A Dynamic Model of Housing Demand
Simon J. Hviid
Student i.d.: 20071490
Department of Economics and Management
Aarhus University
IMSQE Master’s Thesis
Supervisor: Rune M. Vejlin
2012
Abstract
This thesis estimates a dynamic model of demand for housing in Denmark. The
model builds on the assumption that the demand function can be characterized by
a discrete choice of the households. This model inverts the market share of buys
in Danish neighborhoods for different household types to determine the marginal
willingness to pay for several locally non-traded amenities.
The model is estimated using household-level data from a new and very rich
dataset. The estimates are compared to a hedonic equilibrium model and a comparable static model of demand, and it is found that the dynamic model performs
significantly better in terms of correcting for various biases.
Acknowledgement
First of all, I would like to thank Rune Vejlin for excellent guidance in theoretical as well
as data and programming related issues. Thereafter, Statistics Denmark has provided
brilliant support with data matters. I would like to direct a special thanks to Realdania
and the Rockwool Foundation that both have financially supported the project, such that
the key data could be acquired.
1 INTRODUCTION
1
Introduction
Over the past four decades various techniques toward estimating demand functions for
houses and marginal willingness to pay estimates for a range of local amenities has been
applied. The first approach was the hedonic technique formulated and applied by Rosen
(1974) that defined the hedonic equilibrium prices and thereby laid the ground for the use
of hedonic techniques to value all sorts of neighborhood amenities such as school quality,
air quality, crime rates and ethnic composition. The advantage of these techniques is that
they have used observed prices and behavior to extract the demand function by revealed
preferences using simple least squares. But these results come at the cost of an assumption
that the housing market is a perfect market.
The conventional hedonic techniques implicitly assume that there is no cost associated
with moving and that households can move whenever this implies a higher instantaneous
utility. However, this is very unrealistic and therefore many papers have turned to residential sorting models when modeling the demand of local amenities. The class of residential
sorting models first identify the move-stay decision of the household and deduct the value
from living in different regions, cities or neighborhoods. Secondly, these models decompose the single-period utilities and extract marginal willingness to pay estimates of locally
non-traded amenities. The literature has recognized and identified both moving costs that
are monetary one-time costs and psychological costs from searching for a new house and
leaving a well-known location. If the moving costs are large enough to withhold just a
single household from moving, then the implication from not modeling these costs is that
the marginal value of local amenities will be estimated with a downward bias. The idea
is that the model that assumes a perfect housing market will believe that the household
do not move because their marginal value of an amenity equals the marginal cost of the
amenity, when in reality they do not move because the fixed costs from moving outweigh
the benefit. Despite that this bias has been known since Ellickson (1971) the literature
using hedonic techniques is still growing.
As most researchers have recognized the impact from moving costs on the residential
sorting model, as many have realized that the housing market is more dynamic in its nature
3
1 INTRODUCTION
than it is static. One of the main sources of the dynamic behavior of the housing market
is again the moving costs as these costs significantly affects the households wealth and
thereby it affects the future consumption possibilities of the household. This implies that
households move infrequently and that the current period location decision is expected to
influence the utility of the household for at least some periods to come. This last effect
is magnified as the local amenities are not constant but evolving over time, such that the
household’s decision will in part be based on the current level of the amenities and in part
on the expected future levels. This last insight is essential to understand, as it can explain
why households move to neighborhoods, that does not currently have particular favorable
amenities, but relatively high prices. Furthermore, the housing market assess dynamic
features as the household needs change over the life-cycle. In particular this matter when
the number of family members change, either as children are born, or when couples are
divorced, or a family member dies.
Even though many have recognized these dynamic aspects of the housing demand
function, most papers have estimated static models, that give biased estimates value of
local amenities. The biases that result from a static model depend in size and direction
on the time-series property of the amenity considered. If an amenity is mean reverting
then a static model will understate the true marginal value as the household will seem
not to respond to in increase in the level of the amenity, when in practice, the household
knows that the level is only temporarily above normal. At the same time the model will
overestimate the value of positively persistent amenities as the price increases are actually
discounting of the high expected future levels.
It is not only of theoretical interest to identify households’ marginal valuation of different local amenities, but also of high interest in order to determine the optimal public
investment level into improving the level of some amenities. Consider for example the case
of school quality. For many years there have been an ongoing political debate on whether
or not to invest further in the quality of public schools by e.g. increasing the education
of teachers which is in the end associated with greater costs for tax-payers. Therefore it
is important to determine the unbiased valuation of school quality to ensure that the cost
4
1 INTRODUCTION
of such policy changes does not outweigh the benefit.
This thesis will estimate a dynamic model of demand for houses and neighborhoods
that build directly on the model developed by Bayer, McMillan, Murphy, and Timmins
(2011). The ideas behind this model extend the insight from many other models of residential sorting and utilize the evolution of the theory known from industrial organization
on durable goods.
In this model the state variables, including observed and unobserved household and
neighborhood characteristics, are allowed to vary over time and the household will form
expectation about these changes through the history of these values. Secondly, the households will form expectations about the evolution of house prices as households will not
only see housing as a consumption good, but also as a financial asset, which they expect
to sell at some future point in time.
As a separate contribution to the literature, the model of Bayer, McMillan, Murphy,
and Timmins (2011) recovers estimates of the marginal value of wealth, when estimating the move-stay decision. This result stems from the monetary cost of moving which
has been parameterized to household characteristics. This key contribution is then used
to convert marginal utility estimates into estimates of marginal willingness to pay. Furthermore this insight is used to overcome the endogeneity of user costs of housing in the
decomposition of the single-period utility.
The model in this thesis will be related to a hedonic pricing model as presented by
Bishop and Timmins (2011) and use ideas from this literature to improve the estimate
of the evolution of neighborhood prices. This procedure overcomes some endogeneity
problems that are not tackled in the original model, as many locally non-traded amenities
seem to be correlated with the house attributes, i.e. included in the error term.
The thesis will use Danish register based data from 1992 to 2005 to estimate the model.
The data is collected from various sources but the key registers are IDA and BBR that
contain individual demographic and socioeconomic information and house characteristics
respectively. The data set includes very detailed information on the wealth of individuals
5
1 INTRODUCTION
and in particular it contains a public valuation off all properties. By adjusting the public
valuation by differences to actual sales prices it is possible to observe how the market
price of houses change over time. The greatest advantage of this data is that it is rich
enough to enable researchers to follow households over time, year by year, and in this
way it is possible to get around a large amount of the technical assumption in Bayer,
McMillan, Murphy, and Timmins (2011), especially it improves the estimates of the movestay decisions of households.
The estimation of the model finds that households on average are willing to pay for
the six locally non-traded amenities included. These cover the fraction of ethnic Danes,
school quality, sexual offences, property and violent crimes, and population density. Only
the sexual offences might be insignificant. For example is it found that households will pay
DKK 43,000 to increase the fraction of ethnic Danes of a neighborhood by 1 percentage
point and that are willing to pay DKK 800 to reduce the number of property criminals
per 1000 inhabitants by one. The model is also estimated for different types of households
and it is found that households with low wealth are much more willing to pay to avoid non
ethnic Danes, which high wealth households does not seem to care about. Furthermore it
is found that as expected households that include home living children are willing to pay
more for school quality and for avoiding violent crimes. The model estimates are found
robust to changes in the subjective rate of time preference, but less robust to changes in
the realtor fee.
When considering the two primary corrections proposed to Bayer, McMillan, Murphy,
and Timmins (2011) I find that both significantly improve the estimates of the marginal
willingness to pay. That is cleaning house prices and measuring realtor fees as a linear
function in house prices in stead of average neighborhood prices.
Lastly, the thesis compares the hedonic equilibrium model and the dynamic model of
demand by inserting the marginal willingness to pay estimates into the equilibrium condition in Rosen’s second stage, and find that these differ significantly from the marginal
equilibrium costs estimated in the first step of Rosen’s model. This reveals that the use
of hedonic models is unreasonable as these models rely on assumptions that are too strict.
6
2 HEDONIC PRICING MODELS
The thesis is organized in the following way. Section 2 and Section 3 provides an
overview of the theory and current strands in the literature regarding hedonic pricing models and dynamic models of demand for differentiated durable goods respectively. Thereafter section 4 will describe the data used and the sample selection. Section 5 formally
sets up the model and states the assumptions used and then Section 6 goes through the estimation procedure of the model. In Section 7 the results will be presented and discussed.
Section 8 will present robustness checks of the mode, discuss some of the implication of
the model specification, and compare the hedonic equilibrium model with the dynamic
demand model estimated. Finally, Section 9 will conclude the thesis.
2
Hedonic Pricing Models
This section will give an introduction to the theory and limitations of hedonic equilibrium
pricing models. Especially, it focus on the two-step procedure proposed by Rosen (1974)
that obtains estimates of marginal willingness to pay of heterogeneous individuals for
attributes of differentiated products.
In the literature, the paper by Rosen (1974) is referred to as the paper that initiated
the use of hedonic techniques to value locally non-traded amenities, however papers such
as Court (1939) and Griliches (1961) use hedonic techniques to estimate the values of the
attributes of various differentiated products1 .
