LET, Transport Economics Laboratory (CNRS, University of Lyon, ENTPE)

Insight into apartment attributes and location

with factors and principal componentsapplying oblique rotation

LET, Transport Economics Laboratory(CNRS, University of Lyon, ENTPE)

17th Annual ERES conference, 2010, Milano, SDA

Bocconi

Alain Bonnafous Marko Kryvobokov Pierre-Yves Péguy

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17th Annual ERES conference, 2010, Milano, SDA Bocconi

1. Introduction

Methods not focusing on price as dependent variable – an alternative or a complement to hedonic

regression:

• Factor Analysis (FA)• Principal Component Analysis (PCA)• Others…

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1. Introduction

Two ways of PCA application in a hedonic price model:

• PCA + clustering (submarkets) => hedonic price model Example: Bourassa et al. (2003): - citywide hedonic model with dummies for submarkets - hedonic models in each submarket - the best result: clusters based on the first two

components load heavily on locational variables

• PCA (data reduction) => hedonic price model Des Rosiers et al. (2000): principal components are

substitutes for initial variables

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1. Introduction

Selection of the methodology based on the aim (Fabrigar et al., 1999):

• FA (explains variability existing due to common factors) – for identification of latent constructs underlying the

variables (structure detection) • PCA (explains all variability in the variables) – for data reduction

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1. Introduction

Selection of the rotation method (Fabrigar et al., 1999):

• Methodological literature suggests little justification for using orthogonal rotation

• Orthogonal rotation can be reasonable only if the

oblique rotation indicates that factors are uncorrelated

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1. Introduction

• Aim 1: identification of latent construct underlying our variables with FA

• Aim 2: data reduction with PCA

• Rotation: oblique (non-orthogonal)

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2. Data preparation

14

38

9

5

4

13

116

10

715

12

1

2

0 1 20.5 Kilometers

apartments

IRISes

Boundary of Lyon and Villeurbanne

15 centres

Location of apartments: central part of the Lyon Urban Area

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2. Data preparation

Lyon

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2. Data preparation

• 4,251 apartment sales• 1997-2008• Location data for IRIS (îlots regroupés pour

l'information statistique)• Count variables as continuous variables• Categorical variables as continuous variables (Kolenikov and Angeles, 2004)• Skew < 2• Kurtosis < 7 (West et al., 1995)

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2. Data preparation

Description Mean Minimum MaximumStd. deviatio

nSkew

Kurtosis

Transaction price, Euros 122,235.90

20,276.00 500,000.00

69,979.67 1.45 2.93

Count for year of transaction 6.87 1 12 2.87 -0.10 -0.88

Apartment area, square metres

68.63 18 196 25.98 0.78 1.51

Number of rooms 3.05 1 8 1.19 0.26 -0.18

Floor 2.84 0 18 2.25 1.35 3.85

Construction period 5.12 1 7 1.75 -0.50 -0.73

State of apartment 2.79 1 3 0.47 -2.14 3.87

Number of cellars 0.69 0 2 0.50 -0.43 -0.88

Descriptive statistics of apartment variables

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2. Data preparation

Descriptive statistics of location variables

Description Mean Minimum MaximumStd. deviation

SkewKurtosi

s

Percentage of low income households

29.42 10.24 52.12 5.78 -0.10 -0.05

Percentage of high incomehouseholds

12.58 4.34 28.77 2.92 0.51 0.68

Travel time to Stalingrad 11.31 1.41 24.43 4.85 0.43 -0.25

Travel time to Louis Pradel 11.18 2.22 29.36 5.35 0.62 0.01

Travel time to Bellecour-Sala

10.99 0.45 31.28 4.96 0.89 0.79

Travel time to Jussieu 10.44 0.45 30.36 5.18 0.72 0.01

Travel time to Charles Hernu 11.19 0.45 26.17 5.37 0.35 -0.66

Travel time to Les Belges 11.00 0.45 27.48 5.34 0.49 -0.44

Travel time to Villette Gare 10.68 0.45 29.25 5.35 0.37 -0.81

Travel time to Part-Dieu 10.62 0.45 29.36 5.24 0.46 -0.71Travel times are calculated with the MOSART transportation model for the a.m. peak period, public transport by Nicolas Ovtracht and Valérie Thiebaut

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3. Factor analysis

• Principal axes factoring – the most widely used method (Warner, 2007)• The standard method of non-orthogonal

rotation – direct oblimin• Of 8 apartment variables, 5 are included• Of 15 variables of travel times, 8 are included • 4 factors with Eigenvalues > 1 • Correlation between Factor 1 and Factor 4 is -

0.52 (the choice of non-orthogonal rotation is right)• Continuous representation: interpolation of

factor scores to raster

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3. Factor analysisCommunalities and factor loadings

