Welfare gains of the poor: An endogenous Bayesian approach ... · Seminario Institucional: Escuela...

Welfare gains of the poor:An endogenous Bayesian approach with

spatial random effects

Andres Ramırez HassanUniversidad EAFIT

Santiago Montoya BlandonUniversidad EAFIT

Seminario Institucional: Escuela de EstadısticaUniversidad Nacional sede Medellın

February 16, 2015

Introduction Energy Market Equivalent Variation Econometric Approach Results Conclusions References

Outline

1 Introduction

2 Energy Market

3 Equivalent Variation

4 Econometric Approach

5 ResultsDataDemand EstimationWelfare Gains

6 Conclusions

3 / 60


Outline

1 Introduction

2 Energy Market




6 Conclusions

4 / 60


Novelty

Bayesian simultaneous equations system with spatialrandom effects suited to handle spatial dependence,heterogeneity, endogeneity and statistical inferenceassociated with complicated non-linear functions ofparameter estimates.Full conditional posterior distributions available along withGibbs sampler implementation.

5 / 60


Application

Evaluate welfare implications on poor households, measuredthrough the Equivalent Variation, caused by the electricity pricechanges in the province of Antioquia (Colombia), after EPMacquired EADE in 2006. Spatial effects, endogeneity betweenprice and electricity demand, unobserved heterogeneity andstatistical inference on the Equivalent Variation are allconsidered. Important motivation for the application: electricityservices represent a significant share of households’ budget,and this fact is prominent on the poor population (Gomez-Lobo,1996, Ruijs, 2009, You and Lim, 2013).

6 / 60


Methodology

Spatial effects to perform statistical inference based oncross-sectional areal data have a long tradition in spatialstatistics (Cressie, 1993, Ripley, 2004, Anselin, 1988).Mostly Frequentist with some noteworthy exceptionsfounded on Bayesian methods (LeSage, 1997, 2000, Parentand LeSage, 2008, LeSage and Pace, 2009, LeSage andLlano, 2013).Endogeneity has been treated from the spatial perspective(SAR), and recently from the regressors (Rey and Boarnet,2004, Kelejian and Prucha, 2004, Fingleton and Gallo,2008, Crukker et al., 2013, Liu and Lee, 2013).

7 / 60


Methodology

Unobserved heterogeneity has been tackled using aBayesian approach. References involve simultaneouslyspatial effects and unobserved heterogeneity, but they donot take into consideration recursive endogeneity (LeSage,2000, Smith and LeSage, 2004, LeSage et al., 2007, Parentand LeSage, 2008, Seya et al., 2012, LeSage and Llano,2013).

8 / 60


Applications

Microeconomic foundation has been developed to greatextent but not completely using a logarithmic demandfunction in an increasing-block pricing scheme.Welfare implications based on estimates that account forspatial characteristics, endogeneity and unobservedheterogeneity are scarce in literature.Acton and Mitchell (1983), Gomez-Lobo (1996), Dodonovet al. (2004), Lundgren (2009), Ruijs (2009), You and Lim(2013) examine welfare implications from price changes inutilities. All impacts on lowest income groups are found tobe considerable, even those associated with small changes.

9 / 60


Outline

1 Introduction

2 Energy Market




6 Conclusions

10 / 60


Characteristics

General characteristics of the Colombian energy market that arekey to understanding the price variations we seek to analyze:

Division by socioeconomic strata.Regulator establishes subsidized subsistence electricityconsumption for strata one, two and three. Subsistenceconsumption differs depending on elevation.Each company has a reference tariff over its market.Generation, transport at country level, distribution at marketlevel and commercialization define tariffs. Same referencetariff is not equal to same average prices for strata andmunicipalities.

11 / 60


Tariff unification process

EPM acquires EADE in 2006.EPM serves an urban market with large population density,EADE attended rural areas with low population density.First: low tariffs for EPM’s market, high tariffs for EADE’smarket. Now: huge decreases in tariffs for rural areas (up to17.53%), slight increase on urban areas.Welfare implications are considerable for rural municipalitiesand poorest households.

