Spatial Regression

Morning:

• Motivation

• Specification

  • Spatial dependence
  • Spatial heterogeneity

Afternoon:

• Diagnostics
• Estimation
• Software implementation

Theory-driven

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)

Data-driven

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

• 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 fixed effects

Y = α_r + Xβ + ε

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

Spatial regimes

Y = α_r + Xβ_r + 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

• 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