Hierarchical Trend Model
Marc K. Franckemfrancke@ortax.nl 1Department of Econometrics
Vrije Universiteit Amsterdam
2OrtaX
Amsterdam
Outline
1 Introduction
2 State-Space Model
3 The Hierarchical Trend Model
4 Results
Introduction
Use of the Hierarchical Trend Model (HTM) is operational for more than 10 years.
property tax (Law WOZ)
performance index for housing corporations
number of valuations a year is approx. 1.2 million objects only minor changes in model specification
Introduction
Strengths of the HTM are:
temporal dependence of selling prices (30 years) (structural time series model),
sophisticated way of modeling the housing characteristics (nonlinear model),
Statistical Model
The HTM is a parametric statistical model. A statistical model
imposes a priori a structure on the data and requires distributional assumptions
The statistical framework enables
to test for model assumptions and
to compare competing statistical models by the likelihood (no ad hoc training samples and criteria like MAPE).
State-Space Model
The HTM is an example of a State-Space model (SSM): State-space model can be seen as a regression model with time varying parameters
allow forflexible functional forms
yt = Ztαt + εt, εt ∼ N(0, σ2Ht),
αt+1 = Ttαt +Rtηt, ηt ∼ N(0, σ2Qt),
α1 = a0+A0β +R0η0, β ∼ N(β0, σ2Σ),
yt is an (nt× 1) vector of observations
αt is an unobserved state vector
εt, ηt,and β are uncorrelated
The matrices Zt, Tt, Ht,Rt and Qt are called system
Examples of State-Space Models
Simple State-Space Modelyt = Ztαt + εt, εt ∼ N(0, σ2I),
αt+1 = αt + ηt, ηt ∼ N(0, σ2ηI),
For ση =0, this is a regression model yt =Ztα + εt
For ση >0, time-varying coefficients
When Zt =1, then for
ση= ∞, a dummy variable model
ση=0, αt is constant in time
State-Space Model and the Kalman filter
State-space models are estimated by the Kalman filter Important features of the Kalman filter
It producesrecursivepredictions of next period’s
observations based on information until now.
Thepredictedvalues are compared to theactualvalues. The reliability of the predictions is provided by the
likelihood, the ultimate measure of the quality of forecasts.
Kalman filter provides alsooptimal revisionof the estimates
The Hierarchical Trend Model
Dependent variable yt : ln(Selling Prices) at time t
Time dependent part:
trends per market segment
Time independent part:
Market Segments
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
Price development
In the sameMS houses have the same price development
The trend forMSjk is the sum of
Generaltrend: µt
Districttrend: ϑj,t (in deviation from the general trend)
House type clustertrend: λk ,t (” ”)
Houses in the sameMS have the same price development
(µt+ ϑj,t+ λk ,t)
Time index t is in months
Time Trends
General Trend: local linear trend model (linear model with varying level and slope)
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, σλ2I)
ϑt+1= ϑt + ωt, ωt ∼ N(0, σϑ2Sϑ),
σ2η σ2ζ Model
∞ Dummy variable model
Location
Location is modeled by
Neighbourhoods in the same district have thesame trend
but can havedifferent levels
modeled by random effects ψ ∼ N(0, σψ2Sψ), where Sψ is
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, σϑ2Sϑ), λt+1 = λt + ςt, ςt ∼ N(0, σλ2I), φ ∼ N(0, σφ2Sφ)the scalar µt thegeneraltrend
the vector ϑt thedistricttrend
the vector λt thehouse type clustertrend
the vector φ random effects forneighbourhood
Outline
1 Introduction
2 State-Space Model
3 The Hierarchical Trend Model
4 Results
Data
Data Heerlen
2658 observations from January ’01 until December ’04 52 neighbourhoods: 6 districts
Outline
1 Introduction
2 State-Space Model
3 The Hierarchical Trend Model
4 Results
Data
Estimates
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
Model results
Fit of model measured by σ, can be interpreted as a standard deviation (sd) of 12%
The sd of the general trend σµis monthly 0.629%,
corresponding with 2.2% a year
The sd of the district trend σϑis monthly 0.401%,
corresponding with 1.4% a year
The sd of the house type cluster trend σλ is negligible
The sd of the the neighbourhood effect is σφis 10.7%
Outline
1 Introduction
2 State-Space Model
3 The Hierarchical Trend Model
4 Results
Data Estimates
Model Extensions
Heteroskedasticity (over for example house type) Regression parameters varying over
time space
Summary
Hierarchical Trend Model (HTM) is aState-Space model.
State-Space model is aparametricmodel.
State-Space model allows forflexible functional forms.
(stochastic trends)
TheHTM can be characterized as as Hedonic PriceModel
Time SeriesModel SpatialModel
