Spatial Modeling Lee Rivers Mobley Ph D

Spatial Modeling Lee Rivers Mobley, Ph. D.

Spatial Modeling I. Why or When Does It Matter? II. Some Basics of Spatial Modeling III. Examples of Interesting Research Applications

I. Spatial Modeling: When or Why Does it Matter? · Maps Are Powerful Tools for Summarizing Information · There May Be Dangers Inherent In Using 'Convenient' Spatially-Referenced Data (County data, MSA data, State data)

Maps Are Powerful Tools for Summarizing Information • Can Results From Regional Studies be Generalized? • Where are the Urban Areas? • Where Is the Spatial Interaction? · What is the Distribution of Physician Group Practices? · Where do PPOs Locate?

An Example Of Spatial Interaction … Interaction Between the Hefindahl Statistic and Managed Care Penetration in the SF area: And How It Varies By Unit of Aggregation: HFPA (Left), County (Right)

An Example Of Spatial Interaction … Interaction Between the Hefindahl Statistic and Managed Care Penetration in the LA area: And How It Varies By Unit of Aggregation: HFPA (Left), County (Right)

Where Are The Large vs. Small Physician Practices Located?

Q: Where do PPOs Locate? A: Even Where Docs Don’t

What Are The Dangers Inherent In Using 'Convenient' Spatially-Referenced Data? · Observed Spatial Interaction Can Depend Upon the Unit of Analysis. This Is The Most Important Reason To Model Space Carefully. · Typically, Analyses are Conducted Using Spatial Units Which are Mismatched With the Underlying Economic/Social Activity which can cause bias and inefficiency in estimated regression coefficients

Other Problems. . . · More Generally, Unmodeled Spatial Heterogeneity Can Cause Bias in Estimated Regression Parameters - an omitted variables effect · A Simpler Problem derives from positive spatial autocorrelation - the redundant information makes OLS estimators inefficient, and produce misleading standard errors

Solutions. . . · The Appropriate Level of Aggregation, or Extent of Spatial Interaction, Can Be Revealed With Spatial Modeling. · Problems Associated With Autocorrelation, MAUP and Unmodeled Spatial Heterogeneity can be Solved by Careful Modeling

II. Some Basics Of Spatial Modeling

Some Basics of Spatial Modeling

Spatial Regression Models • Based on work by Luc Anselin • All models described here can be estimated using Space. Stat software: • http: //www. spacestat. com/

What’s Different About Spatial Regression? • Spatial regression is concerned with the analysis of spatially-referenced data. • Spatial regression differs from classical regression in that the observations analyzed are not independent. • Observations are correlated with others that are spatially proximate, resulting in spill-over of information from location to location (positive correlation).

Causes of Spatial Spillovers • The MAUP problem: a mismatch between the spatial scale of the phenomenon under study and the spatial scale at which it is measured. • The mismatch causes spatial measurement errors and spatial autocorrelation between these errors in adjacent locations (Anselin, 1988).

More Causes of Spatial Spillovers • Unmodeled spatial heterogeneity : the failure to include as explanatory variables measures fully modeling the spatial environment (regimes, drift). • Simple spatial autocorrelation in the errors due to regular patterns of similarity among neighbors, which result in redundant information.

Consequences of Misspecification • Either the MAUP or unmodeled heterogeneity problems can lead to biased and inconsistent estimators. • Simple spatial autocorrelation: regression parameters are unbiased, but inefficient (as a special case of a non-spherical error variance covariance matrix).

Proper Accounting for Spillovers • In the presence of simple spatial autocorrelation, the spatial stochastic process is in the error term. The spatial error model yields efficient (and unbiased) parameter estimates: y = X + , where = W + , and W is the weighed average of the errors in specified neighboring locations

Proper Accounting for Spillovers • When the spatial stochastic process is in the dependent variable, the spatial lag model (with or without regimes) yields unbiased parameter estimates: y = Wy + X + , where = W + • Wy is a weighted average of the values of y within a specified neighborhood of y.

Specification of Spatial Weights · Neighborhood sets may be defined in Space. Stat based on contiguity (common boundary), isotropy (within a distance band), or k-nearest neighbors. · These approaches produce exogenous weights, which is important. · Using k-nearest neighbors is preferred to isotropic measures, because variation in the number of neighbors can lead to heteroskedasticity.