12 / 60


Outline

1 Introduction

2 Energy Market




6 Conclusions

13 / 60


Equivalent Variation

There is Consumer Surplus (CS), Compensating Variation(CV) and Equivalent Variation (EV). Which one to pick?CS leaves income effects out but they are implied in theproblem. CV does not rank alternatives well if preferencesare not homothetic and income changes. EV encompassesboth.Increasing-block prices create non-linear budget constraints(see Figure 1)

14 / 60



Figure 1: Example of Equivalent Variation with price changes on bothtiers, and new and virtual consumption on the second tier

x xe x1

y0

e(p0, u1)

e(p20, u1)

EV

slope p10

slope p11

slope p20

slope p21

u1

x

xa

15 / 60



From Varian (1992), EV can be expressed in terms ofexpenditure functions as

EV (p0,p1, y0) = e(p0,u1)− e(p1,u1) = e(p0,u1)− y0 (1)

Subscripts represent time, superscripts the block, x subsistenceconsumption. We consider two cases:

(i) u1 = V ((p11,1), y0)

(ii) u1 = V ((p21,1), y0 + (p2

1 − p11)x)

16 / 60



For (i):EV (p0,p1, y0) = e(p1

0,u1)− y0 (2)

For (ii):

EV (p0,p1, y0) = e(p20,u1)− (p2

0 − p10)x − y0 (3)

Hausman (1981) developed a method that relates theeconometric specification of a Marshallian demand function tothe definitions stated above. We model the demand as:

x(p, y) = pαy δ1ez′δ (4)

17 / 60



The method shown by Hausman (1981) leads to the followingexpressions for the EV. For (i):

EV (p0,p1, y0) =

[1− δ1

1 + α

(p1(1+α)

0 − p1(1+α)1

)ez′δ + y1−δ1

0

] 11−δ1−y0

(5)For (ii):

EV (p0,p1, y0) =

[1− δ1

1 + α

(p2(1+α)

0 − p2(1+α)1

)ez′δ+

(y0 + (p2

1 − p11)x)1−δ1

] 11−δ1 − (p2

0 − p10)x − y0

(6)

18 / 60


Outline

1 Introduction

2 Energy Market




6 Conclusions

19 / 60


Model

We propose an endogenous Bayesian approach usingsimultaneous equations with spatial random effects to estimatea model facing recursive endogeneity (as an instrumentalvariable problem). The specification of our model is:

ln xi = z′1iδ + α ln pi + u1i (7)ln pi = z′1iφ + αsz2i + u2i (8)

where u1i and u2i are the idiosyncratic error terms associatedwith the demand and price of each municipality, respectively.Following Greenberg (2008), we assume that(u1i ,u2i)

′ ∼ N (0,Σ), Σ = σij, such that σ12 captures theendogeneity of the system.

20 / 60


Model

The previous model can be estimated from the reduced formequations:

ln xi = π0 + π1 ln yi + π2 ln psi + π3alti + π4 ln urbi + π5EADEi+

µ1i + vi

ln pi = φ0 + φ1 ln yi + φ2 ln psi + φ3alti + φ4 ln urbi + αsEADEi+

µ2i + vi

(9)

21 / 60


Model

Where µ1i = u1i + αu2i and µ2i = u2i , such that(µ1i , µ2i)

′ ∼ N (0,Ω),

Ω =

[σ2

1 + α2σ22 + 2ασ12 σ12 + ασ2

2σ12 + ασ2

2 σ22

]=

[ω11 ω12

ω12 ω22

]and vi captures the spatial interactions between neighboringmunicipalities.

