• Data-driven / Econometrician approach: test statistics to pick up the presence of spatial effects in (nonspatial) regression residuals

  • Spatial autocorrelation tests (e.g. Moran's I):

    • Classical test with a correction in the estimation of the variance
    • Null of spatial randomness; alternative unspecified

  • Maximum Likelihood tests (e.g. LM error/lag)

    • Based on likelihood function
    • Null/alternative considering the (un)constrained model (λ ≠ 0 and/or ρ ≠ 0)
    • Robust versions

• Model estimation  ≠  Model specification

• OLS is fine for some specifications but others require specific estimators. Some OLS assumptions are violated:

  • Spatial Lag: endogeneity caused by Wy on the RHS
  • Spatial Error: VC matrix of error term ε ≠ σ²I

• Although there are more methods (e.g. Bayesian), this are the most commonly used in this context.

  • Maximum Likelihood
  • IV - GMM

Details can get "pretty" technical  →  only overview & intuition

Maximum Likelihood

• Based on multivariate normal density (normality assumption)
• Nonlinear optimization
• Hard to scale up because of matrix inversion  →  Small(er) samples

Maximum Likelihood

Spatial lag

$$ ln(L) = -(\dfrac{N}{2}) ln(2\pi) - (\dfrac{N}{2}) ln \sigma^2 + ln |I - \rho W| - \\ \dfrac{(y - \rho Wy - X\beta)' (y - \rho Wy - X\beta)}{ (2\sigma^2)} $$

Maximum Likelihood

Spatial error (FGLS)

$$ln(L) = -(\dfrac{N}{2}) ln(2\pi) - (\dfrac{N}{2}) ln \sigma^2 + ln |I - \lambda W| - \dfrac{e_L'e_L}{2\sigma^2}$$

with:

e_L = (I − λW)(y − Xβ)

IV - GMM

• Modern approach (Late 90's, 00s)
• Fast but relies on assymptotics  →  Large datasets

Spatial Lag

• Instrumental variables (IV)
  • Deals with the endogeneity of Wy
  • Kelejian & Prucha, 2004 proves that the optimal instruments are X, WX, WWX...

IV - GMM

Spatial Error

• General Method of Moments (GMM)
  • Consistent residuals from nonspatial regression
  • Solve system of equations for a set of moments to consistently and efficiently estimate λ
  • Plug the estimate in IV/OLS results through spatial Cochrane–Orcutt filtering (Y^* = Y − λWY)
  • Re-run model with filtered variables

• Rich range of proprietary and open source alternatives
• Most part of larger projects (Python scientific, R...)

• Code-Command line

  • Matlab Toolbox (LeSage)
  • PySAL
  • R (spdep, sphet, splm)
  • Stata (spivreg)

• GUI

  • OpenGeoDa
  • GeoDaSpace

Basics

• Luc Anselin's online lectures freely available at https://geodacenter.asu.edu/eslides

Advanced (cross-section)

• Family of Kelejian & Prucha papers (KP-1998/99, 2004, 2008, 2010)

Panels

• Anselin, L.; Le Gallo, J. and Jayet, H. (2008) Spatial Panel Econometrics (link)
• Lee, L. F. and Yu, J. 2010, Some recent developments in spatial panel data models (link)

Non-parametric

• McMillen, D. (2012) Perspectives on Spatial Econometrics: Linear Smoothing with Structured Models. Journal or Regional Science, 52 (2): 192-209

Discrete choice

• Fleming, M. (2004). Techniques for estimating spatially dependent discrete choice model (link)