Download A Conditionally Parametric Probit Model of Micro

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