22 / 60


Likelihood

Setting z′i = (ln yi , ln psi ,alti , ln urbi ,EADEi),

π′ = (π1, π2, π3, π4, π5), φ′ = (φ1, φ2, φ3, φ4, αs) andv′ = (v1, v2, . . . , vn).The likelihood function, f(ln x , ln p|Ω,π,φ, π0, φ0,v), is :

|Ω|−N/2

(2π)N/2exp

−12

N∑i=1

(ln xi − z′iπ − π0 − vi , ln pi − z′iφ− φ0 − vi )Ω−1(

ln xi − z′iπ − π0 − viln pi − z′iφ− φ0 − vi

)

23 / 60


Prior: CAR Process

We assume the spatial random effect follows as prior distributionan improper (intrinsic) Conditionally Autoregressive (CAR)structure (Besag et al., 1991):

vi |vi∼j ∼ N

∑i∼j

wijvj∑i∼j wij

,σ2

v∑i∼j wij

The joint distribution of the improper CAR isv ∼ N (v, σ2

v (IN −Wn)−1) where Wn is the contiguity matrix (Wall,2004).

24 / 60


Prior: CAR Process

Why CAR over SAR? Literature can be found in favor of eachone (Banerjee et al., 2004, Parent and LeSage, 2008, Darmofal,2009, Chakraborty et al., 2013, for the CAR) and (Smith andLeSage, 2004, LeSage et al., 2007, LeSage and Llano, 2013,for the SAR). Our decision is based on:

Inherent heteroscedasticity and therefore higher levels ofheterogeneity (Cressie, 1993).Markovian characteristics in space. Spatial heterogeneity isdue to local variation, rather than a global spatial pattern, asin the SAR (Anselin, 2003).Computational convenience and intuitiveness (Banerjeeet al., 2004).

25 / 60


Prior: Location Parameters

To complete our Bayesian specification, we set the other priorsas follow: the elasticities and semi-elasticities have aMultivariate Normal distributions with mean vector 0 andprecision matrices 0.001I5. This implies vague prior informationwhere there is no effect of each control variable on dependentvariables.

26 / 60


Prior: Scale Matrix

In addition, we assume a Wishart distribution for Ω−1 with 3degrees of freedom and scale matrix I2. Setting the degrees offreedom to p + 1, where p is the dimension of the covariancematrix, the Wishart form reduces to π(Ω−1) ∝ |Ω−1|−(N+1)/2,which is a diffuse prior used by Savage that emerges usingJeffrey’s invariance theory (Zellner, 1996). Thus, a priori there isno endogeneity, and the fat-tailed prior will guaranteerobustness of outcomes regarding this prior distribution (Berger,1985).

27 / 60


Prior: Scale parameter

Finally, 1/σ2vi

, which is the precision parameter of the CARcomponent, has a prior Gamma distribution with shape andscale parameters equal to 0.5 and 0.0005, respectively. Thisparametrization is frequently found in literature (Kelsall andWakefield, 1999, Thomas et al., 2007), and expresses the priorbelief that the standard deviation for the spatial random effectsis centered around 0.05 with a 1% prior probability of beingsmaller than 0.01 or larger than 2.5.

28 / 60


Prior: Scale parameter

Those specific values of the hyperparameters are inspired in thefact that we have two different sources of stochastic variability inour model, µil , l = 1,2 and (vi). As a consequence, both setsof hyperparameters of the prior distributions of these randomeffects cannot imply arbitrarily large variability, since theseeffects would be unidentifiable; we try to identify two randomeffects using a single observation at each spatial unit. Thus, it isnecessary to fix the hyperparameters of the Gamma distribution,given that we use a diffuse prior distribution for Ω−1 (Banerjeeet al., 2004).

