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A Conditionally Parametric Probit Model of Micro-Data Land Use in Chicago Daniel McMillen Maria Soppelsa Overview • Residential v. Commercial/Industrial Land Use in Chicago, 2010 • A conditionally parametric (CPAR) approach produces smooth estimates over space • Target points chosen using an adaptive decision tree approach (Loader, 1999) • Interpolation from 182 target points to all 583,063 individual parcels in the data set Estimation Procedures • Case (1992). Special From for W • McMillen (1992). EM Algorithm • Pinkse and Slade (1998). GMM for spatial error model. • LeSage (2000). Bayesian approach • Klier and McMillen (2007). Linearized version of GMM probit/logit for spatial AR model. GMM Probit • 𝑦 ∗ = 𝜌𝑊𝑦 ∗ + 𝑋𝛽 + 𝑢 = 𝐼 − 𝜌𝑊 • 𝑣 = 𝐼 − 𝜌𝑊 −1 𝑢 −1 • 𝐸𝑣𝑣 ′ = 𝜎 2 𝐼 − 𝜌𝑊 ′ 𝐼 − 𝜌𝑊 𝜎𝑖2 • 𝑋𝑖∗ ≡ 𝑋𝑖 /𝜎𝑖 , • 𝑣𝑖∗ = 𝑦𝑖 −Φ 𝑋𝑖∗ 𝛽 Φ 𝑋𝑖∗ 𝛽 1−Φ 𝑋𝑖∗ 𝛽 ϕ 𝑋𝑖∗ 𝛽 β, ρ to minimize 𝑣 ∗ 𝑍 𝑍 ′ 𝑍 −1 ′ ∗ 𝑍𝑣 −1 𝑋𝛽 + 𝑣 𝐸𝑣𝑖2 ≡ Linearized GMM Probit 1. Standard probit: 𝛽 2. 2SLS regression of e on 𝑒 + 𝑔𝑋𝛽 on 𝑔𝑋 and 𝑔𝑊𝑋𝛽, where 𝑦𝑖 − Φ 𝑋𝑖 𝛽 𝑒𝑖 = ϕ 𝑋𝑖 𝛽 Φ 𝑋𝑖 𝛽 1 − Φ 𝑋𝑖 𝛽 𝑑𝑒𝑖 𝑔𝑖 = − 𝑑𝛽 3. 𝑝𝑖 = Φ 𝑋𝑖 𝛽 𝜎𝑖 . Requires inversion of 𝐼 − 𝜌𝑊 ′ 𝐼 − 𝜌𝑊 CPAR Probit • 𝑛 𝑗=1 𝑤𝑗 𝑦𝑗 𝑙𝑛Φ 𝛽 𝑙𝑜, 𝑙𝑎 ′ 𝑥𝑗 + 1 − 𝑦𝑗 𝑙𝑛 Φ −𝛽 𝑙𝑜, 𝑙𝑎 ′ 𝑥𝑗 • 𝑤𝑗 = kernel weight function, distance between observation j and target point. • Straightforward extension of “GWR” – a special case of locally weighted or locally linear regression. • Applications: – McMillen and McDonald (2004) – Wang, Kockelman, and Wang (2011) – Wren and Sam (2012) Spatial AR v. LWR Data • Individual parcels in Chicago, 2010 • Major Classes: 1. Vacant Land (33,139) 2. Residential, 6 units or fewer (728,541, 539,975 after geocoding) 3. Multi-Family Residential (11,529) 4. Non-Profit (316) 5. Commercial and Industrial (50,508, 43,088 after geocoding) 6. “Incentive Classes” (1,487) Explanatory Variables • Distance from parcel centroid to: 1. CBD 2. Lake Michigan 3. EL line 4. EL stop 5. Rail line 6. Major street 7. Park 8. Highway Rogers Park Descriptive Statistics Variable Mean Std. Dev. Min Max Residential Lot 0.926 0.262 0.000 1.000 Distance from CBD 7.518 3.433 0.022 17.006 Distance from Lake Michigan 4.116 2.716 0.005 12.321 Distance from EL Line 1.358 1.277 0.001 6.265 Distance from EL Stop 1.214 1.081 0.001 6.265 Distance from Rail Line 0.428 0.294 0.001 1.997 Distance from Major Street 0.080 0.057 0.000 0.508 Distance from Park 0.233 0.153 0.000 2.999 Distance from Highway 1.476 1.027 0.011 4.809 Probit Models, Probability Residential Standard Probit Variable Intercept Distance from CBD Distance from Lake Michigan Distance from EL Line Distance from EL Stop Distance from Rail Line Distance from Major Street Distance from Park Distance from Highway Log-likelihood Pseudo-R2 Coef. Std. Error 0.061 0.046 0.132 0.007 -0.095 0.007 0.002 0.013 -0.091 0.013 0.626 0.014 8.748 0.070 -1.099 0.020 0.212 0.007 -131518.9 0.144 CPAR Probit Mean Std. Dev. 0.351 1.008 0.101 0.266 -0.086 0.308 -0.423 1.168 0.511 1.263 0.649 0.686 11.570 6.427 -0.881 0.994 0.048 0.351 -120714.1 0.215 Probability of Residential Land Use: Standard Probit Probability of Residential Land Use: CPAR Probit, 10% Window Size Difference, CPAR Probability – Standard Probit Probability Kernel Density Estimates for CPAR Coefficients LWR Estimates of CPAR Coefficients Marginal Probabilities Marginal Probabilities Marginal Probabilities Marginal Probabilities Marginal Probabilities Marginal Probabilities Marginal Probabilities Marginal Probabilities Rogers Park Rogers Park, n = 3,193 Intercept Standard Coef Std. Err. 49.979 11.999 GMM Coef Std. Err. 42.977 12.592 CBD -1.804 0.462 -1.549 0.480 Lake Michigan -7.621 1.672 -6.555 1.814 -0.726 5.314 EL Line -3.324 0.651 -2.901 0.723 -4.449 9.934 EL Stop 3.127 0.654 2.698 0.739 6.593 9.706 Rail Line 1.906 0.395 1.659 0.428 1.675 4.059 Major Street 7.123 0.837 5.992 1.346 15.900 9.561 Park -1.797 0.514 -1.594 0.525 Highway -7.207 1.743 -6.197 1.809 Metra Stop 0.038 0.216 0.024 0.178 ρ pseudo-R2 0.084 0.155 0.167 0.084 CPAR Mean Std. dev. 0.025 2.445 0.343 Correlations, Predicted Probabilities Standard GMM CPAR 1 0.57 0.99 GMM 0.57 1 0.57 CPAR 0.99 0.57 1 Standard Standard Probit Probabilities CPAR Probit Probabilities Standard Probit: Southwest CPAR – Standard: Southwest Standard Probit: Southeast CPAR – Standard: Southeast Standard Probit: Northwest CPAR – Standard: Northwest Standard Probit: Northeast CPAR – Standard: Southeast