Variable Communality

Factors

Structure matrix Pattern matrix

1 2 3 4 1 2 3 4

Price 0.56 -0.18 0.86 -0.08 -0.05 -0.12 0.86 -0.12 <0.01

Area 0.53 0.03 0.82 0.07 -0.13 0.10 0.83 0.04 0.02

Construction period 0.34 0.08 0.04 -0.78 -0.13 0.01 0.06 -0.77 -0.08

Condition 0.14 0.02 0.08 -0.40 -0.04 <0.01 0.09 -0.41 -0.01

Cellars 0.18 0.04 0.18 0.37 -0.12 0.01 0.14 0.36 -0.11

% low income households 0.85 -0.49 -0.12 0.01 0.93 -0.01 <-0.01 -0.04 0.93

% high income households 0.86 0.50 0.10 -0.02 -0.94 0.03 -0.01 0.03 -0.93

Travel time to Bellecour-Sala 0.96 0.68 -0.07 -0.22 -0.60 0.49 -0.07 -0.18 -0.34

Travel time to Les Belges 0.98 0.95 -0.12 -0.15 -0.48 0.95 -0.05 -0.10 0.01

Travel time to Jussieu 0.99 0.94 -0.09 -0.17 -0.63 0.82 -0.05 -0.12 -0.20

Travel time to Part-Dieu 0.99 0.95 -0.02 0.04 -0.54 0.92 0.03 0.09 -0.07

Travel time to Louis Pradel 0.98 0.87 -0.14 -0.28 -0.55 0.77 -0.09 -0.23 -0.15

Travel time to Charles Hernu 0.98 0.93 -0.04 0.06 -0.41 >0.99 0.04 0.10 0.11

Travel time to Villette Gare 0.99 0.91 -0.00 0.09 -0.52 0.90 0.05 0.14 -0.05

Travel time to Stalingrad 0.96 0.88 -0.07 -0.00 -0.37 0.95 0.01 0.04 0.12

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3. Factor analysisRaster map of Factor 1: high income households farther from centres

14

3

9

116

101

2

0 1 20.5 Kilometers

-1.870418668 - -1.310576412

-1.310576411 - -0.750734157

-0.750734157 - -0.190891902

-0.190891902 - 0.368950354

0.368950354 - 0.928792609

0.928792609 - 1.488634864

1.488634865 - 2.04847712

2.048477121 - 2.608319375

2.608319376 - 3.168161631

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3. Factor analysis

14

3

9

116

101

2

0 1 20.5 Kilometers

-4.226680756 - -3.416438474

-3.416438473 - -2.606196192

-2.606196191 - -1.79595391

-1.795953909 - -0.985711628

-0.985711628 - -0.175469346

-0.175469346 - 0.634772937

0.634772937 - 1.445015219

1.44501522 - 2.255257501

2.255257502 - 3.065499783

Raster map of Factor 4: low income households closer to centres

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3. Factor analysis

0 1 20.5 Kilometers

-1.828741908 - -1.095964061

-1.09596406 - -0.363186214

-0.363186214 - 0.369591634

0.369591634 - 1.102369481

1.102369482 - 1.835147328

1.835147329 - 2.567925175

2.567925176 - 3.300703022

3.300703023 - 4.033480869

4.03348087 - 4.766258717

Raster map of Factor 2: big and expensive apartments

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3. Factor analysis

0 1 20.5 Kilometers

-1.748729944 - -1.222226858

-1.222226857 - -0.695723772

-0.695723772 - -0.169220686

-0.169220686 - 0.3572824

0.3572824 - 0.883785486

0.883785486 - 1.410288572

1.410288573 - 1.936791658

1.936791659 - 2.463294744

2.463294745 - 2.989797831

Raster map of Factor 3: older apartments in bad condition

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4. PCA of location attributes

• Data reduction: - two variables for income groups - 15 variables of travel times to centres• Direct oblimin rotation• 3 principal components with Eigenvalues > 1 • Correlation between Principal Components are 0.54, -0.50 and -0.32 (the choice of non-orthogonal rotation is right)• Continuous representation

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4. PCA of location attributesRaster map of Principal Component 1: centres of Lyon

14

38

9

4

13

1165

10

715

12

1

2

0 1 20.5 Kilometers

-1.509705424 - -0.928679758

-0.928679758 - -0.347654091

-0.347654091 - 0.233371576

0.233371576 - 0.814397242

0.814397242 - 1.395422909

1.39542291 - 1.976448576

1.976448577 - 2.557474242

2.557474243 - 3.138499909

3.13849991 - 3.719525576

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4. PCA of location attributesRaster map of Principal Component 2: centres of Villeurbanne

14

38

9

4

13

1165

10

715

12

1

2

0 1 20.5 Kilometers

-1.669591546 - -1.222203546

-1.222203545 - -0.774815546

-0.774815546 - -0.327427546

-0.327427546 - 0.119960454

0.119960454 - 0.567348454

0.567348454 - 1.014736454

1.014736455 - 1.462124454

1.462124455 - 1.909512454

1.909512455 - 2.356900454

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5. Conclusion and perspective

• Oblique rotation is found to be applicable for real estate data

• The results are intuitively easy to interpret• Separate factors are formed for apartment

attributes and location• Factor 4 highlights the existence of a problematic

low income area in the central part of Lyon (similarly to the finding of Des Rosier et al. (2000) in the Quebec Urban Community)

• With PCA a more complex spatial structure is detected

• Perspective: clusters of factors/principal components as proxies of apartment submarkets?

LET, Transport Economics Laboratory (CNRS, University of Lyon, ENTPE)

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Transcript of LET, Transport Economics Laboratory (CNRS, University of Lyon, ENTPE)