29 / 60


Posterior: Scale Matrix

π(Ω|π,φ, π0, φ0, v, σ2v ,Data) ∼ Wκ(ω, Ω), where ω = ω + N and

Ω =

Ω−1 +N∑

i=1

(ln xi − z′iπ − π0 − viln pi − z′iφ− φ0 − vi

)(ln xi − z′iπ − π0 − vi , ln pi − z′iφ− φ0 − vi )

−1

30 / 60


Posterior: Location parameters demandequation

To sample π, we use f (ln xi , ln pi |Θ) = f (ln xi | ln pi ,Θ)f (ln pi |Θ)where Θ = (Ω,π,φ, π0, φ0,v).In particular,ln xi | ln pi ,Θ ∼ N (z′iπ + π0 + vi + ω12

ω22(ln pi − z′iφ− φ0 − vi), ψ11)

where ψ11 = ω11 −ω2

12ω22

. In addition, the prior is Nk (π,Π).

31 / 60


Posterior: Location parameters demandequation

Then, π(π|Ω,φ, π0, φ0,v, σ2v ,Data) ∼ Nk (π, Π) where

Π =[

Π−1 + ψ−111

∑Ni=1 ziz′i

]−1and

π = Π(

Π−1π + ψ−111∑N

i=1 zi (ln xi −ω12ω22

(ln pi − z′iφ− φ0 − vi )− π0 − vi ))

.

32 / 60


Posterior: Location parameters priceequation

We follow the same procedure to deduce the conditionalposterior distribution of φ, that is, we usef (ln xi , ln pi |Θ) = f (ln pi | ln xi ,Θ)f (ln xi |Θ).In particular,ln pi | ln xi ,Θ ∼ N (z′iφ + φ0 + vi + ω12

ω11(ln xi − z′iπ − π0 − vi), ψ22)

where ψ22 = ω22 −ω2

12ω11

. Additionally, the prior is Nk (φ,Φ).

33 / 60


Posterior: Location parameters priceequation

Thus, π(φ|π,Ω, π0, φ0,v, σ2v ,Data) ∼ Nk (φ, Φ) where

Φ =[

Φ−1 + ψ−122

∑Ni=1 ziz′i

]−1and

φ = Φ(

Φ−1φ+ ψ−122∑N

i=1 zi (ln pi −ω12ω11

(ln xi − z′iπ − π0 − vi )− φ0 − vi ))

.

34 / 60


Posterior: constant demand equation

Regarding the posterior distribution of the constant term π0,using as prior an improper uniform distribution and givenf (ln xi , ln pi |Θ) = f (ln xi | ln pi ,Θ)f (ln pi |Θ), we obtain thatπ0 ∼ N (π0, ψ11) whereπ0 = ln xi − ω12

ω22(ln pi − z′iφ− φ0 − vi)− vi − z′iπ.

35 / 60


Posterior: constant price equation

In a similar way, using as prior an improper uniform distributionfor φ0, and the fact thatf (ln xi , ln pi |Θ) = f (ln pi | ln xi ,Θ)f (ln xi |Θ), we obtain thatφ0 ∼ N (φ0, ψ22) whereφ0 = ln pi − ω12

ω11(ln xi − z′iπ − π0 − vi)− vi − z′iφ.

36 / 60


Posterior: spatial effects

Using the fact that f (ln xi , ln pi |Θ) = f (ln pi | ln xi ,Θ)f (ln xi |Θ) towrite the likelihood function and the improper CAR formulation,π(vi |v−i ,π,Ω, π0, φ0, σ

2v ,Data) is N (ξ, η) where

η =

[(σ2

vwi+

)−1+ N

(ψ22(

ω12ω11

+1))−1

]−1

and ξ equal to

η

( σ2v

wi+

)−1 ∑i∼j

(wij

wi+

)vj +

ψ22(ω12ω11

+ 1)2

−1

N∑i=1

1ω12ω11

+ 1

(ln pi −ω12

ω11(ln xi − z′i π − π0)− φ0 − z′i φ)

37 / 60


Posterior: scale parameter

Given that 1/σ2v has as a prior G(α, β), its conditional posterior is

G(α, β) where α = α + 1/2 and

β = β +(wi+

2

) (vi −

∑i∼j

(wijwi+

)vj

)2.

