Hedonic regression data for real estate prices Quantile regression models Bayesian inference Results Modeling Real Estate Data using Semiparametric Quantile Regression Alexander Razen Department of Statistics University of Innsbruck September 9th, 2011 Alexander Razen Modeling Real Estate Data using Quantile Regression
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Hedonic regression data for real estate pricesQuantile regression models
Bayesian inferenceResults
Modeling Real Estate Data usingSemiparametric Quantile Regression
Alexander RazenDepartment of StatisticsUniversity of Innsbruck
September 9th, 2011
Alexander Razen Modeling Real Estate Data using Quantile Regression
Hedonic regression data for real estate pricesQuantile regression models
Bayesian inferenceResults
Overview
1 Application: Hedonic regression data for real estate prices
2 Quantile regression models
3 Bayesian inference
4 Results
Alexander Razen Modeling Real Estate Data using Quantile Regression
Hedonic regression data for real estate pricesQuantile regression models
Bayesian inferenceResults
Hedonic regression data for house prices in Austria
Variable of primary interestHouse price per square meter
CovariatesStructural characteristics, like the floor space area, the plotarea, the age, the equipment etc.Locational characteristics at different levels, like the buyingpower index (municipal), the share of academics(municipal), the real estate price index (district), etc.
Alexander Razen Modeling Real Estate Data using Quantile Regression
Hedonic regression data for real estate pricesQuantile regression models
Bayesian inferenceResults
Hedonic regression data for house prices in Austria
The term Xβ contains the linear effects.The functions fi are possibly nonlinear functions of thecovariates.The function g describes a spatial district effect.
Alexander Razen Modeling Real Estate Data using Quantile Regression
Hedonic regression data for real estate pricesQuantile regression models
Bayesian inferenceResults
Hedonic regression data for house prices in Austria
GoalDetermining the conditional quantiles of the distribution of thehouse prices
Approaches
Mean regression based on a normal distributionassumptionQuantile regression
Alexander Razen Modeling Real Estate Data using Quantile Regression
Hedonic regression data for real estate pricesQuantile regression models
Bayesian inferenceResults
Overview
1 Application: Hedonic regression data for real estate prices
2 Quantile regression models
3 Bayesian inference
4 Results
Alexander Razen Modeling Real Estate Data using Quantile Regression
Hedonic regression data for real estate pricesQuantile regression models
Bayesian inferenceResults
Linear quantile regression
Linear modelGiven: Observations yi , xi1, . . . , xip for i = 1, . . . ,n from themodel
y = Xβ + ε.
Assumptions for a particular quantile ϕ:
εiiid∼ F
Qϕ(εi) := F−1(ϕ) = 0
Then:Qϕ(y) = Xβ
Alexander Razen Modeling Real Estate Data using Quantile Regression
Hedonic regression data for real estate pricesQuantile regression models
Bayesian inferenceResults
Linear quantile regression
Loss function
ρϕ(u) =
{uϕ if u ≥ 0u(ϕ− 1) if u < 0
Empirical loss of an estimation y :
n∑i=1
ρϕ(yi − y)
Alexander Razen Modeling Real Estate Data using Quantile Regression
Hedonic regression data for real estate pricesQuantile regression models
Bayesian inferenceResults
Linear quantile regression
Regression quantile
β = arg minβ∈Rp
{n∑
i=1
ρϕ(yi − x ′iβ)
}
Estimation of the conditional quantile:
Qϕ(y) = X β
Alexander Razen Modeling Real Estate Data using Quantile Regression
Hedonic regression data for real estate pricesQuantile regression models
Bayesian inferenceResults
Linear quantile regression
Example
y = 1 +35
x + ε, ε ∼ N (0,0.5)
Alexander Razen Modeling Real Estate Data using Quantile Regression
Hedonic regression data for real estate pricesQuantile regression models
Bayesian inferenceResults
Nonlinear and spatial quantile regression
Nonlinear or spatial model:
y = f (z) + ε = Zγ + ε
Smooth effects: Penalize differences between the coefficientsof adjacent B-splines or the coefficients of neighbouringregions, respectively.
