (MAUP)
  • Scale issues and boundary mismatch

Some of these violate classical assumptions in OLS

Model specification

  • Spatial dependence Vs. spatial heterogeneity
  • Deviations from traditional linear model:

Y = α + Xβ + ε

  • Several approaches:

    • [SH] Spatial fixed effects
    • [SH] Spatial regimes
    • [SD] Exogenous spatial effects
    • [SD] Endogenous variable spatially lagged
    • [SD] Spatial effects in the error term
  • Each has different consecuences on the kind and extent of the spatial effects modelled
  • Some are straightforward to estimate, some require particular estimators

     →  Anselin (2003)

Spatial heterogeneity

  • Account for systematic differences across space without relying on interdependences
  • Typically justified by unobservables that have a clear spatial dimension
  • Econometrically "simpler"

Spatial fixed effects

Y = αr + Xβ + ε

  • Level differences in the outcome Y due to location
  • Needs to be defined ex-ante
  • Non-spatial estimation

Spatial regimes

Y = αr + Xβr + ε

  • Level differences + different effects of exogenous variables in the outcome Y due to location
  • Needs to be defined ex-ante
  • Non-spatial estimation
  • Similar to running separate regressions by regime ("complete no pooling"), but allows for testing of the differences with a (spatial) Chow test

Spatial dependence

  • Model interdependencies between observations channeled through space
  • Also, potential way to account for MAUP (spatial smoothing)
  • Econometrically more involved, because (most of the times) it violates many OLS assumptions

Exogenous spatial effects

Main equation in matrix form:

Y = Xβ + WXγ + ε

  • Limited spatial extent: after one order of spatial magnitude, their effect dissapears ( ≈  local externalities)
  • Akin to one more exogenous variable  →  Ignoring it is akin to an omitted variable problem (bias and loss of efficiency)
  • Straighforward estimation (OLS) because they are exogenous

Spatial lag model (AR)

Y = ρWY + Xβ + ε

  • Endogenous variable is spatially lagged and included as one more explanatory variable

  • Captures global spatial effects (spatial multiplier):

    Y = (I − ρW) − 1Xβ + (I − ρW) − 1ε


    (I − ρW) − 1Xβ = Xβ + ρWXβ + ρ2WWXβ + . . . 

Spatial lag model (AR)

  • Two main rationales behind its adoption:

    • Theory-driven: compatible with spatial interaction and reaction functions
    • Data-driven: spatial filter to deal with scale problems
  • Its effect is interpreted as the outcome of a simultaneous system, not as a direct causal effect  →  model interdependent decisions
  • Omission induces bias and efficiency issues and, because of the endogeneity induced, estimation requires particular techniques (e.g. ML, IV)

Spatial error model

  • Spatial effects in (uncorrelated) unmodelled shocks
  • Off-diagonal elements of the VC matrix are non-zero and follow a spatial pattern
  • Efficiency problem: OLS estimates remain "on target" but their precission is damaged

Spatial error model

  • AR process:

Y = Xβ + u

u = λWu + ε

-Global effects: u = (I − λW) − 1ε

Spatial error model

  • MA process:

Y = Xβ + u

u = λWε + ε

-Local effects: after two orders of neighbors, the effect washes away

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