Hierarchical trend model

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Hierarchical Trend Model

Marc K. Francke

mfrancke@ortax.nl

1_{Department of Econometrics}

Vrije Universiteit Amsterdam

2_OrtaX

Amsterdam

Advances in mass appraisal methods

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Outline

1 State-Space Model

2 _{The Hierarchical Trend Model}

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State-Space Model

The linear Gaussian State-Space Model:

yt = Ztαt+ εt, εt ∼ N(0, σ2Ht), αt+1 = Ttαt+Rtηt, ηt ∼ N(0, σ2Qt), for t=1, . . . ,T , initialized by α1 = a0+A0β +R0η0, η0 ∼ N(0, σ2Q0), β ∼ N(β0, σ2Σ), yt is an(nt×1)vector of observations

αt is an unobserved state vector

εt, ηt,andβ are uncorrelated

The matrices Zt, Tt, Ht,Rt and Qt are called system

matrices; they may depend on unknown parameters

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Examples of State-Space Models

Simple State-Space Model

yt = Ztαt + εt, εt ∼ N(0, σ2I), αt+1 = αt + ηt, ηt ∼ N(0, σ2ηI),

Forση =0, this is a regression model yt =Ztα + εt

Forση >0, time-varying coefficients

When Zt =1, then for

ση= ∞, a dummy variable model

ση=0,αt is constant in time

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State-Space Model and the Kalman filter

State-Space models can efficiently be estimated by the Kalman filter

Estimates of the state-vector, conditional on the parameters in the system matrices

The likelihood function for the parameters in the system matrices

Extensions

Non-Gaussian disturbances, for example t-distribution Nonlinear relations

yt = Zt(αt) + εt, αt+1 = Tt(αt) +Rtηt,

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Outline

1 State-Space Model

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Market Segments

Dependent variable yt : ln(Selling Prices) at time t

A priori classification ofmarket segments(MS):

Clusters ofneighbourhoods(districts) Clusters ofhouse types

Example for the Heerlen case

House type Description Cluster of House Type

1111 Detached house 1

1113 Detached bungalow 1

1115 Detached converted farmhouse 1

1121 Semi-detached house 1

1123 Semi-detached bungalow 1

1125 Semi-detached converted farmhouse 1 1128 Semid-detached drive-in house 1

1131 Row house 2 1133 Row bungalow 2 1138 Row drive-in 2 1141 Corner house 2 1143 Corner bungalow 2 1171 Linked house 1 1173 Linked bungalow 1

1175 Linked converted farmhouse 1

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Price development

In the same MS houses have the same price development

The trend for MSjk is the sum of

Generaltrend:µt

Districttrend:ϑj,t (in deviation from the general trend)

House type clustertrend:λ_k,t (” ”)

Houses in the same MS have the same price development (µt+ ϑj,t+ λk,t)

Time index t is in months

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Time Trends

General Trend: local linear trend model

µt+1= µt + κt + ηt, ηt ∼N(0, σ2µI) κt+1= κt + ζt, ζt ∼N(0, σ2κI)

Cluster Trends are modeled as random walks

λt+1= λt + ςt, ςt ∼N(0, σ2λI) ϑt+1= ϑt + ωt, ωt ∼N(0, σϑ2I)

σ2_η σ2_ζ Model

∞ Dummy variable model 0 0 Deterministic linear trend

0 Random walk with drift >0 > 0 Level and slope may vary over time

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Outline

1 State-Space Model

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Explanatory variables (X

t

)

Linear specification makes no sense:

y = β1ln x1+ β2ln x2+ α + ,

Y =xβ1

1 ×x

β2

2 ×exp(α + ),

Internal floor space: x1; Lot size: x2

Multiplication of internal floor space and lot size

Nonlinear specification, like for example

y =ln(xβ1 1 exp[z 0_{δ] + β} 2x2) + α + , Y = (xβ1 1 exp[z 0 δ] + β2x2) ×exp(α + ), where z contains variables regarding the internal floor

space x1, like year of building and maintenance

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Internal floor space and Lot size

Forβ1<1, Y is less than proportional to x1

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Lot size

Discontinuities can be circumvented by the use of an exponential function

Corrected lot size

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Lot size and neighbourhood effects

Correction function for lot depends on

location(city center or rural area)

house type(detached house or row house)

Neighbourhoods

Neighbourhoods in the same district have thesame trend

but can havedifferent levels

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Outline

1 State-Space Model

Market Segments Explanatory variables Model Specification 3 _Results Data Estimates Extensions

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The Hierarchical Trend Model

yt =iµt +Dϑ,tϑt +Dλ,tλt+Dφ,tφ +f(Xt, β) + t, t ∼N(0, σ2I) µt+1 = µt+ κt + ηt, ηt ∼ N(0, σµ2), κt+1 = κt + ζt, ζt ∼ N(0, σκ2), ϑt+1 = ϑt + ωt, ωt ∼ N(0, σϑ2I), λt+1 = λt + ςt, ςt ∼ N(0, σ_λ2I), φ ∼ N(0, σ_φ2I)

the scalarµt thegeneraltrend

the vectorϑt thedistricttrend

the vectorλt thehouse type clustertrend

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Outline

1 State-Space Model

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Data Heerlen

2658 observations from January ’01 until December ’04 52 neighbourhoods: 6 districts

15 house types: 2 house type clusters

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Outline

1 State-Space Model

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Model results

Variable Coef T-value

House size in m3 0.7339 49.02 Age -0.0075 -22.69 Year of construction : 1920 - 1945 -0.3980 -25.85 Year of construction : 1900 - 1920 -0.3971 -13.11 Year of construction < 1900 -0.2559 -4.34 Poor Maintenance -0.3112 -5.20 Good Maintenance 0.0795 3.45 Lot size in m2 0.1230 10.85 Number of ’dormers’ 3.7688 3.88 Garage in m3 0.1501 7.87 Carport in m2 0.2975 3.12 Sun room in m2 0.1315 2.50 Cellar in m3 0.0913 4.79 Detached house 0.1074 8.90 Linked house 0.0597 4.53 Corner house 0.0292 4.18 Bungalow 0.1354 8.08

An increase of house size with 10%, gives an increase of value with 7.3%

Difference between ’poor’ and ’good’ maintenance is 0.4, approximately 49% of the house value excluding the land

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Model results

Fit of model measured byσ, can be interpreted as a

standard deviation (sd) of 12%

The sd of the general trendσµis monthly 0.629%,

corresponding with 2.2% a year

The sd of the district trendσϑis monthly 0.401%,

corresponding with 1.4% a year

The sd of the house type cluster trendσλ is negligible

The sd of the cluster trendσφis 10.7%

Variable Coef σ 0.1200 σµ 0.00629 σκ 0.00039 σϑ 0.00401 σ_λ 0.00032 σφ 0.10695 LL 1876.3501

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Outline

1 State-Space Model

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Model Extensions

Heteroskedasticity (over for example house type) Regression parameters varying over

time space

Spatial correlation in district trends

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Summary

Hierarchical Trend Model (HTM) is aState-Space model.

State-Space model is aparametricmodel.

State-Space model allows forflexible functional forms.

(stochastic trends)

The HTM can be characterized as as

Hedonic PriceModel

Time SeriesModel

SpatialModel

Hierarchical Model orMultilevelmodel