T HE M ICRO F OUNDATIONS OF M ACRO S ORTING M ODELS Timothy L. - - PowerPoint PPT Presentation

THE MICRO FOUNDATIONS OF MACRO SORTING MODELS

Timothy L. Hamilton University of Richmond Daniel J. Phaneuf University of Wisconsin October 22, 2012

INTRODUCTION – SORTING MODELS FOR NON-MARKET VALUATION

Households sort a la Tiebout to optimal residential location:  Choice reveals tradeoffs between prices and attributes of locations  Use to quantify preferences for local public goods Examples:  Bayer et al. (2009) – particulate matter pollution  Klaiber and Phaneuf (2010) – open space  Tra et al. (2012) – school quality  Tra (2012) – ozone pollution Many apparent advantages vis-à-vis first stage hedonic modeling…

ASSUMPTION HEAVY MODELING ENVIRONMENT Need assertions on:  Spatial extent of analysis → “macro” choice level vs. “micro” choice level → Bayer et al. ‘macro’ – country-wide choice → Klaiber and Phaneuf ‘micro’ – city-wide choice  Definition of a choice element  Functional form for utility and error distributions  Nature/form of equilibrium conditions Important element of research agenda is examining extent to which advantages are assumption-driven…

CHOICE ELEMENTS AND SPATIAL SCALE Seems that most location choices are two-tiered…  Choose region/city based on labor market, regional amenities, family roots…  Choose neighborhood based on local amenities, schools, commute patterns… Choices are distinct – but interrelated?  Complement/substitute relationship between regional and local public goods? e.g. does regional air quality have higher value if local landscape provides more outdoor opportunities?

RESEARCH QUESTIONS How do estimates of the non-market value of air quality depend on the interconnectedness between macro and micro sorting margins? → revisit Bayer et al. (2009) macro-sorting application to regionally varying air pollution → examine more general model that nests their specification while adding micro margin How can we tractably model the multiple sorting decision margins? → examine ‘two stage budgeting’ assumptions as applied to ‘two stage sorting’ → develop sequential estimation approach to accommodating both margins

FINDINGS/CONTRIBUTIONS Consideration of micro sorting margin matters empirically:  Elasticity of WTP with respect to air quality 0.31 (macro only) vs. 0.48 (macro/micro)  Marginal WTP for air quality $232 vs. $371 Difference due to an implicit omitted variable in macro only model Operationalize sequential estimation of two stage budgeting/two stage sorting model  Micro-level choices aggregated to a “quality adjusted price index” summarizing neighborhood level choice sets  Connect practical use of nested logit model to two stage budgeting concept

OUTLINE OF TALK 1) Introduction 2) Conceptual Basis 3) Empirical Basis 4) Application and Data 5) Estimation/Results 6) Conclusions We are working on finalizing paper for submission – what is still needed? Where should it go? Contribution well-framed?

CONCEPTUAL BASIS

,

max exp( ) . .

C Y H X

ijm m jm jm m ijm C H jm im

U C H X H p C I Y s t

   

         (1) C: numeraire consumption H: consumption of ‘housing services’ Ym, m: regionally (MSA) varying public goods Xjm, jm: neighborhood public goods (deviations from MSA averages) Iim: income in MSA m ijm: idiosyncratic preference shock pjm: price of housing unit in neighborhood j, MSA m lnpjm = lnm + lnjm ↔ pjm=m×jm

Conditional indirect utility for person i, neighborhood j, MSA m:

 

ln ln ln ln ln ln , 1,..., 1,...,

ijm I im Y m X jm H m jm jm m ijm m

V I Y X j J m M                    (2) Note:  I=C+H  Literature-standard additively separable form  jm=1, Xjm=1, Jm=1, jm=0 → collapses to Bayer et al. model

TWO STAGE BUDGETING A restriction on preferences implying:  Consumer first determines expenditures on commodity groups – e.g. food, housing, clothing – conditional on income and commodity group price indices  Consumer subsequently allocates commodity group expenditures among individual products – e.g. food expenditures spent on steak, beer, pizza, … Empirically convenient restriction often used in demand system estimation

PROPOSITION  ‘Macro’ stage sorting involves income allocation to broad non- housing (C) and housing (Q) consumption groups  ‘Micro’ stage sorting involves allocation of Q expenditures to house structure (H) and local public goods (X)  Utility function (1) satisfies conditions for this division Proof in paper – relies on establishing ‘price aggregation’ and ‘decentralisability’ (Blackorby and Russell, 1997)

TWO STAGE SORTING MODEL Consistency of (1) with two-stage budgeting allows us to write (2) as: ln ln ln ln , 1,...,

im I i m im Y m H m m im

V I Y j M               where

| |

max{ ln ln }

m

H jm X jm jm i i j m m j J ijm im ij m

X        



       

i m

 is a quality-adjust price (or utility) index for the second stage of sorting

EMPIRICAL BASIS

Start with

 

ln ln ln ln ln ln 1,..., 1,...,

k k ijm I im Y m X jm H m jm jm m ijm m

V I Y X j J m M                    Assume GEV distribution for

1 2

11 1, 21 2 1 ,...,

,..., ,..., ,..., .