38 / 60


Outline

1 Introduction

2 Energy Market




6 Conclusions

39 / 60


Descriptive Statistics

Table 1: Descriptive Statistics

Variable Mean Std. Dev. Min MaxCons. (kWh) 234.874 117.811 26.595 588.937Price (US$) 0.061 0.024 0.027 0.240

Income (US$) 397.085 95.242 230.514 619.227Subs. Price (US$) 0.030 0.006 0.016 0.056

Altitude 29.032% 45.575% 0.000 1.000Urbanization 45.876% 19.917% 10.700% 98.247%

Coverage 77.419% 41.981% 0.000 1.000

Source: Author’s calculations

40 / 60


Figure 2: Average Annual Electricity Consumption per Household(kWh): Province of Antioquia (Colombia) in 2005, Stratum One

[26.6, 142.7)[142.7, 192.4)[192.4, 238.2)[238.2, 305.7)[305.7, 588.9]

41 / 60


Model Estimation

Use of MCMC techniques, specifically Gibbs sampleralgorithms. 10 million updates, 5 million discarded anddraws ever 500.Robustness checks with different contiguity criteria: queen,rook and road length. Base model is road length, due to itseconomic sensibleness.Statistical inference can be performed and hypotheses canbe tested thanks to the Bayesian approach.Convergence diagnostics show convergence of the chainsand therefore validate the results.Global and local spatial dependence can be observed andis significant.

42 / 60


Table 2: Global spatial correlation tests

Road Length ContiguityValue Moran I Geary C

Statistic -0.00032 0.82708P-Value 0.02825 0.00098

Queen ContiguityValue Moran I Geary C

Statistic 0.37807 0.50262P-Value 1.50e-13 2.07e-09

Rook ContiguityValue Moran I Geary C

Statistic 0.37717 0.50747P-Value 3.15e-13 2.43e-09Source: Author’s calculations

43 / 60


Figure 3: Local Moran’s I test p-valuesp > 0.1p < 0.1p < 0.05p < 0.01

44 / 60


Estimation Results

Table 3: Summary of structural parameter posterior estimates

Road Length Contiguity

Parameter Mean Median 90% HPD Interval R01 = P(θ∈(0,∞))P(θ∈(−∞,0]))Lower Upper

Constant 1.913 1.940 -1.539 5.393 4.663Price -0.886 -0.882 -1.449 -0.278 0.014

Income 0.301 0.297 -0.054 0.636 11.920Subs. Price 0.123 0.120 -0.215 0.449 2.560

Altitude 0.139 0.137 -0.041 0.304 9.235Urbanization 0.571 0.566 0.410 0.724 908.090Source: Author’s calculations

45 / 60


Convergence Diagnostics

Table 4: Stationarity and Convergence diagnostics


Parameter Heidelberger Heidelberger Gewekec Rafteryd(1st Part/p-value)a (2nd Part)b

Constant 0.887 0.064 0.758 1.46Price 0.923 -0.047 0.820 1.10

Income 0.414 0.035 -0.740 2.74Subs. Price 0.909 0.067 -0.311 1.11

Altitude 0.581 0.052 -0.515 1.08Urbanization 0.871 0.024 -0.407 1.05

Notes: a Null hypothesis is stationarity of the chain, b Half-width to meanratio, c Mean difference test z-score, d Dependence factor

46 / 60


Main Results

Table 5: Equivalent Variation as share of income by Total,Metropolitan Area and Rest

Road Length ContiguityEquivalent Mean Median 90% HPD IntervalVariation Lower UpperM. Area 0.126% 0.141% 0.006% 0.209%

Rest 0.940% 0.655% 0.257% 2.006%Total 0.874% 0.630% 0.005% 1.913%


47 / 60


Main Results

Table 6: Equivalent Variation as share of income for representativemunicipalities


Municipality Mean Median 90% HPD IntervalLower Upper

Medellın 0.137% 0.139% 0.081% 0.194%Barbosa 0.008% 0.008% 0.006% 0.009%

San Jose de la Montana 0.313% 0.313% 0.307% 0.319%Santa Fe de Antioquia 2.297% 2.297% 1.956% 2.619%


Low income households expend on average 1.13%, 1.74% and4.65% of their income in pension, health care and education.