Interpretation of the Spatial Parameters • Spatial error model: the coefficient is interpreted as a nuisance parameter. • Spatial lag model: the coefficient has been interpreted in the literature as a measure of the extent of spatial spillovers, copy-catting, free riding, competition, collusion, or diffusion.

Specification Testing: Distinguishing between error and lag structures • One-Directional Tests · Rao score test for error autocorrelation : RS · Rao score test for lag autocorrelation: RS The alternative for both tests is 'no spatial autocorrelation'

Specification Testing: Distinguishing between error and lag structures • Multi-Directional (Robust) Tests · Rao 'robust' score test for error autocorrelation : RS * · Tests for spatial error in the presence of spatial lag · Rao 'robust' score test for lag autocorrelation: RS * · Tests for spatial lag in the presence of spatial error

Methodology for Proper Diagnosis of Error Process (Anselin and Bera, 1998) • When RS is more significant than RS , and RS * is significant while RS * is not, then the lag model is most likely the correct error structure. • When RS is more significant than RS , and RS * is significant while RS * is not, then the error model is most likely the correct error structure.

Examples Using Maps • The following slides depict census tracts in the Columbus, Ohio metropolitan area. The variable to be explained is crime rate. • In the first slide, we see an Arc. View theme map of crime rate, which shows clustering at the city center (and evidence of significant spatial autocorrelation).

Modeling Spatial Heterogeneity • In the second slide we see a binary variable designating whether or not a tract is near the city center, CP. • The binary variable seems to do a good job mimicking the spatial pattern inherent in crime rates.

OLS vs Spatial Regression Model • The OLS model was estimated to explain crime rates using income and house values in Columbus neighborhoods. • Based on the diagnostics, a spatial lag model was deemed more appropriate. • Remaining Heteroskedasticity in the lag model led to the addition of regimes - the ‘best model’.

OLS vs. Spatial Regression • The next slide plots (on the right) the residuals from a standard OLS model, which does not account for spatial autocorrelation or heterogeneity. • On the left are plotted residuals from the best model discovered using spatial diagnostics: lag error with regimes.

Informing the Naked Eye • It is hard to determine with the naked eye whether or not the residual patterns are random (the ultimate goal in modeling). • The next slides depict local significance tests for remaining autocorrelation in the residuals from the two models. • The local (Moran) tests are accomplished using Space. Stat with Arc. View.

III. Examples of Interesting Research Applications ·Published Studies ·Potential Applications

Published Studies: Innovation Spillovers • Jaffe (1989) American Economic Review – “there is only weak evidence that spillovers are facilitated by geographic coincidence of universities and research labs within the state” – this conclusion was decisively refuted by Anselin, Varga, and Acs (1997) Journal of Urban Economics

Debunking the No-Spillover Myth • Anselin et al. (1997) used a spatial econometric approach to carefully model spatial interaction • They find a positive and significant relationship between university research and innovative activity, both directly, and indirectly through spillovers on private sector R&D

Spatial Reaction Functions • Brueckner (1998) Journal of Urban Economics employs a spatial lag model to explain strategic reactions among cities to their neighbors’ tax policy. (Raising taxes can deter attraction of new businesses). • In the Boston area, 1980, cities adopted copycat strategies (the lag parameter estimate was a significant positive number).

Potential Applications: Lag Model • The spatial reaction function approach may be useful in modeling hospitals’ strategic adoption of services. • Strategic behavior may help leverage market power against managed care. • In markets with higher managed care penetration, we might expect to see more networking (sharing) - hence a negative lag parameter.

Potential Applications: Spatial Regimes • The spatial regimes model may be appropriate to describe efficiency variation among hospitals in markets with different market pressures. • This model may also be appropriate in determining the balance of power between hospitals and payors - and the possible emergence of Monopsony.

GIS Potential in Spatial Modeling • A GIS program (Arc. View, Atlas GIS) can be used to generate variables which model space. • These variables can be constructed at different levels of aggregation. • Diagnostic checks built into Space. Stat can reveal which level of aggregation is appropriate, and whether mixed scales of aggregation affect results.

Research Potential • Spatial modeling is being used in research on real estate markets, public finance, labor economics, simple marketing, and environmental research. • It has not yet been applied in health economics or finance. • The strategic models have considerable untapped research potential.