M

i i iJ i iJ i M iJ M

            ‘Nested Logit’ specification:  There are M ‘nests’ corresponding to MSAs  There are Jm alternatives in each nest corresponding to neighborhood within the MSA  Household type specific heterogeneity (k=1,…,K)

PROPERTIES OF NESTED LOGIT MODEL Probability of observing (m,j) = Prijm=Prij|m×Prim:

   

1

exp ln ln ln Pr exp ln ln ln

I im Y m H m m im M I in Y n H n n k k m k k n n

I Y I Y IV IV            



        



       

1 | 1

exp exp exp ln / Pr , exp ln /

m m

k k k H jm X jm jm ij m J k k k H jl X lm l k jm J k lm m l l

X X            

 

               

 

 

 

| 1

max{ ln ln ln exp ( } )

m m

k k H jm X jm jm ij m j J J k k i m jm m j k m

V E I E X      

 

         



TWO STAGE BUDGETING/TWO-STAGE SORTING/NESTED LOGIT PUNCH LINE  Literature-standard functional form provides useful structure for sequential estimation  Use micro-sorting data to estimate inclusive value for lower nest ↔ (expectation of) price index in first stage budget allocation  Estimate ‘dissimilarity’ coefficient k via macro-choice conditional logit model ↔ reflects degree of importance of micro-choice set in macro decisions Links upper and lower levels of nested logit to stages of budgeting (though two stage budgeting holds for any error distribution)

APPLICATION AND DATA

Empirical objective follows Bayer et al (2009):  Measure marginal WTP for regionally-varying air pollution (PM10)  Estimate a macro sorting model across 226 MSAs in continental US Data needs:  Neighborhood level residential locations – Census Data Research Center (confidential)  MSA level residential location – IPUMS (public use census micro data)  PM10 emissions – EPA National Emissions Inventory (converted to concentrations using source/receptor matrix)  Other MSA level variables – various sources

CENSUS DATA Confidential micro data:  8.5 million people 1990 and 7.6 million 2000  Observe 23 to 40 year olds locating in 40,416 census tracts  Divide out by household types Public use micro data (IPUMS):  39,058 1990 and 37,165 2000  23 – 40 year olds sorting across 226 MSAs  Observe income, education, household makeup, etc. Use IPUMS data to predict MSA-level housing prices and (potential) household income at all M locations

HOUSEHOLD TYPES

Type Definition (presence of children and education) Type 1 No children in household, No high school degree Type 2 No children in household, High school degree or some college Type 3 No children in household, Bachelor's degree Type 4 No children in household, Graduate or professional degree Type 5 Children in household, No high school degree Type 6 Children in household, High school degree or some college Type 7 Children in household, Bachelor's degree Type 8 Children in household, Graduate or professional degree

EMISSIONS DATA Obtained 1990 and 2000 PM and SO2 emissions from EPA National Emissions Inventory  5903 ‘sources’ identified – i.e. ground level, stacks of different heights  3080 ‘receptors’ identified – correspond to counties Use EPA ‘input-output’ matrix to determine PM10 concentrations at counties in 1990 and 2000 (average to get MSA predictions):  34.56 (18.08) g/m3 1990  29.74.56 (15.34) g/m3 2000  Significant spatial and temporal variability

ESTIMATION AND RESULTS

Estimate the parameters of

1

ln ln , ~

k k im I im m im i m m im

MC V I EV            (S1) where ln , 1,. ., ln .

H m Y m m m

Y m M          (S2) Use ‘BLP’ method:  Estimate (S1) using logit model  Decompose m in (S2) using linear regression  Account for endogeneity of PM10 and m following Bayer et al.

MOVING COSTS We use two specifications for MCim: Bayer et al

bs br im bs i br i

MC D D     Heterogeneous ( ) ( )

bs br bs br im bs i br i bs i i br i i c c

MC D D D D D D           Note:  Difference is that ‘migration costs’ vary with presence of children in latter

COMPUTING THE PRICE INDEX We need

   

| 1

exp Pr exp

m

k jm k ij m J k lm l

 



  … …to compute

 

1

ln exp , 1,..., , 1,...,

m

J k k m jm j k m

IV k K m M 



    



Computation of

k m

 price index requires:  Use of micro level locations to compute ’s  A normalization that allows

k m

 to be compared across all MSA

“Effects Coding”

1

M

J k jm j









 Interpret

k jm

 as deviation from MSA-level grand mean  Normalization implies 1 ln

k jm k jm k q j m qm

s J s 



      





k jm

s  share of type k people selecting neighborhood j in MSA m

INTERPRETING THE PRICE INDEX Note for 1/

k jm m

s J  : 

k jm

  

 

1

ln exp ln

m

J k k m jm m j

IV J 



 



Heterogeneity in shares → variability in local attributes → higher index → more effective choice options → higher utility

Variable Description Mean

• Std. Dev.

