Modeling Real Estate Data using Quantile Regression

Transcription

1 Modeling Real Estate Data using Semiparametric Quantile Regression Department of Statistics University of Innsbruck September 9th, 2011

2 Overview 1 Application: 2 3 4

3 Hedonic regression data for house prices in Austria Variable of primary interest House price per square meter Covariates Structural characteristics, like the floor space area, the plot area, the age, the equipment etc. Locational characteristics at different levels, like the buying power index (municipal), the share of academics (municipal), the real estate price index (district), etc.

4 Hedonic regression data for house prices in Austria Hierarchical semiparametric model p qm = f 1,1 (municipal) + f 1,2 (area) f 1,l (age) + Xβ + ɛ 1 f 1,1 (municipal) = f 2,1 (district) + f 2,2 (buying power) f 2,m (academics) + ɛ 2 f 2,1 (district) = f 3,1 (state) + f 3,2 (real estate index)+ + g(dist) + ɛ 3 f 3,1 (state) = const + ɛ 4 The term Xβ contains the linear effects. The functions f i are possibly nonlinear functions of the covariates. The function g describes a spatial district effect.

5 Hedonic regression data for house prices in Austria Goal Determining the conditional quantiles of the distribution of the house prices Approaches Mean regression based on a normal distribution assumption Quantile regression

6 Overview 1 Application: 2 3 4

7 Linear quantile regression Linear model Given: Observations y i, x i1,..., x ip for i = 1,..., n from the model y = Xβ + ɛ. Assumptions for a particular quantile ϕ: ɛ i iid F Q ϕ (ɛ i ) := F 1 (ϕ) = 0 Then: Q ϕ (y) = Xβ

8 Linear quantile regression Loss function ρ ϕ (u) = Empirical loss of an estimation ŷ : { uϕ if u 0 u(ϕ 1) if u < 0 n ρ ϕ (y i ŷ) i=1

9 Linear quantile regression Regression quantile ˆβ = arg min β R p { n } ρ ϕ (y i x i β) i=1 Estimation of the conditional quantile: Q ϕ (y) = X ˆβ

10 Linear quantile regression Example y = x + ɛ, ɛ N (0, 0.5) 5

11 Nonlinear and spatial quantile regression Nonlinear or spatial model: y = f (z) + ɛ = Z γ + ɛ Smooth effects: Penalize differences between the coefficients of adjacent B-splines or the coefficients of neighbouring regions, respectively. Penalized optimization problem { n } ( ˆγ = arg min ρ ϕ yi z i γ) + λγ K γ γ R d i=1

12 Semiparametric quantile regression Semiparametric model: y = η + ɛ = Xβ + f 1 (z 1 ) f q (z q ) + ɛ Penalized optimization problem n ρ ϕ (y i η i ) + min β,γ k i=1 q λ j γ j K jγ j j=1

13 Overview 1 Application: 2 3 4

14 Bayesian Inference Asymmetric Laplace distribution Density function: p(y µ, σ 2, ϕ) = ( ϕ(1 ϕ) σ 2 exp 1 ) σ 2 ρ ϕ(y µ)

15 Bayesian Inference Assumption: y i ALD(η i, σ 2, ϕ) Joint likelihood: ( p(y η, σ 2, ϕ) 1 (σ 2 ) n exp 1 σ 2 ) n ρ ϕ (y i η i ) i=1 Maximizing this likelihood is equivalent to minimizing the former loss function in the linear case.

16 Bayesian Inference Priors for nonlinear or spatial effects: ( ( ) p γ j τj 2 1 exp 1 ( ) rk(k j ) τj 2 τ 2 j 2 2τ 2 j γ j K jγ j ) variance parameter, governs the smoothness of the respective function.

17 Bayesian Inference Representation of an asymmetric Laplace distribution: Y = D η + 1 2ϕ ϕ(1 ϕ) V + W 2 σ 2 ϕ(1 ϕ) V V, W independent random variables with exponential and normal distributions respectively: ( p v σ 2) ( ) = σ 2 exp σ 2 v and W N (0, 1) Important feature for MCMC-inference.

18 Overview 1 Application: 2 3 4

19 Floor space area:

20 Floor space area:

21 Floor space area:

22 Real estate price index:

23 Unexplained spatial effects:

24 Model selection: Quantile Method G1... G5 10% Quantile reg Mean reg % Quantile reg Mean reg % Quantile reg Mean reg % Quantile reg Mean reg % Quantile reg Mean reg

25 Conclusion Efficient based on the ALD Individual marginal effects for each quantile Superior to mean regression

26 Thank you!