48 / 60


Main Results

Figure 4: Equivalent Variation Posterior Distribution

0 1 2 3 4

0.0

0.5

1.0

1.5

Den

sity

49 / 60


Main Results

Figure 5: Equivalent Variation Posterior Distribution[0%, 0.4%)[0.4%, 0.6%)[0.6%, 0.7%)[0.7%, 1.4%)[1.4%, 3.3%]

50 / 60


Outline

1 Introduction

2 Energy Market




6 Conclusions

51 / 60


Conclusions

Introduced spatial random effects into an endogenousBayesian framework with simultaneous equations anddeduced the complete conditional posterior distributions.Controlled for spatial dependence, endogenous regressors,weak instruments and unobservable heterogeneitysimultaneously.Estimated a system of equations for electricity and find theaverage price, income, substitute and urbanization ratedemand elasticities to be -0.88, 0.30, 0.12 and 0.57,respectively.

52 / 60


Conclusions

Estimated the Equivalent Variation for the total of theprovince to be 0.87% in average and 0.63% in median.The welfare gains of the Metropolitan Area amount only to0.13%.Municipalities that are not part of the Metropolitan Areagained in average 0.94%The less urban and poorest municipalities, increased theirwelfare well above 2% of the initial income.The complete analysis would be extremely difficult tohandle, were it not for the Bayesian approach that allowedus great flexibility and structure.

53 / 60


References I

Acton, J. P. and Mitchell, B. M. (1983). Welfare analysis of electricity ratechanges. RAND Corporation.

Anselin, L. (1988). Spatial econometrics: methods and models, volume 4.Springer.

Anselin, L. (2003). Spatial Externalities, Spatial Multipliers and SpatialEconometrics. International Regional Science Review, 26(2):153–166.

Banerjee, S., Carlin, B. P., and Gelfand, A. E. (2004). Hierarchical modelingand analysis for spatial data. Chapman & Hall / CRC Press.

Berger, J. O. (1985). Statistical decision theory and Bayesian analysis.Springer.

Besag, J., York, J., and Mollie, A. (1991). Bayesian image restoration, withtwo applications in spatial statistics. Annals of the Institute of StatisticalMathematics, 43(1):1–20.

54 / 60


References II

Chakraborty, A., Beamontec, M. A., Gelfand, A. E., Alonso, M. P., Gargallo,P., and Salvador, M. (2013). Spatial interaction models with individual-leveldata for explaining labor flows and developing local labor markets.Computational Statistics and Data Analysis, 58(1):292–307.

Cressie, N. (1993). Statistics for spatial data. Jhon Willey & Sons, INC,United States, revised edition.

Crukker, D. M., Egger, P., and Prucha, I. R. (2013). On two-step estimation ofa spatial autoregressive model with autoregressive disturbances andendogenous regressors. Econometric Reviews, 32(5-6):686–733.Forthcoming.

Darmofal, D. (2009). Bayesian spatial survival models for political eventprocesses. American Journal of Political Science, 53(1):241–257.

Dodonov, B., Opitz, P., and Pfaffenberger, W. (2004). How much do electricitytariff increases in Ukraine hurt the poor? Energy Policy, 32(7):855–863.

55 / 60


References III

Fingleton, B. and Gallo, J. L. (2008). Estimating spatial models withendogenous variables a spatial lag and spatially dependent disturbances:finite sample properties. Papers in Regional Science, 87(3):319–340.

Gomez-Lobo, A. (1996). The welfare consequences of tariff rebalancing inthe domestic gas market. Fiscal Studies, 17(4):49–65.

Greenberg, E. (2008). Introduction to Bayesian Econometrics. CambridgeUniversity Press.

Hausman, J. A. (1981). Exact consumer’s surplus and deadweight loss.American Economic Review, 71(4):662–676.

