Analysis and prediction of apartment prices in inner city Stockholm

82 Pages·2014·2.83 MB·English

Analysis and prediction of apartment
prices in inner city Stockholm
by
Robert Hu and Emil Sjo¨gren
Supervisor:
Gunnar Englund
A thesis submitted in partial fulfillment for the
degree of Bachelor of Science in Engineering Physics
in the
Department of Mathematical Statistics
KTH
May 2014
Abstract
In this thesis, an attempt will be made to find a model that predicts apartment prices
in inner city Stockholm. Lately, apartment prices have steadily increased and it has
become rather popular to invest money in apartments due to the high potential of value
increase. Of course not all apartments will increase in value, and we hope to find the
factors that determine the value of an apartment that is about to be sold. To find these
factors, we will combine the approaches of regression and time series analysis, creating
a linear regression model and a time series model, and then taking the mean of their
predictions.
StatisticsofapartmentsalesduringthetimeperiodofAugust2012toFebruary2014will
be used as data, and when finding the models explicitly, large data handling softwares
such as SPSS, Excel and Matlab will be used. The thesis consists of an introduction
to the general theory behind the two approaches, followed by our specific case. Lastly,
an attempt will be made to draw some general conclusions from our findings, and the
accuracy of our model will be tested by predicting the selling price of a few apartments
that have been or are about to be sold and comparing the predictions with the actual
price.
Sammanfattning
I denna uppsats kommer det f¨ors¨okas att hitta en modell f¨or att f¨orutse l¨agenhetspriser
i Stockholm innerstad. P˚a senare tid har l¨agenhetspriserna vuxit stadigt och det har
blivit popul¨art att investera pengar i l¨agenheter. Det ¨ar en h¨og chans att v¨ardet p˚a en
k¨opt l¨agenhet ¨okar, men detta ¨ar inte n¨odv¨andigtvis normen. Denna uppsats ¨ar ¨agnad
˚atattidentifieradefaktorersomp˚averkarf¨ors¨aljningsprisetf¨orl¨agenheter. F¨oratthitta
dessafaktorerkommeranalytiskamodelleranv¨andasochdessamodellerkommerbaseras
p˚a statistik av l¨agenheter s˚alda mellan augusti 2012 och februari 2014. Modellerna
som kommer anv¨andas ¨ar linj¨ar regression och tidsserier. N¨ar modellernas koefficienter
och faktorer ska definieras explicit, kommer SPSS, Excel och Matlab anv¨andas. Efter
att de b˚ada modellerna definierats, kommer medelv¨ardet av b˚ada modeller representera
predikatorn. V˚armodellsnogrannhetkommerpr¨ovasgenompredikeraf¨ors¨aljningspriset
avenl¨agenhetp˚amarknadenochj¨amf¨oradettameddetfaktiskapriset. F¨orhoppningsvis
kommerdennamodellhatillr¨ackligth¨ogprecisionf¨orattf¨orutsp˚al¨agenhetspriserresten
av˚ar 2014.
Acknowledgements
We would like to thank our supervisor Gunnar Englund for his commentary and sug-
gestions.
We would also like to thank booli.se for providing us with data of apartment sales.
Andfinallywewouldliketothankourfellowstudentsatthedepartmentofmathematical
statistics for providing great discussion and tips.
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Contents
Abstract i
Sammanfattning ii
Acknowledgements iii
List of Figures vi
Abbreviations viii
Symbols ix
1 Introduction 1
2 Theoretical Background 3
2.1 Modeling Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.1.1 Linear Regression. . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.1.2 Ordinary Least Squares . . . . . . . . . . . . . . . . . . . . . . . . 4
2.1.2.1 Harmonic Regression . . . . . . . . . . . . . . . . . . . . 5
2.1.3 Variable Transformations . . . . . . . . . . . . . . . . . . . . . . . 6
2.1.3.1 Dummy Variables . . . . . . . . . . . . . . . . . . . . . . 6
2.1.3.2 Logarithmic Transformation . . . . . . . . . . . . . . . . 6
2.1.4 Model Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1.4.1 R2-Value . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1.4.2 Bayesian Information Criterion . . . . . . . . . . . . . . . 7
2.1.5 Time Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.1.6 Theory of Distributions and Hypothesis . . . . . . . . . . . . . . . 10
2.1.6.1 The Hypothesis’s for the Linear Model . . . . . . . . . . 11
2.2 Problem Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2.1 Heteroscedasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2.1.1 Breusch-Pagan Test . . . . . . . . . . . . . . . . . . . . . 13
2.2.2 The Remedies for Heteroscedasticity . . . . . . . . . . . . . . . . . 14
2.2.2.1 Reformulating the Model . . . . . . . . . . . . . . . . . . 14
2.2.2.2 White’s Consistent Variance Estimator . . . . . . . . . . 14
2.2.3 Endogeneity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2.3.1 Sample Selection Bias . . . . . . . . . . . . . . . . . . . . 16
2.2.3.2 Simultaneity . . . . . . . . . . . . . . . . . . . . . . . . . 16
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Contents v
2.2.3.3 Missing Relevant Covariates . . . . . . . . . . . . . . . . 16
2.2.3.4 Measurement Errors . . . . . . . . . . . . . . . . . . . . . 16
2.2.4 The Remedies for Endogeneity . . . . . . . . . . . . . . . . . . . . 17
2.2.4.1 Instrumental Variables . . . . . . . . . . . . . . . . . . . 17
2.2.4.2 The Bootstrap . . . . . . . . . . . . . . . . . . . . . . . . 18
2.2.5 Multicollinearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2.6 The Remedies for Multicollinearity . . . . . . . . . . . . . . . . . . 19
2.2.6.1 Variance Inflation Factor . . . . . . . . . . . . . . . . . . 19
3 Model and Experimental setup 21
3.1 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4 Experiment and Data Analysis 25
4.1 The Linear Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.2 The Time Series Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
5 Discussion and Conclusion 61
5.1 The Linear Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
5.2 The Time Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
5.3 The Combined Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
5.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
A Matlab code 67