Mean

• Std. Dev.

Mean

• Std. Dev.

ln(ρ) log price of housing services 8.146 0.304 8.448 0.258 0.302 0.139 Γ1 MSA Index: Type 1 5.730 1.099 5.974 1.165 0.244 0.409 Γ2 MSA Index: Type 2 5.583 1.266 5.443 1.315

• 0.139

0.681 Γ3 MSA Index: Type 3 5.911 1.067 5.925 1.101 0.014 0.364 Γ4 MSA Index: Type 4 5.522 1.045 5.747 1.066 0.225 0.270 Γ5 MSA Index: Type 5 5.390 1.014 5.743 1.037 0.353 0.298 Γ6 MSA Index: Type 6 5.476 1.169 5.312 1.260

• 0.164

0.585 Γ7 MSA Index: Type 7 5.658 1.116 5.892 1.229 0.234 0.440 Γ8 MSA Index: Type 8 5.278 1.091 5.738 1.135 0.460 0.310 1990 2000 Change

SUMMARY OF PRICES AND PRICE INDICES

RESULTS First BLP step:

Variable Parameter Coef. t-stat Coef. t-stat Income βI 1.770 83.515 1.912 90.098 MC_State μbs

• 2.890
• 211.925
• 2.668
• 134.465

MC_Region μbr

• 1.363
• 96.744
• 1.182
• 58.556

MC_State * Region νbs

• 0.229
• 8.502

MC_Region * Children νbr

• 0.137
• 4.921

Model with MSA Prices 1 2 Variable Parameter Coef. t-stat Coef. t-stat Income βI 1.757 82.833 1.895 89.166 MC_State μbs

• 2.898
• 212.379
• 2.674
• 135.093

MC_Region μbr

• 1.359
• 96.474
• 1.179
• 58.466

MC_State * Region νbs

• 0.239
• 8.836

MC_Region * Children νbr

• 0.137
• 4.908

Model with Logit Index 1 1

Variable Parameter Coef. t-stat τ Coef. t-stat τ Income βI 1.555 58.776 1.673 63.330 MC_State μbs

• 2.894
• 211.887
• 2.670
• 134.273

MC_Region μbr

• 1.358
• 96.376
• 1.173
• 58.132

MC_State * Region νbs

• 0.230
• 8.512

MC_Region * Children νbr

• 0.145
• 5.200

MSA Index: Γ1 τ1 0.661 6.763 0.659 0.718 7.305 0.672 MSA Index: Γ2 τ2

• 0.043
• 1.767

0.489 0.005 0.202 0.501 MSA Index: Γ3 τ3 1.086 23.080 0.748 1.137 23.874 0.757 MSA Index: Γ4 τ4 1.871 15.360 0.867 1.955 15.302 0.876 MSA Index: Γ5 τ5 0.634 10.346 0.653 0.724 11.524 0.673 MSA Index: Γ6 τ6

• 0.242
• 11.762

0.440

• 0.191
• 9.403

0.452 MSA Index: Γ7 τ7 0.238 6.001 0.559 0.293 7.357 0.573 MSA Index: Γ8 τ8 0.707 10.867 0.670 0.771 11.614 0.684 Model with Gamma Index 1 2

Second BLP step: Bayer et al. Moving Costs ↓ Heterogeneous Moving Costs ↓

Price Only Restricted Index Unrestricted Index Δ ln(PM10)

• 0.5420**
• 0.6757**
• 1.1079***

(0.2575) (0.3373) (0.3935) Dependent Variable: Δθ + .25Δln(ρ) Price Only Restricted Index Unrestricted Index Δ ln(PM10)

• 0.4063
• 0.5527
• 0.8185**

(0.2598) (0.3411) (0.3533) Dependent Variable: Δθ + .25Δln(ρ)

Note:  Elasticity of WTP 0.31 Bayer et al. model  Elasticity of WTP 0.48 preferred micro/macro model An omitted variable story: ln , 1,. ., ln .

H m m Y m m

Y m M           In price-only model more variability ‘goes into’ m…  …and resides in m Correlation between (change in) PM10 and (change in) characteristics

• f the micro choice sets determines existence/sign of bias

MARGINAL WILLINGNESS TO PAY ESTIMATES

No MC Interaction With MC Interaction Baseline Model $232.22 $161.14 Restricted Index Model $291.62 $221.16 Unrestricted Index Model $540.08 $371.05

CONCLUSIONS

We find evidence that characteristics of neighborhood choice set can affect MSA-level choices (and quantification of tradeoffs)  operates as an omitted variable not addressed by standard instrument  we isolate using a structural approach + additional data on micro level sorting  conceptual two stage budgeting framework integrated with practical nested logit specification Follow up questions:  How does macro choice affect micro-level valuation?  A non-structural approach to addressing omitted variable bias?

And now?  Polishing paper and submission (where – an environment or urban journal?)  Spatially explicit cost of living indices  Application to ‘quality adjust income’ distribution

QUESTIONS? COMMENTS?