Rosen’s two-step procedure has been applied in two ways in the literature thereafter
and both of these use variations in implicit prices to recover the marginal willingness to
pay function. The first procedure use multi-market data and assume that preferences are
homogeneous across markets. The other specify a non-linear relation for the equilibrium
market price of the differentiated products. Here, I will present a simple model following
Epple (1987) in applying the latter procedure.
Let the market price of a house, j, with observed attributes and amenities xj be given
1
These two papers in particular consider automobile prices
7
2 HEDONIC PRICING MODELS
by the following quadratic hedonic price function,
pj (xj ) = a0 + a1 xj +
a2 2
x + j
2 j
(1)
The need for this quadratic specification, is motivated by a simple diminishing marginal
utility observation that e.g. the utility of an extra square meter is falling when the house
size increase.
The marginal change in the market price implied by a change in an amenity can be
thought of as the marginal cost of that amenity and will be determined as,
p0j (xj ) ≡
∂pj (xj )
= a1 + a2 x j .
∂xj
(2)
The first step of Rosen’s procedure is basically to estimate (1) to recover the marginal cost
in (2). There is a range of problems related to Rosen’s procedure and the most of these
problems relate to the second step described below. However, even in this first step there
is a potentially large problem related to omitted variables bias as it is difficult empirically
to identify and measure the entire set of explanatory variables, that is correlated with
the locally non-traded amenities. See e.g. Palmquist (2006) that provide an overview of
the most common problems in estimating hedonic property value models using Rosen’s
two-step procedure.
The second step of Rosen’s procedure intends to recover the demand function of the
consumers. To do this define the utility function of consumer i with characteristics zi as
U (xj , zi ) = b0 + b1 xj +
b2 2
x + b3 xj zi + νi xj + (Ii − pj (xj ))
2 j
(3)
Here Ii denotes income of the individual. From this quadratic specification of the utility
function it follows that the heterogeneity of consumers is two dimensional. The first
heterogeneity stems from individual characteristics, which is observed, and the second
stems from the unobserved idiosyncratic shock to preferences over amenities, νi .
As in any other market, the consumer will have a demand for amenities as long as the
marginal utility is greater than zero, hence, from the first order condition the following
8
2 HEDONIC PRICING MODELS
equilibrium relation is obtained2
p0j (xj ) = b1 + b2 xj + b3 zi + νi
(4)
This is the marginal willingness to pay for the amenity xj of household i as any higher
prices would imply a negative effect on utility. In hedonic equilibrium it is clear that the
marginal cost from (2) and the marginal willingness to pay from (4) should equal, hence
the second step of Rosen’s procedure has been to regress the right hand side in (2) on the
right hand side of (4). Even though it seems simple, this step is not trivial. To see this
first realize that this equilibrium condition also identify the equilibrium quantity of the
amenity as
xj =
1
[(b1 − a1 ) + b3 zi + vi ]
a2 − b 2
(5)
This equation points to an essential point; the equilibrium level of the amenity of the
house which an individual will buy is directly correlated with the idiosyncratic shock.
This has been referred to as the Bartik-Epple endogeneity problem as both Epple (1987)
and Bartik (1987) identify this implicit endogeneity of Rosen’s second step.
After the identification of the endogeneity, many researchers has contributed with
various solution methods of which most of them rely on identifying good instruments,
using e.g. cross-market data (See Kahn and Lang, 1988) with an assumption about crossmarket homogeneity of preferences, or of conditional expectations as Ekeland, Heckman,
and Nesheim (2004). A recent approach by Bishop and Timmins (2011) change the second
step and estimate a version of (5) to cross-market data by maximum likelihood. In this
way they avoid the need for instrumental variables as they point out and utilize that
the endogeneity stems from the econometric specification of the problem, but is not a
fundamental issue of the equilibrium conditions of the problem.
However, this is a very recent approach that can only be applied with cross market
data. Therefore, acknowledging that proper instruments are very hard to identify, most
2
In most papers an equivalent relation is obtained for the supply side of the market, to obtain estimates
of the marginal willingness to accept. I will neglect this, as this paper will focus only on the demand side
of the housing market.
9
3 DYNAMIC DEMAND FOR DURABLE GOODS
papers has relied on the hedonic price function in the first step of Rosen’s procedure, and
neglected the second step all together. Bishop and Timmins (2011) provides a long list
of papers that turn to this approach and point to Deacon, Brookshire, Fisher, Kneese,
Kolstad, Scrogin, Smith, Ward, and Wilen (1998) that postulate that until 1998 no hedonic
model has been able to estimate the second stage of Rosen’s model successfully.
3
Dynamic Demand for Durable Goods
The aim of this section is to give an overview of the most recent advances in the Industrial Organization literature on empirical dynamic demand models of durable goods.
For a broader survey on developments in dynamic Industrial Organization models see
Aguirregabiria and Nevo (2010).
Over the years the literature has changed from using static models, to using more
dynamic models of demand in recovering the demand function for durable goods. This
shift has been motivated by the acknowledgement that the demand for durable goods is
dynamic in its nature for three primary reasons.
First of all, the consumers decision today will most likely affect the single-period utility
of at least some periods to come, simply because the product is durable. This effect is
enhanced by the general assumption of the presence of transaction costs associated with a
resale decision. One of the first to consider these transaction costs in a market for durable
goods was Akerlof (1970) when exploring the effect of adverse selection. These transaction
costs make it costly to re-optimize the choice of product3 in each period and will therefore
influence future utility. In general, transaction costs have not been restricted to be of a
financial character, but are in different ways measured in units of utility.
Secondly, the current decision of the consumer is affected by today’s holdings and
by the expectations to the evolution of those state variables, which have an effect on
future prices, future attributes of products, and on the future single-period utilities of the
3
A neighborhood can be thought of as the unit of a product, which will be the basis for the model of
this paper.
10
3.1 A Simple Static Model
3 DYNAMIC DEMAND FOR DURABLE GOODS
consumer.
Thirdly, the consumer’s needs and tastes evolve over the life cycle, for example in the
case of cars, you will need a practical car, such as a van or a station wagon when you have
children, whereas when you retire you might only need a two-seater or an other small car.
It is important to note that the durability of the product is not sufficient to necessitate
the building of a dynamic model framework. Imagine that there is no transaction costs
associated with a resale, that there is no uncertainty about future prices and that the
consumer only holds a single good, then the choice of the consumer is identical to simply
renting the product over the next period and in this case the rental rate will be the known
depreciation in price over the next period.
3.1
A Simple Static Model
To illustrate the challenges of dynamic models, and at the same time stress their importance, I will outline a rather naive baseline static model for demand of durable goods (see
Aguirregabiria and Nevo, 2010).
Assume that the aim is to estimate a static demand system of J products, that is a
function of a J-dimensional price vector, p, and a K-dimensional vector, x, that contains
characteristics of each product. Letting q be a J-dimensional vector that denote the
quantity demanded of each product, then the system to be estimated will be
q = D(p, x)
(6)
Clearly there is a dimensionality problem embedded in this specification as even by assuming that the demand is linear in all of its parameters, the dimensions of the matrix
containing cross price elasticities alone will be too large to make it possible to estimate
the system. This is what is often referred to as the curse of dimensionality. Though
many solutions has been suggested over the years, the literature seems to have settled for
the approach using discrete choice models. McFadden (1974) first formalized this idea
and it has been enhanced up until Berry, Levinsohn, and Pakes (1995). Even though 17
years has past, this still seem to be the frontier model in the static demand Industrial
11
3.1 A Simple Static Model
3 DYNAMIC DEMAND FOR DURABLE GOODS
Organization literature of differentiated durable products. The idea is to model demand
as a consumer decision of consuming one unit of J + 1 different consumption possibilities,
where the (J + 1)th option is a no-consumption option. This model is based on Lancaster
(1971) that first modeled consumer preferences over products as a function of personal
characteristics and of product attributes.
Out of I consumers, consumer i gets the following utility from choosing product j at
time t
ui,j,t = xj,t βi − pj,t αi + ξj,t + εi,j,t
(7)
Here, ξj,t is an unobserved product characteristic and εi,j,t is an idiosyncratic stochastic shock to the consumers perceived quality of the product. Already at this point, an
assumption is made, that help to reduce the dimensions of the state space, βi is a Kdimensional vector of marginal utilities of the K product attributes, this imply that a
product is defined as a bundle of characteristics, hence, the dimension of the model is
determined by the number of attributes and not the number of products. This effectively
ties down the dimensionality. The last parameter, αi is of particular interest, as it can be
interpreted as the marginal utility of wealth as it determines the dis utility of paying the
resale price and thereby re-optimizing holdings4 .
An other common way do deal with dimensionality in demand models has become
to follow what McFadden (1978) defines as the inclusive value. The inclusive value is
the expected utility of a consumer, before having observed the vector of product specific
idiosyncratic shocks, εi,t , conditioned on the consumer choosing the alternative that yields
the highest utility when observing the shocks. Assuming that the idiosyncratic shocks are
i.i.d. Type 1 Extreme value then, conditional on buying a product, the inclusive value
e
denoted vi,t
will be,

e
vi,t
= ln 
J
X

e{xj,t βi −pj,t αi +ξj,t } 
(8)
j=1
The advantage of the inclusive value is that the consumer only needs to form expectations
about how the inclusive value transitions in the future, instead of forming expectation
4
An extended version of this will play a central role in the dynamic model of this paper.
