Morning:

- Motivation
Specification

- Spatial dependence
- Spatial heterogeneity

Afternoon:

- 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

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

*Y* = *α* + *X**β* + *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)

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

*Y* = *α*_{r} + *X**β* + *ε*

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

*Y* = *α*_{r} + *X**β*_{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

Main equation in matrix form:

*Y* = *X**β* + *W**X**γ* + *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

*Y* = *ρ**W**Y* + *X**β* + *ε*

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

Captures

**global**spatial effects (spatial multiplier):*Y*= (*I*−*ρ**W*)^{ − 1}*X**β*+ (*I*−*ρ**W*)^{ − 1}*ε*↓

(

*I*−*ρ**W*)^{ − 1}*X**β*=*X**β*+*ρ**W**X**β*+*ρ*^{2}*W**W**X**β*+ . . .

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 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

**AR process**:

*Y* = *X**β* + *u**u* = *λ**W**u* + *ε*

-Global effects: *u* = (*I* − *λ**W*)^{ − 1}*ε*

**MA process**:

*Y* = *X**β* + *u**u* = *λ**W**ε* + *ε*

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

Spatial Data, Analysis and Regression - A mini course by Dani Arribas-Bel is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.