Kelejian, H. H. and Prucha, I. R. (2004). Estimation of simultaneous systemsof spatially interrelated cross sectional equations. Journal ofEconometrics, 118:27–50.

56 / 60


References IV

Kelsall, J. E. and Wakefield, J. C. (1999). Discussion of ’Bayesian models forspatially correlated disease and exposure data’ by best et al. In Bernardo,J. M., Berger, J. O., Dawid, A. P., and Smith, A. F. M., editors, BayesianStatistics 6: Proceedings of the Sixth Valencia International Meeting, page151. Oxford University Press.

LeSage, J. P. (1997). Bayesian estimation of spatial autoregressive models.International Regional Science Review, 20:113–119.

LeSage, J. P. (2000). Bayesian estimation of limited dependent variablespatial autoregressive models. Geographical Analysis, 32(1):19–35.

LeSage, J. P., Fischer, M. F., and Scherngell, T. (2007). Knowledge spilloversacross Europe: Evidence from a Poisson spatial interation model withspatial effects. Papers in Regional Science, 86(3):393–422.

57 / 60


References V

LeSage, J. P. and Llano, C. (2013). A spatial interaction model with spatiallystructured origin and destination effects. Journal of GeographicalSystems, 15:265–289.

LeSage, J. P. and Pace, R. K. (2009). Introduction to Spatial Econometrics.Taylor & Francis, Boca Raton.

Liu, X. and Lee, F.-L. (2013). Two stage least squares estimation of spatialautoregressive models with endogenous regressors and manyinstruments. Econometric Reviews. Forthcoming.

Lundgren, J. (2009). Consumer welfare in the deregulated Swedishelectricity market. Frontiers in Finance and Economics, 6(2):101–119.

Parent, O. and LeSage, J. P. (2008). Using the variance structure of theconditional autoregressive spatial specification to model knowledgespillovers. Journal of Applied Econometrics, 23:235–256.

58 / 60


References VI

Rey, S. J. and Boarnet, M. G. (2004). A taxonomy of spatial econometricmodels for simultaneous equation systems. In Anselin, L., Florax, R. J.,and Rey, S. J., editors, Advances in Spatial Econometrics: Methodology,Tools and Applications, pages 99–119. Springer.

Ripley, B. (2004). Spatial Statistics. Jhon Wiley & Sons, United States,second edition.

Ruijs, A. (2009). Welfare and distribution effects of water pricing policies.Environmental and Resource Economics, 43(2):161–182.

Seya, H., Tsutsumi, M., and Yamagata, Y. (2012). Income convergence inJapan: A Bayesian spatial Durbin model approach. Economic Modelling,29:60–71.

59 / 60


References VIISmith, T. E. and LeSage, J. P. (2004). A bayesian Probit model with spatial

dependences. In LeSage, J. P. and Pace, R. K., editors, Advances inEconometrics: Volume 18: Spatial and Spatiotemporal Econometrics,pages 127–160. Elsevier Ltd.

Thomas, A., Best, N., Lunn, D., Arnold, R., and Spiegelhalter, D. (2007).GeoBUGS User Manual version 1.3.

Varian, H. R. (1992). Microeconomic Analysis. W.W. Norton & Company,New York, Third edition.

Wall, M. (2004). A close look at the spatial structure implied by the CAR andSAR models. Journal of Statistical Planning and Inference, 121:311–324.

You, J. S. and Lim, S. (2013). Welfare effects of nonlinear electricity pricing.Submission for the 37th International Association for Energy EconomicsInternational Conference.

Zellner, A. (1996). An Introduction to Bayesian Inference in Econometrics.Wiley, New York, Second edition.

60 / 60

Welfare gains of the poor: An endogenous Bayesian approach ... · Seminario Institucional: Escuela...

Documents

Transcript of Welfare gains of the poor: An endogenous Bayesian approach ... · Seminario Institucional: Escuela...