12
3.2 Dynamic Models
3 DYNAMIC DEMAND FOR DURABLE GOODS
about all state variables before identifying the lifetime utility. In this way, with a reasonable assumption about how expectations are formed, the state space can be reduced
sufficiently to deal with the curse of dimensionality.
This discrete choice model is widely used as the starting point for many papers on
dynamic models for durable goods. However, most papers have used the static approach
following Berry, Levinsohn, and Pakes (1995) as outlined here.
3.2
Dynamic Models
Given that the demand is dynamic, then it is crucial to estimate a model that captures
these properties. If ignoring the dynamic property of demand and setting up a static
model, then one will be faced with biased estimates. This happens as a static model
shuts down the formation of expectations to the future, and incorporate all information
into the estimates of today’s marginal willingness to pay. Bayer, McMillan, Murphy,
and Timmins (2011) gives an excellent overview of the implications of ignoring dynamic
features of demand by identifying the biases associated with estimating a static model.
Besides the econometric flaws of the static model, it is also insufficient to identify the
differences between short run and long run effect from both temporary and permanent
price changes or changes to other state variables.
Reducing the dimensionality of dynamic models is far more complicated than in the
static framework. The approach that project the products into an attribute-space is no
longer sufficient. In fact a forward-looking individual would have to form expectations
about future prices and attributes of each product. Hence, at least (K + 1) · J parameters
would need to be estimated, if all characteristics and prices were assumed to follow first
order Markov processes.
As for static models, it is important to strive a balance between the degree of unobserved heterogeneity allowed for, and the dimension of the problem. In many cases
consumers are restricted to share preferences for the unobserved quality of a product, but
some recent approaches such as Timmins (2007) allow different subsets of consumers to
differ in their perceived quality of the products.
13
3.2 Dynamic Models
3 DYNAMIC DEMAND FOR DURABLE GOODS
To outline the standard framework of analyzing dynamic demand for durable goods I
will follow a model by Gowrisankaran and Rysman (2009) and use the notation from the
simple static model.
Here the single period utility5 that consumer i gets from consuming product j at time
t is
ui,j,t = ufi,j,t − pj,t αi + εi,j,t ,
ufi,j,t = xj,t βi + ξj,t
(9)
(10)
As in the static model there is an outside option that yield the single-period utility
ui,j,t = ufi,0,t + εi,0,t ,
ufi,0,t =



0
(11)
if no holding
(12)


uf
i,ĵ,t̂
if last buy was product j at time t
This definition of the outside option is one of the sources that implies that the model will
have a dynamic nature because past decisions affect the current period utility. This is one
of the main differences from the static model.
The literature has turned to rely on a simple but effective assumption for dynamic
models of demand for durable products.
Assumption A1 The consumer can hold at most a single product at any point in time.
The idea behind this assumption is to realize that a product is characterized in two
dimensions, quality and quantity. The assumption then forces the model only to extract
information about the demand for qualitative characteristics of the product and not consider the quantitative aspects of demand. This assumption is widely used in papers that
consider demand for houses and automobiles, which seem fairly reasonable. One way
to relax his assumption could be to allow for e.g. depreciations of the product, which
5
Gowrisankaran and Rysman (2009) refers to this as conditional indirect utility
14
3.2 Dynamic Models
3 DYNAMIC DEMAND FOR DURABLE GOODS
opens the second quantity-dimension and the model will then have some features from
the models of storable goods, which are not covered in this section6 .
In dynamic models, the consumers wish to maximize lifetime utility Ui,t at time t such
that the purchase decision is made to maximize
Ui,t = ui,j,t + Et
" T
X
#
β
τ −t
(13)
ui,j,τ
τ =t+1
By imposing Assumption A1 and additionally that (i) there is no resale market and (ii)
the dynamic control problem of the consumer has an infinite horizon, such that T = ∞,
then the Bellman equation of the problem reads
h
h
Vi (εi,t , ufi,0,t , Ωt ) = max
ui,j,t + βEt Vi (ufi,j,t+1 , Ωt+1 )|Ωt
J
ii
(14)
{j}0
Letting Ωt contain all state variables at time t, i.e. prices, product characteristics etc.,
then
h
i
E Vi (ufi,j,t , Ωt )|Ωt =
Z
Vi (ufi,j,t , Ωt )dFε (εi,t )
(15)
Hence, the expectation is taken with respect to future prices, attributes of products and
with respect to the idiosyncratic error.
The primary difficulty lies within the estimation of the Bellman equation and it is here
that the curse of dimensionality still has bite. The Assumption A1 helps to deal with the
dimensionality of the holdings vector, primarily because that consumers do not have to
form expectations about the future interaction utilities, i.e. complementarities of holding
more than one product.
The holdings vector is only one side of the dimensionality problem and the state space
will still be too large to estimate, therefore the dynamic inclusive value has become a
common tool used to reduce the dimensionality further. With definitions as above, start
by defining
h
e
vi,j,t
= ufi,j,t − αi pj,t + δE Vi (ufi,j,t+1 , Ωt+1 )|Ωt
6
i
(16)
See e.g. Aguirregabiria and Nevo (2010) for a review of the literature on dynamic demand for storable
products.
15
3.2 Dynamic Models
3 DYNAMIC DEMAND FOR DURABLE GOODS
Then, still assuming that the idiosyncratic shock is i.i.d. Type 1 Extreme Value, the
dynamic inclusive value will be given by

e
vi,t
(Ωt ) = ln

J
X
e
vi,j,t 

e
(17)
j=1
The important difference between the inclusive value from the static model and the dynamic inclusive value is that the latter is expressing the expected lifetime value, whereas
the static version only measures the expected single period utility. This important distinction implies that the dynamic model also captures the endogenous future behavior of
all agents.
Following Gowrisankaran and Rysman (2009) it is possible to make an assumption
referred to as the Inclusive Value Sufficiency assumption. This assumption is the key to
the reduction of the state space’s dimensionality.
e
e
e
(Ωt )
|vi,t
|Ωt = F vi,t+1
Assumption A2 F vi,t+1
This implies that the dynamic inclusive value is a sufficient statistic to determine the
transition probabilities of all state variables measured under one. Therefore the consumers
only have to form this single expectation. In practice, most papers use some sort of
autoregressive model to determine the dynamics of the dynamic inclusive value according
to which the expectations are formed. This significantly eases the computational burden.
With this model framework at hand it is now possible to estimate and extract information about the single-period utility of the consumer. However, in the literature there
has been a broad division between whether to use consumer level or market level data in
the estimation procedure. Effectively, the major difference in the two approaches is that
models using consumer level data, can follow consumers over time, which is an obvious
advantage in a dynamic setting (See Rust, 1987), and calculate market shares by the
observed shares. But often such data is hard to come by. As models based on market
data only observe aggregated numbers, the model is estimated by a number of simulated
consumers, thereby obtaining the transitional dynamics (See e.g. Berry, Levinsohn, and
Pakes (1995) or Gowrisankaran and Rysman (2009) for an application of this procedure.).
16
4 DATA DESCRIPTION
Either way, the estimated dynamic inclusive values can be inverted to identify the
single period utilities. After recovering these estimates of the mean single-period utility,
these are used as the left hand side variable in a decomposition that recovers the marginal
utility of the observed product attributes. This has been done in many papers, but see e.g.
Berry, Levinsohn, and Pakes (1995) or Hotz and Miller (1993) for examples. Assuming
that the product specific error term ξj,t is normally distributed the decomposition will
look something like,
uj,t = γ0 + γt xj,t + ξj,t
(18)
Here γt is a K-dimensional vector of marginal utilities. It is in general assumed that
the error term is identical across individuals for the same product, such that there is not
allowed for agents to have different preferences over products. However, some papers like
Timmins (2007) have relaxed this assumption somewhat and allow for different perceived
marginal utilities across different types of consumer. This follows a projection of individuals into a type space, based on individual characteristics. The approach does however
require consumer level data.
4
Data Description
This thesis will estimate a model using consumer level data, and this section will describe
the large amount of data sources that are drawn upon. When presenting the data, some
emphasis will be put on explaining why a dynamic model of neighborhood demand will
be suitable.
4.1
Individual-level Data
The model will be estimated using Danish data. The first dataset that is used is the
Integrated Database for Labor Market Research (IDA) which is a Danish register based
database that match employers and employees. IDA contains the Danish population aged
15-74. For this model the focus will not be on the employer-employee data, but on the
17
4.1 Individual-level Data
4 DATA DESCRIPTION
large amount of socio-economic factors that is part of IDA. These factors are primarily
marital status, wealth, yearly pre-tax income, children etc.. All of the variables used are
measured ultimo of the year.
IDA contains a family variable, that link spouses/couples and their children living at
home. This variable will define the unit of observation as a family. Even though only
one of the family members owns/buys/sells a house, then it will be considered as a family
decision. The family identification number is a social security number of the oldest female
in the family, whenever the family is not a single living person. Therefore, the model
is actually modeling the choices of females’ families and of single men, when considering
the family over time. This feature will be ignored in the model of this paper, but it is
important to keep in mind, as there are some underlying dynamics of family formation
in the data base that might drive some of the results of the model. This is particularly
important when the families form expectations about the future states.