Bibliography 72
List of Figures
4.1 First Residual Plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.2 Second Residual Plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.3 First Regression. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.4 Second Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.5 Third Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.6 Reduced Model Residual Plot . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.7 Residual vs.”living area” . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.8 Residual vs. ”Rooms” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.9 Extended Model Residual Plot . . . . . . . . . . . . . . . . . . . . . . . . 30
4.10 Extended Model Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.11 Consistent Estimates Extended Model . . . . . . . . . . . . . . . . . . . . 32
4.12 Residual Plot Reduced Model (no outliers) . . . . . . . . . . . . . . . . . 33
4.13 Residual vs. ”livingarea” (no outliers) . . . . . . . . . . . . . . . . . . . . 33
4.14 Residual vs. ”Rooms” (no outliers) . . . . . . . . . . . . . . . . . . . . . . 34
4.15 Residual Plot Extended Model (no outliers) . . . . . . . . . . . . . . . . . 34
4.16 Results Extended Model (no outliers) . . . . . . . . . . . . . . . . . . . . 35
4.17 White’s Estimator (no outliers) . . . . . . . . . . . . . . . . . . . . . . . . 35
4.18 Effect of β’s in % . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.19 Time Series of Raw Data . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4.20 Time Series Season and Trend Removed . . . . . . . . . . . . . . . . . . . 38
4.21 ACF plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.22 Cubic OLS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.23 Linear OLS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.24 Magnitude Plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.25 Harmonic Regression Results . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.26 Second Harmonic Regression Results . . . . . . . . . . . . . . . . . . . . . 43
4.27 Plot of Full Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.28 Plot without Outliers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.29 Trend Function Quadratic Model . . . . . . . . . . . . . . . . . . . . . . . 45
4.30 Trend Function Linear Model . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.31 Trend Removed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.32 Magnitude Plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.33 First Harmonic Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.34 Second Harmonic Regression . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.35 Detrended and Deseasonalized data . . . . . . . . . . . . . . . . . . . . . . 48
4.36 ACF Plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.37 Full Time Series Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
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List of Figures vii
4.38 Plot of logy in a Time Series . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.39 First Regression of Trend . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.40 Second Regression of Trend . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.41 Third Regression of Trend . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.42 Data with Trend Removed 1 . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.43 Data with Trend Removed 2 . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.44 Harmonic Regression of log(y) 1 . . . . . . . . . . . . . . . . . . . . . . . 53
4.45 Harmonic Regression of log(y) 2 . . . . . . . . . . . . . . . . . . . . . . . 54
4.46 log(y) with everything removed . . . . . . . . . . . . . . . . . . . . . . . . 54
4.47 sample ACF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.48 Full Logarithmic Time Series Model . . . . . . . . . . . . . . . . . . . . . 56
4.49 Logarithmic Timeseries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.50 Logarithmic Timeseries No Trend and Season . . . . . . . . . . . . . . . . 57
4.51 Logarithmic Time Series ACF . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.52 Cubic Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.53 Linear Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.54 First Harmonic Regression of log(y) . . . . . . . . . . . . . . . . . . . . . 59
4.55 Second Harmonic Regression of log(y) . . . . . . . . . . . . . . . . . . . . 59
4.56 Last model vs. Raw data . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
5.1 Predicted Prices Using the Linear Model . . . . . . . . . . . . . . . . . . . 62
5.2 End Results Time Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
5.3 Predictions Combined Model . . . . . . . . . . . . . . . . . . . . . . . . . 65
5.4 Comparison of Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
Abbreviations
OLS Ordinary Least Squares
SE Standard Error
SSR Sum of Squared Residual
SSE Sum of Squared Error
MSS Mean Sum of Squares
BLUES Best Linear Unbiased EStimator
BIC Bayesian Information Criterion
2SLS 2-Stage Least Squares
ACF Auto Correlation Function
ACVF Auto Co Variance Function
IID Identical and Indepentdently Distributed
WN White Noise
VIF Variance Inflation Factor
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Symbols
X Matrix containing all covariates
x The i:th column in X
i i
β Coefficient vector for regression
β The i:th coefficient in β
i
y Regression vector
yˆ Predicted regression vector
eˆ Error vector after OLS
eˆ Error vector of the restricted regression
∗
βˆ Least-squares coefficient vector
βˆ The i:th coefficient in βˆ
i
a ·b The inner product between vectors a and b
i i i i
|a | The norm of the vector a
i i
XT The transpose of the matrix X
σ The standard error
f Degrees of freedom
k The amount of covariates
n The amount of data points
s The estimated standard error s = 1 |eˆ|2
1−k−1
H The null hypothesis
0
H The alternative hypothesis
1
α The error rate for which the true hypothesis H is falsely rejected
0
E(X) The expected value of the random variable X
Var(X) The variance of the random variable X
ix