Penalized optimization problem
γ = arg minγ∈Rd
{n∑
i=1
ρϕ(yi − z ′iγ
)+ λγ ′Kγ
}
Alexander Razen Modeling Real Estate Data using Quantile Regression
Hedonic regression data for real estate pricesQuantile regression models
Bayesian inferenceResults
Semiparametric quantile regression
Semiparametric model:
y = η + ε = Xβ + f1(z1) + . . .+ fq(zq) + ε
Penalized optimization problem
minβ,γk
n∑
i=1
ρϕ (yi − ηi) +
q∑j=1
λjγ′jK jγ j
Alexander Razen Modeling Real Estate Data using Quantile Regression
Hedonic regression data for real estate pricesQuantile regression models
Bayesian inferenceResults
Overview
1 Application: Hedonic regression data for real estate prices
2 Quantile regression models
3 Bayesian inference
4 Results
Alexander Razen Modeling Real Estate Data using Quantile Regression
Hedonic regression data for real estate pricesQuantile regression models
Bayesian inferenceResults
Bayesian Inference
Asymmetric Laplace distributionDensity function:
p(y |µ, σ2, ϕ) =ϕ(1− ϕ)
σ2 exp(− 1σ2 ρϕ(y − µ)
)
Alexander Razen Modeling Real Estate Data using Quantile Regression
Hedonic regression data for real estate pricesQuantile regression models
Bayesian inferenceResults
Bayesian Inference
Assumption:yi ∼ ALD(ηi , σ
2, ϕ)
Joint likelihood:
p(y |η, σ2, ϕ) ∝ 1(σ2)n exp
(− 1σ2
n∑i=1
ρϕ(yi − ηi)
)
Maximizing this likelihood is equivalent to minimizing the formerloss function in the linear case.
Alexander Razen Modeling Real Estate Data using Quantile Regression
Hedonic regression data for real estate pricesQuantile regression models
Bayesian inferenceResults
Bayesian Inference
Priors for nonlinear or spatial effects:
p(γ j |τ2
j
)∝ 1(
τ2j
) rk(K j )2
exp
(− 1
2τ2jγ ′jK jγ j
)
τ2j variance parameter, governs the smoothness of the
respective function.
Alexander Razen Modeling Real Estate Data using Quantile Regression
Hedonic regression data for real estate pricesQuantile regression models
Bayesian inferenceResults
Bayesian Inference
Representation of an asymmetric Laplace distribution:
Y D= η +
1− 2ϕϕ(1− ϕ)
V + W
√2
σ2ϕ(1− ϕ)V
V ,W independent random variables with exponential andnormal distributions respectively:
p(
v |σ2)= σ2exp
(σ2v
)and W ∼ N (0,1)
Important feature for MCMC-inference.
Alexander Razen Modeling Real Estate Data using Quantile Regression
Hedonic regression data for real estate pricesQuantile regression models
Bayesian inferenceResults
Overview
1 Application: Hedonic regression data for real estate prices
2 Quantile regression models
3 Bayesian inference
4 Results
Alexander Razen Modeling Real Estate Data using Quantile Regression
Hedonic regression data for real estate pricesQuantile regression models
Bayesian inferenceResults
Results
Floor space area:
Alexander Razen Modeling Real Estate Data using Quantile Regression
Hedonic regression data for real estate pricesQuantile regression models
Bayesian inferenceResults
Results
Floor space area:
Alexander Razen Modeling Real Estate Data using Quantile Regression
Hedonic regression data for real estate pricesQuantile regression models
Bayesian inferenceResults
Results
Floor space area:
Alexander Razen Modeling Real Estate Data using Quantile Regression
Hedonic regression data for real estate pricesQuantile regression models
Bayesian inferenceResults
Results
Real estate price index:
Alexander Razen Modeling Real Estate Data using Quantile Regression
Hedonic regression data for real estate pricesQuantile regression models
Bayesian inferenceResults
Results
Unexplained spatial effects:
Alexander Razen Modeling Real Estate Data using Quantile Regression
Hedonic regression data for real estate pricesQuantile regression models