In Denmark there was a wealth tax from 1986 to 1997, and there was established an
automatic reporting system from banks and mortgage institutions during that period that
was managed by SKAT. The system was kept running after the abolishment of the wealth
tax, which is why the wealth data used is perceived to be of very god quality. These
wealth informations are of key importance as wealth will be the only endogenous variable
of the household in the model framework. One lack in the wealth information is that
savings in pension funds are not observed. This might constitute a problem for people
close to retirement as their effective wealth will be underestimated.
In the sample selection a data set called EJER is used to identify individuals that
own part of a property approved for permanent residence, i.e. not summer houses and
similar. The dataset also contains firms, but only families that are successfully linked to
IDA through PNR/VNR7 will be used.
Finally, I use a dataset, EJSA, that contains all sales that are registered from 1992
to 2010. This dataset makes it possible to identify families that choose to move and it
7
It is not possible to distinguish between a person’s and firm’s identification number in EJER and
EJSA.
18
4.2 House Data
4 DATA DESCRIPTION
contains sales price, sales date, and take over date. EJSA is the only dataset used that is
not cross sectional, but collecting all observations from the latest year. Due to some lag
in the reporting system for sales there is some timing issues when merging with EJER,
such that not all buyers are listed as owners of the house they buy.
Characteristic
Obs.
Mean
Std. Dev.
Min.
Max.
Wealth
15,428,712
545,829
1,533,710
-39,865,996
79,987,862
Income
15,428,712
485,468
500,258
-203,597,926
805,387,346
Children at Home 15,428,712
37.55%
48.42%
0
1
9.24%
28.96%
0
1
Buyers
15,428,712
Table 1: Summary Statistics of Household Characteristics
The sample will be limited to post 1991, as this is the first year that EJSA covers, and
end in 2005 after which it has not been possible to link all data. In Table 1 there is listed
some summary statistics of the family characteristics. These characteristics will later be
used to discretize the type space of households.
4.2
House Data
With individual level data on houses, including sales prices, it will be possible to clean
the house prices from the effects of house-specific characteristics. To do this two registries
are merged, BOL and BBR, where the BBR-registry is effectively replacing BOL after
2003. The two datasets contain a large amount of descriptive variables for houses, around
100 variables in both, but due to differences in definitions, many missing values, and
non-quantifiable measures, only eight of these are used in the estimation.
The BOL/BBR registries only contain house-specific information, so to get data on
the plot size the dataset is merged with EJMT that include this information.
The value of a house is measured as the sum of the public valuation of the plot and
of the house. These informations are obtained from the dataset EJVK, that is managed
by SKAT. The reason why the plot and house values are measured individually is that
19
4.2 House Data
4 DATA DESCRIPTION
Denmark has both a property tax and a land tax, where the property tax is recovered
to the state and the land tax is set by and recovered to municipalities. However, these
numbers are updated only every second year, and the algorithm for measuring house
values are not publicly available. Due to the lack of market information about property
values, the public estimates are corrected by the fractional deviation between the public
valuation and actual sales prices from EJSA. This correction is done for each year and
for each neighborhood, where neighborhood is defined below. For neighborhoods in which
there are many sales per year, this is very reasonable, however, in some neighborhoods,
only few or no sales are observed, in which case the correction might distort prices and
therefore no correction is done.
Characteristic
Obs.
Mean
Std. Dev.
Min.
Max.
Price
15,428,712
1,445,111
2,463,059
570
39,999,860
Baths
15,428,712
1.18
0.49
0
97
WCs
15,428,712
1.48
0.60
0
91
Floors
15,428,712
1.18
0.75
0
90
Rooms
15,428,712
4.68
1.63
0
465
House Size
15,428,712
137.23
50.88
0
16,490
Business Area
15,4287,12
0.94
8.60
0
2,500
Basement Size
15,428,712
21.02
161.48
0
56,712
Plot Size
15,428,712
31,509
873,170
0
303,994,267
House Age
15,428,712
52.23
40.16
0
1005
Table 2: Summary Statistics of House Characteristics
In Table 2 there is presented summary statistics of the house characteristics for the
sample period 1992 to 2005.
20
4.3 Neighborhoods
4.3
4 DATA DESCRIPTION
Neighborhoods
The aim of this paper is to determine the demand function by the marginal willingness to
pay for locally non-traded amenities. Therefore a subdivision of the geographical space
in the sample is needed. The Rockwool Research Foundation have financed such a subdivision at two levels. The smaller level clusters houses in groups of 150, and the larger
one clusters houses into groups of approximately 600. When doing this, some weight has
been put on the degree at which the neighborhoods are physically connected, it will for
instance be avoided that a neighborhood is divided by a highway or other such natural
divisions.8
In this paper, the large neighborhoods has been chosen as the geographical unit to
ensure a fair amount of sales in each neighborhood. This implies that the sample is
divided into 2296 neighborhoods that contain approximately 2300 inhabitants.
4.4
Local Amenities
The challenge is to identify locally non-traded amenities of the neighborhoods, that possibly explain the large differences in the demand for neighborhoods. The character of these
can be either dynamic or static as long as there is sufficient heterogeneity to explain the
cross sectional expectation of future states.
4.4.1
Measuring school quality
One of the measures to consider is the quality of public schools, which is allowed to evolve
over time. The calculations of this measure is based on IDA that contain the highest
educational level of individuals and a dataset containing all pupils in Denmark. Using
9th grade pupils as the best measure of the output of a public school, the measure will
be based on the probability that the pupil will attend high school at some later point in
time. The measure is constructed as a year-specific school fixed effect on the probability
that pupils attend high school. Private schools are neglected, as there is a selection effect
8
See Damm and Schultz-Nielsen (2008) for details on the neighborhoods.
21
4.4 Local Amenities
4 DATA DESCRIPTION
that will not be modeled. What you buy is the quality of the public school. Let Kt denote
the number of public schools in Denmark in year t and let di,k,t be a dummy variable for
pupil i going to school k. As high school attendance is a binary variable, denoted yi,t for
student i, the fixed effect is modeled in a logit model where there is included controls for
a variety parents socio-economic factors such as yearly income, degree of unemployment,
education level, marital status et cetera. Denote the mth of these M parent characteristics
by Yi,m,t .
Let Ai,t be the unobserved ability of pupil i and assume that this is linear in parental
characteristics and the school fixed effect, such that the binary choice of high school
attendance will be determined as



1
PK
PM
m=1 βm+K Yi,m,t + i,t > 0,
yit = 
PM

0 ifA = β + PK β d
i,t
0,t
m=1 βm+K Yi,m,t + i,t ≤ 0.
k=1 k i,k,t +
ifAi,t = β0,t +
k=1
βk di,k,t +
then it is straight forward to form the likelihood function and obtain the school fixed
effect9
As the datasets does not contain information regarding the home of individuals at
ages below 15 years, the school quality can only be linked to neighborhoods through parents PNR. This implies that pupils with parents that are divorced will possibly link some
neighborhoods to schools far away, therefore neighborhoods which are linked to multiple
schools will be measured by the mean school quality, neglecting single observations. The
motivation for doing this instead of taking a simple average is that neighborhoods consisting of only 600 households are very unlikely to have many 9th grade students. In this case
these falsely linked pupils might imply a large measurement error for some neighborhoods.
4.4.2
Crime
Whereas high school quality is expected to positively influence the neighborhood quality,
crimes are oppositely expected to lower perceived quality. From the database AFG that
contain all convictions in Denmark from 1980, it is possible to link these to the neighborhoods and measure crimes as the number of convicted inhabitants per 1000 inhabitants.
9
For a specification of the applied estimation procedure see e.g. Wooldridge (2002).
22
4.4 Local Amenities
4 DATA DESCRIPTION
Notice that the year in which the conviction will be measured is the year of the conviction
and not necessarily the year of the actual crime. The variables in AFG make it possible to
distinguish between types of crime to a very specific extend. In this paper this information
is used to divide convictions into three subgroups, sexual offenses, violent crime and property crime. Considering all of Denmark, the amount of convictions for the two first types
have been increasing over the sample period from 500 and 5,000 convictions respectively
to 1,500 and 15,000 convictions respectively. Oppositely the amount of convictions within
property crime fell from 45,000 to 25,000 over the sample period.
4.4.3
Neighbors
After having linked individuals to neighborhoods, it is possible to identify various characteristics of inhabitants in the different neighborhoods.
IDA contains a measure of the ethnicity of individuals such that it is possible to calculate the share of inhabitants in a neighborhood that are ethnic Danes. Hence, the fraction
of neighbors that are ethnic Danes will be included as a neighborhood characteristic, as it
is assumed that households would prefer living next to neighbors with the same cultural
background as its own.
The educational background of the neighborhood is measured by the first sub-grouping
of the variable hffsp that ranks all educations by the prescribed study period. The variable
will be on the interval [10, 70] and increasing in prescribed study period10 .
Furthermore, I include such measures as mean income, mean wealth and mean unemployment degree. The latter is a register-based measure that sums over the number of
days of unemployment during the latest year calculated per thousands.
Table 3 presents summary statistics for the neighborhood characteristics, however not
all of these will be used in the estimation procedure as some prove insignificant.
From these statistics if is clear that there is significant cross sectional variation, which
is necessary in order to identify differences in demand. If a variable were to be uniform
10
See http://www.dst.dk/Statistik/dokumentation/times3/emnegruppe/emne/variabel.aspx?sysrid=372481&
timespath=19%7C1013%7C for documentation
23
4.4 Local Amenities
Amenity
4 DATA DESCRIPTION
Obs.
Mean
Std. Dev.
Min.
Max.
Sales
31,836
60.35
48.91
0
1741
Price
31,836
1,784,903
3,784,098
0
91,140,844
Mean Plot Size
31,836
4.33
10.54
0
363.91
Inhabitants
318,36
1,653
736
297
5,715
Mean Income
31,836
221,634
59,367
14,108
908,503
Mean Wealth
31,836
178,643
187,181
-282,853
3,155,791
Unemployment Degree 31,836
5.66%
2.92%
0.13%
31.50%
Ethnic Danes
31,836
92.55%
7.68%
18.92%
100.00%
Educational Level
31,836
26.78
4.37
14.17
43.37
School FE
31,836
0.10
0.54
-7.61
6.52
Sexual Offenses
31,836
0.25
0.50
0
11.46
Violent Crime
31,836
2.42
2.32
0
33.44
Property Crime
31,836
10.83
9.51
0
131.19
Population Density
31,836
1,052.71
1,750.57
13.32
8,011.29
Table 3: Summary Statistics of Neighborhood Amenities
24
4.5 Sample Selection
4 DATA DESCRIPTION
across neighborhoods, this variable would not be identified as a positively or negatively
valued amenity in demand, even though it might be. For example, it is not possible to
estimate the demand for public health insurance, as the level of this is the same across
Denmark, however, imagine that some region did not provide public health insurance,
then it is very likely that this would be a positively valued locally non-traded amenity.
Also the time series properties are important to consider in order to identify the need
for a dynamic model. If the amenities were constant over time then the need to build a
dynamic model would be reduced. However, a static model applied to a dynamic context
would be biased in its estimates of households marginal willingness to pay. In particular,
such a static model would overestimate the demand for strongly persistent amenities,
such as the characteristics of your neighbors. Oppositely, the simple static model would
underestimate the willingness to pay for mean reverting amenities, which is the case for
the school fixed effect and the three measures of crimes. These biases stem from the
expectation of the consumers that know these time series properties of the amenities in
the real world.
In Appendix A0 the distributions of these neighborhood amenities are illustrated.
4.5
Sample Selection
To provide an idea about the properties of the final dataset, this subsection will briefly
go through how this dataset has been created by merging the raw datasets.
The starting point of the sample is IDA. Restricting IDA to 1992-2005, IDA is merged
with BOL/BBR by the municipality number, kom, and then the house number in municipality, bopikom, in which approximately 10 thousand houses are not matched and deleted
per year. When merging the new set with EJER a lot of observations are lost from EJER,
which is firms. There is 56% of the individuals that does not own a house, however, these
are not deleted yet, as some of them might be first-time buyers, that are not registered
yet. This problem arise, as there are some timing issues in EJER, which also implies that
the total share of owners for some houses in some years are above 100%.11 Next, when
11
In these cases the owner shares are scaled down such that they add up to 100%.
25
4.5 Sample Selection
4 DATA DESCRIPTION
merging with EJMT, EJVK and EJSA only very few observations are lost. The dataset is
then shrunk to individuals that are either owning a house or appear in EJSA as a house
buyer. The reason why buyers that are not listed as owners are allowed is that it seems
to be EJER that contains the timing issues.
The sample file provided by The Rockwool Research Foundation identifying the neighborhoods contain all individuals that live in a house in 2003. This uniquely links social
security numbers to the neighborhood identification numbers. However, the dataset is
only available for 2003. By the PNR it is possible to link houses to to neighborhoods as
IDA contain this link. The identifier for the house is municipality number and then house
in municipality. There are two challenges that shrinks the sample size. The first is that
not all neighborhoods existed in 2003, such that they either died out before 2003 or emerge
in the years after. Likewise, not all neighborhoods that exist for the rest of the sample
period. This last observation leads to the second challenge: the municipality reform. The
reform in 2007 merged the 271 previous municipalities into the 98 municipalities of today.
The problem arise, as the municipality numbers (kom) change, such that houses can no
longer be identified. Therefore the sample is cut off already from 2006 and later, as the
variables are dated ultimo of the year. As Bornholm, for some reason, merged before 2003,
the problem is opposite for this part of Denmark. Altogether this implies that Bornholm
and a few other neighborhoods are deleted from the sample, such that the sample shrinks
from 2296 to 2274 neighborhoods.
Finally, some observations are perceived to be outliers and deleted. This identification
is based on income, wealth and house prices, which reduce the sample by 0.5%.
Altogether, the data used provide sufficient heterogeneity to estimate a flexible demand
system, however, some heterogeneity will be sacrifices regarding families, in order to deal
with the dimensionality problem.
26
5 THE MODEL
5
The Model
This section will formally set up a model equivalent to the one proposed by Bayer, McMillan, Murphy, and Timmins (2011). The aim is to obtain the Bellman equation needed
to back out the choice specific value function. From there it will be easy to obtain the
single-period utility of the household from living in a specific neighborhood.
First, it is necessary to introduce some notation. At time t every household makes
a choice between moving and staying. Here household will choose to move if there is a
value gain after deducing moving costs. Denote the decision variable of a household by
di,t , where the subscript t denotes the time period for which the decision is made, and
i identifies the household in question. The choice variable identifies not only whether
to move or not, but also to which neighborhood the household will move. Let J be the
number of neighborhoods and let 0 denote the outside option, that is if a household chooses
to either leave the housing market, e.g. if choosing to rent instead of owning a house, or
if the household moves to a neighborhood located outside Denmark12 . If the household
chooses to stay in the current location, then the choice will be J + 1. Hence, the choice
will be in the set di,t = j ∈ {0, 1, . . . , J + 1} and the number of choice options is J + 2.
For each point in time there are two observable state vectors and a state variable
{xj,t , zi,t , hi,t }. Here, j identifies the neighborhood and xj,t is a vector of neighborhood
specific attributes that are allowed to evolve over time. Such attributes will cover school
quality, crime rates etc.. The vector zi,t contains household specific characteristics such
as wealth, income, social status and number of children. The last variable, is the current
location of the household, which is equivalent to the neighborhood chosen one period ago,
hence, hi,t ∈ {0, 1, . . . , J}.
In addition to the three observed state variables, there are two unobservable variables
in the model. The first is an unobserved quality of the neighborhood, which is a collection
of amenities that are not possible to quantify and not embedded in xj,t denoted by ξj,t .
The other is an idiosyncratic stochastic shock to the value of a household associated with
12
In practice this option also include moving to Bornholm and a few other neighborhoods, which has
been deleted in the sample selection.
27
5 THE MODEL
each decision option, εi,j,t .
Before proceeding, define Ωi,t = {zi,t , xt , ξt , zi,t−1 , xt−1 , ξt−1 , . . . }, that is an information
set that contain all information sets about household and neighborhoods, that can possibly
help forecasting future characteristics of the household and neighborhoods.
Neglecting the cost of moving, let ui,j,t = u (zi,t , xj,t , ξj,t , εi,j,t ) be the single period
utility function of the individual household from living in neighborhood j. Then, let
ci,t = c(zi,t , xhi,t ,t ) denote the single period costs associated with a move decision. This is
zero if the household chooses not to move. These costs of moving are allowed to cover both
financial and psychological factors that affect the single period utility of the household. As
Bayer, McMillan, Murphy, and Timmins (2011) points out, this specification allows the
moving costs to be only a function of the characteristics of the neighborhood that is left.
This assumption implies that the actual single period utility function can be represented
by uci,j,t = ui,j,t −ci,t I[j 6= J +1]. Hereby it is implicitly assumed that the utility is additive
separable in the cost of moving. They further assume that realtor fees are uniform across
neighborhoods and proportional to the average value of the houses in the neighborhood.
As will be argued in the Evaluation below, this is highly unreasonable for the Danish
housing market. This model will instead assume that the realtor fees are linear in the
value of the house as described below.
The transition probabilities of the model are Markovian and denoted,
qi,t = q(Ωi,t+1 , hi,t+1 , εi,t+1 |Ωi,t , hi,t , εi,t , di,t )
(19)
Assuming that the subjective rate of time preference, β, is constant implies that the
household’s infinite horizon objective function is the expectation,13
Ui,t =
uci,j,t
+ Et
" ∞
X
β
τ −t c
u
(xj,τ , ξj,τ , εi,τ , hhi,t ,τ ) #
Ωi,t , hi,t , εi,t , di,t
(20)
τ =t+1
This is an autonomous problem as it does note depend directly on time. Assuming that
13
The models robustness toward alternative specifications of both the subjective discount factor and
the realtor fees will be examined in Section 8 below.
28
5.1 Assumptions
5 THE MODEL
{d?i,t }∞
t=τ is the optimal control makes it possible to define the lifetime value function as
Vt (Ωi,t , hi,t , εi,t ) = max
Et
∞
{di,τ }τ =t
"∞
X
β
τ −t c
u
(xj,τ , ξj,τ , εi,τ , hi,τ ) #
Ωi,t , hi,t , εi,t , di,t
(21)
τ =t
As the time horizon is infinite, the time subscript on the value function can be dropped.
Standing at time t then choosing an optimal control will yield a total reward over all
subsequent periods equal to V (·) and the immediate reward will be the single period
utility, such that the Bellman equation for the household becomes
i
h
V (Ωi,t , hi,t , εi,t ) = max uci,j,t + βEt [V (Ωi,t+1 , hi,t+1 , εi,t+1 |Ωi,t , hi,t , εi,t , di,t )]
di,t
(22)
Given that that the lifetime value function is appropriately bounded, this Bellman equation will be a contraction mapping in V (·)14
Before we can proceed to the estimation of the model it is necessary to impose some
distributional assumptions.
5.1
Assumptions
In order to extract the single period utility from the Bellman equation, it is necessary to
make a few assumptions about the observed and unobserved state variables. These assumptions are inspired by Rust (1987) that defined Additive Separability and Conditional
Independence to cope with the unobserved state variables.
Additive separability
It will be assumed that the single period utility function, ui,j,t , and the idiosyncratic
error term, εi,j,t , are additive separable. Hence, it follows from the definitions above that
the non-moving cost single period utility will be given as
uci,j,t = u(xj,t , ξj,t , zi,t ) − c(zi,t , xhi,t )I[j6=J+1] + εi,t
14
(23)
See section 12.7 in Sydsaeter, Hammond, Seierstad, and Strom (2008) for a range of alternative
boundedness conditions that suffice to make the equation a contraction mapping in V (·)
29
5.1 Assumptions
5 THE MODEL
Conditional Independence
To obtain conditional independence it is assumed that the idiosyncratic stochastic
shock at time t, εi,j,t , does not have any predictive power on the realization of the state
variables in the subsequent period, {Ωi,t+1 , hi,t }. Thereby it is possible to break the
conditional probability of the realizations of future states in equation (19) into three
multiplicative transition densities. Hence, the following representation of the density
function will hold,
q (Ωi,t+1 , hi,t+1 , εi,j,t+1 |Ωi,t , hi,t , εi,j,t , di,t ) = qω (Ωi,t+1 |Ωi,t , di,t )qh (hi,t+1 |di,t )qε (εi,t+1 ) (24)
In particular, note that the current periods location, hi,t , will have no predictive power
against any future state, as this information is fully captured by the current periods
locations decision, di,t . Furthermore, the current location decision, di,t , is sufficient to
perfectly predict the future location state, hi,t+1 , hence, this density function is a binary
function that either takes the value 1 if di,t = hi,t+1 and the value 0 otherwise.
The Idiosyncratic Stochastic Shock
For easing the calculations to come, and to follow the standard assumption for dynamic
models (See Berry (1994) for an application in a comparable model framework.) and
assume that the idiosyncratic stochastic shock, εi,j,t , is independent identically distributed
Type 1 Extreme Value with the cumulative density function
Fε (εi,j,t ) = e−e
−εi,j,t
(25)
It is important that this error term is independently distributed both across families and
across neighborhoods. This assumption makes it possible to identify the ’market’ share
of households that chose a given neighborhood in a logit-type formula. Appendix A1
illustrates the Type 1 Extreme Value distribution. This will be utilized in the Stage 1 and
Stage 2 of the estimation procedure.
30
6 ESTIMATION
Given these assumptions it is now possible to obtain the choice-specific value function,
"
vjc (Ωi,t , hi,t )
= ui,j,t − ci,t I[j6=J+1] + βEt log
J+1
X
vkc (Ωi,t+1 ,hi,t+1 )
e
k=0
!
#
Ωi,t , di,t = j
(26)
From here it follows that also the choice-specific value function can be divided into two
additive components, the first being a choice specific value function, neglecting the cost
associated with a move, and the other being the potential moving cost incurred by the
optimal location decision, vjc (Ωi,t , hi,t ) = vj (Ωi,t ) − c(zi,t , xhi,t )I[j6=J+1] . Here the value
function is not a function of the current location as it follows from the right hand side of
equation (26) that only the potential moving cost is a function hereof. Hence,
"
vj (Ωi,t ) = ui,j,t + βEt log
J+1
X
e
k=0
vkc (Ωi,t+1 ,hi,t+1 )
!
#
Ωi,t , di,t = j
(27)
This is the fundamental equation that will be estimated in the subsequent section.
The equation is used to subtract informations about the average single period utility that
households get from living in neighborhoods with particular compositions of the various
locally non-traded amenities.
6
Estimation
Based on the model presented above, the estimation will proceed in four stages in order
to recover the model primitives and thereby determine the willingness to pay for a range
of locally non-traded amenities. Except for Stage 0, these steps follow Bayer, McMillan,
Murphy, and Timmins (2011) very closely, but with some corrections and a few adjustments to make the model fit the Danish data the best possible.
The estimation procedure will be organized as follows. First, Stage 0 will estimate
Rosen’s first step, to identify equilibrium price dynamics, and at the same time it will
make a normalized index for neighborhood prices. Then, Stage 1 will start the estimation of the dynamic model of demand by identifying the choice specific value function
of each household, type, and neighborhood combination, which is done as a function of
shares that move to the neighborhoods. Stage 2 will then estimate a binary choice model
31
6.1 Stage 0 - Hedonic Price Regression
6 ESTIMATION
between moving and staying, and at the same time recover estimates of the marginal
value of wealth and calculate the normalized value function. Hereafter, Stage 3 will use
the neighborhood prices and the normalized value function to estimate dynamics of the
model. These dynamics are used in a simulation procedure to extract information about
future expectations of the state variables to identify the single-period utilities of household
types and neighborhood combinations. The last Stage 4 will then use the marginal value
of wealth to extract the disutility of user costs of housing from the single period utility,
in this way handling the endogeneity of the model, and then decompose the recovered
single period utilities to identify the marginal willingness to pay for the various locally
non-traded amenities.
6.1
Stage 0 - Hedonic Price Regression
The aim of this initial stage is two-fold and the notation will be slightly different compared
to the rest of the model as in this stage the subscript i will not only identify the household,
but also the house that the household owns.
First of all, this stage will estimate a version of the first step of the Rosen (1974)
procedure, that include both house-specific attributes denote xi,t and locally non-traded
amenities denoted xj,t . For the sake of later comparison, I will rely on an assumption
that the hedonic price function is linear in the locally non-traded attributes, but allow
for non-linearities in the house-specific attributes by including squared terms for these.
Hence, assuming that the error term is i.i.d. N (0, σp2 ) the following hedonic price function
is estimated using ordinary least squares
pi,t (xi,t ) = a0 + a1 xi,t + a2 x2i,t + a3 xj,t + i,t
(28)
Following this simple estimation, the resulting estimate â3 gives the marginal equilibrium
hedonic price of local amenities, that is not to be confused with the marginal willingness
to pay, cf. Section 2. The aim is to compare this result with the outcome of the dynamic
model of demand in order to identify the importance and need for a dynamic model of
demand.
32
6.2 Stage 1 - The Choice Specific Value Function
6 ESTIMATION
Secondly, this stage will estimate a version of (28) that neglect the neighborhood
specific amenities. That is
pi,t (xi,t ) = b0 + b1 xi,t + b2 x2i,t + ζi,t
(29)
where ζi,t is assumed i.i.d. N (0, σζ2 ). One insight from this estimation is that it identifies
the omitted variables bias that has been one of the main problems that the first step of
Rosen’s procedure suffers from. However, the main purpose of doing this is to clean house
prices for house-specific attributes, such that the subsequent estimation stages will use
the empirical distribution of ζi,t to forecast future house prices. The intuition behind this
is that by successfully extracting the house-specific value of a house leaves only the value
of the neighborhood in the residual, hence in the later stages ζi,t will be interpreted as
the price of the neighborhood. To look a little forward, this is what will be used to form
expectations about future states in Stage 3 of the estimation.
6.2
Stage 1 - The Choice Specific Value Function
In this stage of the estimation procedure only the households that have chosen to move
will be considered. These are used to determine the choice specific value functions as
these households will choose to move to the household that yields the highest expected
choice specific lifetime value, vjc . The idea is to utilize that conditional on moving, the
monetary moving cost, ci,t (zi,t , hi,t ), will only be an additive constant to the households
neighborhood choice, as it is assumed only to be a function of the characteristics of the
neighborhood left, hence it can be neglected. Then the households will choose,
d?i,t = arg max
[vj (Ωi,t ) + εi,j,t ]
J
(30)
{j}0
Before proceeding, the households are divided into groups based on multidimensional
household characteristics. This discretization of the type space allows the estimation
procedure to circumvent the curse of dimensionality, as it is impossible to identify the
probability that a single household will move to a specific neighborhood without clustering
households into groups of households that are alike. Hence, letting τ denote the household
33
6.2 Stage 1 - The Choice Specific Value Function
6 ESTIMATION
type implied by the household characteristics zi,t , then the value function of the household
τ
from moving to neighborhood j will be given by vj,t
= vj (Ωi,t ). This discretization of the
type-space will affect the single period utility of the household in a comparable way, such
that uτj,t will denote the type-specific single period utility. This changes the value function
specification in equation (27) to,
"
τ
vj,t
=
uτj,t
+ βEt log
J+1
X
τ
e
vkt+1 (Ωi,t+1 )−cτt+1 (zi,t+1 ,xhi,t )I[k6=J+1]
k=0
!
#
Ωi,t , di,t = j
(31)
h
i
Now, the type specific optimal choice will be determines as d?i,t = arg max{j}J0 vjτ + εi,j,t ,
conditioned on the household having decided to move. Since the idiosyncratic stochastic
shock is identically independently distributed Type 1 Extreme Value, then the probability
that households of type τ will choose to move to a certain neighborhood j at time t, as a
function of the set of value functions, will be given by
evj,t
τ
Pj,t
(v τt ) = PJ
τ
vk,t
k=0 e
(32)
These probabilities are calculated by using the share of the N1 inside buyers that belong
to the type Z τ who choose a specific neighborhood, i.e.
PN1
τ
P̃j,t
=
i=1 I[di,t =j] · I[Zi,t ∈Z τ ]
PN1
i=1 I[Zi,t ∈Z τ ]
(33)
This is estimated for all the inside buyers, N1 , that is for all buyers that does not choose to
move to somewhere outside the Danish housing market, which is independent of whether
the household owned a house in Denmark beforehand, was renting or lived abroad. Oppositely, the outside shares will be based on the N2 buyers and sellers in a given year t,
but not taking account of the potential entrants to the housing market, that choose not
to buy in Denmark,
PN2
τ
P̂0,t
=
i=1 I[di,t =0] · I[Zi,t ∈Z τ ]
PN2
i=1 I[Zi,t ∈Z τ ]
(34)
Then the inside shares are corrected down to reflect the possibility that the type might
leave the Danish housing market. In this way the final estimates of the transition proba-
34
6.3 Stage 2 - The Costs of Moving
6 ESTIMATION
bilities are calculated as15
τ
τ
τ
P̂j,t
= 1 − P̂0,t
· P̃j,t
(35)
With these transition probabilities at hand, it is possible to recover the value function
for each type, year and neighborhood combination. It is possible to determine an estimate
of the unnormalized value function as,
τ
τ
ṽj,t
= log(P̂j,t
)−
J
1 X
τ
log(P̂k,t
)
J + 1 k=0
(36)
However, these value functions are only unique across neighborhoods up to an additive
constant. Hence, the average lifetime value across neighborhoods will be equal across
τ
τ
types. Therefore it is useful, to define the unnormalized value function as ṽj,t
= vj,t
− mτ ,
to be able to allow for differences in mean lifetime value across types. Intuitively, these
unnormalized value functions rank the neighborhoods attractiveness only for each type
τ
, and the normalizand period. The identification of the normalized value function, vj,t
ing constant, mτt , will be handled in Stage 2 of the estimation. These will then ensure
consistency in the value functions across types.
6.3
Stage 2 - The Costs of Moving
This second stage of the estimation procedure will use the estimated unnormalized value
functions to determine the decision about whether to move from the current house or not.
The approach utilizes the dynamic property of the housing market that stem from two
sources, which both are evident from the type specific value function presented in equation
(31). The first is that moving is associated with some costs in the present period, with
longer term effects on wealth. Secondly, as it is very costly for the household to re-optimize
in every period, there is an effect on the current period utility as well as for the single
period utilities of the periods to come of the decision today.
15
As it follows from the normalized value function below, zero-shares will constitute a problem, hence,
it is assumed that there exists a strictly positive lower bound, µ ∼ 0, on the probability of any type
τ
τ
moving to any neighborhood, i.e. P̂j,t
= max[µ, P̂j,t
]
35
6.3 Stage 2 - The Costs of Moving
6 ESTIMATION
Households will only move if the lifetime value, normalized or unnormalized, is bigger
when moving than when staying. This fact is used to estimate the moving costs, as it is
known for all households whether or not they chose to move in a given year. In practice
the moving costs are decomposed into two terms, financial moving costs, f (xhi,t ,t ), that
are assumed only to depend on the neighborhood that is left and psychological moving
costs, p(zi,t ) that are a function of the characteristics of the family. Hence,
c(zi,t , xhi,t ) = f (xhi,t ,t ) + p(zi,t )
(37)
In practice, this term is calculated as 4% of the value of the house sold. Thereby I
assume that realtor fees are linear in the value of the house sold. This differ from Bayer,
McMillan, Murphy, and Timmins (2011) that assume that the realtor fee is linear in the
average price of houses in the neighborhood left. This is not consistent with the Danish
housing market, which is much more house dependent and dependent on how much sellers
are willing to participate in selling the their house. In section 8 the models robustness
regarding different assumptions on the realtor fee is investigated.
The only endogenous characteristic of the household in this model is the wealth. If
a household choose to move, then this will have a negative impact on the wealth that
potentially changes the type of the household. If the household is of type τ before the
move, then it will be of type τ̄ after the realtor fee, f (xhi,t ,t ), have been paid to the realtor.
τ
τ
Define yi,t
as the binary decision variable of the household, where yi,t
= 1 is equal to
τ
choosing not to move and yi,t
= 0 is a choice of moving to the neighborhood that yields
the highest lifetime utility. This choice can be modelled as,
#
"
τ
yi,t
=I
τ
vJ+1,t
+ εi,J+1,t >
τ̄
max
[vk,t
J
{k}0
+ εi,k,t ] − p(zi,t )
(38)
As the normalized value function is yet unknown, substitute in for the unnormalized value
function and the normalizing constant to obtain the following form of the decision rule,
"
τ
yi,t
=I
τ
ṽJ+1,t
#
+ εi,J+1,t >
τ̄
max
[ṽk,t
J
{k}0
+ εi,k,t ] −
(mτt
−
mτ̄t )
− p(zi,t )
(39)
Here, the difference between the two normalizing constants, (mτt −mτ̄t ), can be measured as
the baseline difference in value between being of type τ and being of the type with the lower
36
6.3 Stage 2 - The Costs of Moving
6 ESTIMATION
wealth, τ̄ . Instead of estimating these baseline differences using dummy variables in the
estimation procedure hereafter I follow Bayer, McMillan, Murphy, and Timmins (2011)
and parametrize the differences to financial moving costs, f (xhi,t ,t ), and to household
characteristics, zi,t , which are assumed be multiplicative, such that In a similar way,
the psychological moving costs are parametrized to household characteristics, assuming
linearity in the arguments, hence,
0
p(zi,t ) = zi,t
γp
(40)
As in the first stage, it is used that the idiosyncratic stochastic shocks are i.i.d. Type 1
Extreme Value. Therefore, it is possible to identify the probability of no move as16 ,
τ
τ
P (yi,t
= 1) =
eṽJ+1,t
τ
eṽJ+1,t +
PJ
k=0
0
τ̄
(41)
0
eṽk,t −f (xhi,t ,t )·zi,t γf −zi,t γp
Knowing each households move-stay decision for each year, it is possible to form a likelihood function that can be used to estimated the financial and psychological cost components. The likelihood function is formed from the joint density function
N
τ
τ
L = ΠN
i=1 f (yi,t |zi,t , xhit ) = Πi=1 Pi,t
y τ i,t
τ
1 − Pi,t
1−yτ
i,t
(42)
To ease the maximization and avoid numerical errors, I turn to the average log-likelihood
function
l̄ =
N
1 X
τ
τ
τ
τ
{yi,t
log Pi,t
+ 1 − yi,t
log 1 − Pi,t
}
N i=1
(43)
With the resulting estimate of the financial costs of moving, γf , it is straight forward
to determine the baseline differences across types from (??). Having both identified the
baseline differences and the unnormalized lifetime values, it is simple to extract the norτ
τ
malized lifetime value for the households by vj,t
= ṽj,t
+ mτt . To make this identification
it is only to choose an appropriate fixed point around which to determine the differences.
Therefore, the mean lifetime utility from being of the lowest wealth-type, across all other
type-characteristics and neighborhoods, is set to zero. Around this measure is the set of
lifetime values unfolded.
16
See Cameron and Trivedi (2005) for a derivation of this result for the bivariate case
37
6.4 Stage 3 - The Single Period Utility
6 ESTIMATION
Because the likelihood function is based on a comparison of different choice options that
are associated with a deduction in wealth, the estimates of the parameters in this stage of
the estimation procedure, can be used to identify the utility that a household would receive
from additional wealth. Hence, the baseline differences, is the average utility gain/loss
associated with changing type through wealth accumulation/financial moving costs, and
will be used in later sections as the determinant of the marginal value of wealth.
6.4
Stage 3 - The Single Period Utility
The main purpose of the third stage of the estimation procedure is to recover the typespecific single period utility by estimating the transition probabilities to obtain the expectation in equation (31).
In the first two stages there is obtained estimates of the normalized value function,
moving costs and the marginal value of wealth. These estimates will now be utilized
to determine how the state variables transition. The section will follow Assumption 2
from Section 3 in assuming that the household only considers how the normalized lifeτ
transitions and only use information about todays’ states to predict these
time value, vj,t
transitions. Furthermore, the households need to form expectations about how future
prices of the house that they currently own, will transition. This is essential as wealth is
the only endogenous determinant of type, therefore, the dynamics of the property value
will determine the potential wealth gain and the financial moving costs associated with
a move in the future, and hence, it will determine future types. These effect are central
features from the dynamics of this model. With these simple assumptions it is possible to
determine the transitional dynamics of the lifetime value and house prices as follows.
With estimates of all type-neighborhood-year combinations of the lifetime value, the
dynamics is determined by an autoregression including the time varying attributes of the
neighborhood that potentially affect the future lifetime value of the neighborhood. Here is
used local crime rates, public school quality and ethnic composition of the neighborhood.
The time series properties of these attributes, such as persistence or mean-reversion, are
essential to identify and explain the marginal willingness to pay, which is obtained in
38
6.4 Stage 3 - The Single Period Utility
6 ESTIMATION
the last stage of the estimation. Lags of the life time value is included in the regression
to control for dynamics in unobserved neighborhood heterogeneity, hence, also allowing
unobserved neighborhood attributes to influence the transitional dynamics of the neighborhood quality.
τ
= ρτ0 +
vj,t
L
X
τ
+
ρτ1,l vj,t−1
L
X
τ
x0j,t−l ρτ2,l + ςj,t
(44)
l=1
l=1
As for the outside option, where there is no data available on neighborhood characteristics, the transition probabilities are only allowed to depend on lagged lifetime values.
Equivalent to the inside options, this allows unobserved attributed to affect the transitional dynamics where the known amenities are implicitly included as a subset of the
unobserved characteristics.
τ
v0,t
=
ρτ0
+
L
X
τ
τ
ρτ1,l v0,t−1
+ ς0,t
(45)
l=1
τ
is assumed i.i.d. N (0, σς2j ) ∀ j ∈ {0, . . . , J}.
Here ς0,t
When turning to the transitions of house prices, the transitional dynamics will be
determined from a regression like in (44). But, instead of using the actual house prices.
the residual from the regression in Stage 0 will be used. cf. Frisch-Waugh’s Theorem17
this will identify the dynamics of the house prices, that are cleaned from house specifics
which is equivalent to the dynamics of the neighborhood price. The motivation for this
approach is to eliminate the endogeneity from larger household investments, which is not
explicitly modeled in this framework. In this way it is possible to extract a neighborhood
specific error term, and thereby obtain the empirical distribution of house prices, that
households are faced with, when making a move-stay decision.
In practice, I form a neighborhood price index, ζ, that is the average value of houses,
cleaned for house-specific characteristics. This index is regressed on lagged neighborhood
amenities and lags of itself to obtain the empirical distribution and thereby the transition
probabilities of neighborhood prices.
ζj,t = %0 +
L
X
x0j,t−l %1,l +
l=1
17
L
X
l=1
See Lovell (2005)
39
0
ζj,t−l
%2,l + νj,t
(46)
6.5 Stage 4 - The Marginal Willingness to Pay
6 ESTIMATION
Using the estimates from this autoregression and combining these with the fitted values
from Stage 0 of the reduced regression (29) gives estimates of the expected value of the
house in the next period.
Et [pi,j,t+1 ] = b̂0 + x1,i,t b̂1 + x21,i,t b̂2 + %̂0 +
L
X
x0j,t−l %̂1,l +
L
X
l=0
0
ζj,t−l
%̂2,l
(47)
l=1
This expectation suffices to perfectly predict the expectation of future wealth and type as
well. In the following simulation procedure, only the mean prices of each neighborhood
will be estimated, therefore the subscripts i will be neglected.
As assumed, the households will use the empirical distribution, from the set of regressions above to form expectations about the future. Thus, there is enough information to
calculate the single period utility of a type τ household living in neighborhood j in the
year t as,
"
uτj,t
=
τ
vj,t
− βEt log
J+1
X
τ
e
vkt+1 (Ωi,t+1 )−cτt+1 (zi,t+1 ,xhi,t )I[k6=J+1]
k=0
!
#
Ωi,t , di,t = j
(48)
In order to estimate the expectation of future lifetime values, a large number, R, of
simulations is created for each type, neighborhood and year combination. This yield
series of estimated single period utilities uτj,t (r), where r denotes the simulation number.
The draws in each simulation is taken from the empirical distribution of the error terms
from the regressions above, to match the empirical expectations of households. The draw
on νj,t+1 is important to determine the future type, τt+1 , and thereby moving costs, cτt+1 .
Then calculate the mean single period utility as
uτj,t =
R
1 X
uτ (r)
R r=1 j,t
(49)
from which it is possible to extract estimates of marginal willingness to pay for various
neighborhood amenities by a decomposition in the following fourth and last stage of the
estimation procedure.
6.5
Stage 4 - The Marginal Willingness to Pay
This subsection makes a decomposition of the type specific single period utility, to measure
the effect from individual neighborhood amenities on demand. Basically this is done by
40
6.5 Stage 4 - The Marginal Willingness to Pay
6 ESTIMATION
running the following regression
τ
uτj,t = α0 + x0j,t α1 + ξj,t
(50)
In contrast to what is the standard assumption in the literature (See e.g. Berry, Levinsohn, and Pakes, 1995) where the unobserved neighborhood quality, ξj,t , is assumed to be
identical across types, here I will do as Bayer, McMillan, Murphy, and Timmins (2011)
and Timmins (2007) and assume that the unobserved neighborhood quality is allowed to
be differently perceived by the different types.
One of the neighborhood characteristics is the user-cost of housing, denoted ucj,t , which
is calculated as 5% of house value. However, the user costs are endogenous as user-costs are
high in neighborhoods with high house prices, which would be due to high neighborhood
quality equivalent to high neighborhood utility of households. However, having user-cost
as an explanatory variable would falsely lead to the conclusion that user-cost is a coveted
local good.
To handle the endogeneity of this model, but avoiding the difficulty of identifying good
instruments that are correlated with price but not neighborhood quality, I follow the novel
approach by Bayer, McMillan, Murphy, and Timmins (2011). Recalling the estimate of
0
γf , it is possible to subtract the
the marginal value of wealth from Stage 2, that is zτ,t
disutility of paying user-costs from the single period utility on the left hand side of the
decomposition in equation (50). Here, the vector x̃j,t is the amenities of neighborhood j
but excluding the user costs.
0
τ
uτj,t − ucj,t zτ,t
γf = α0τ + x̃0j,t α1τ + ξj,t
(51)
In this way it is straight forward to identify the marginal utility of the neighborhood
amenities in x̃ by least squares estimation.
It is not particularly clear how the estimated marginal utilities in the vector α1τ should
be interpreted. By using the obtained estimate for the marginal utility of wealth from the
third estimation stage as above, it is possible to turn the marginal utilities into marginal
41
7 RESULTS
willingness to pay estimates, denoted wτ , by the transformation
wτ =
α1τ
zτ0 γf
(52)
Hereby, the estimation procedure is completed, as the marginal willingness to pay
estimates are the measures of demand that this thesis would like to uncover. The results
can be found in the subsequent sections.
7
Results
In the following section, the results from the estimation procedure will be presented and
discussed in the chronological order from the estimation. The standard error estimates
that follow the Maximum Likelihood estimation in Stage 3 is calculated numerically using a Delta Method for which the specifics can be found in Appendix 2. The standard
errors presented with all least squares estimates are all robust and calculated as shown in
Appendix 3.
In this and the subsequent section all results are measured in DKK thousands.
7.1
Hedonic Prices
The first estimates to be considered is the results from the first step of Rosen’s two-step
procedure. These estimates correspond to the hedonic equilibrium marginal costs of the
various attributes and these results are presented in Table 4.
From the estimates it is seen that the house value will increase with more than DKK
300,000 if the house contains an additional toilet. Initially also the number of baths were
included but insignificant, probably because of the close relation to the number of toilets.
The value of an additional square meter is just above DKK 6,000, however, the value of
an additional hectare of the plot is actually not considered being a good, which might
lead to the idea that households dislike the maintenance obligations that follow a large
plot. Regarding the number of floors, this is a relatively large estimate, however, the
marginal value of floors is decreasing as the parameter for floors squared is negative. One
42
7.1 Hedonic Prices
7 RESULTS
explanation for this result could be that the number of floors increase where the value of
owning property is the highest, that is in the cities.
Turning to the locally non-traded amenities then there are some very peculiar results.
for the sake if ethnicity, the equilibrium price is decreasing in the fraction of ethnic Danes,
which is the opposite of what would be expected. However, as the average fraction of
ethnic Danes is over 90% with a relatively small standard error, this part cancels some of
the effect from the extremely high constant. For property crimes these positively affect
equilibrium house prices which is also counter intuitive. It is however unclear what the
total effect of crimes will be, as the negative impact from the other types of crimes, sexual
offences and violent crimes, is negative and greater in absolute terms, but these crime
types occur less frequently cf. Table 3.
The measure of population density is determined at the municipal level and included
as a proxy for living near large cities. The positive parameter estimate is in line with the
expectation that equilibrium house prices should increase when approaching larger cities.
As is clear from Table 4, not all characteristics that are squared are included. This
is primarily a result of insignificant parameters, but also a few are excluded as they were
sources of multicollinearity that made the estimates and their standard errors unreasonable. This was for example the case with the number of toilets squared. Furthermore,
there has been included dummies for each year after 1992 to correct the prices from the
inflation, external shocks to the housing market etc.
To illustrate the implication of omitted variables in the equilibrium hedonic price
model, an alternative is estimated that omit all neighborhood related variables. These
variables are chosen to be omitted, as the residuals and the fitted values from this regression will play a central role in the third estimation stage.
As Palmquist (2006) an Brown, Champ, and Boyle (2003) both point out, this first step
of Rosen’s two-step procedure is often cursed by omitted variables bias. This bias is clearly
also present in this naive model of hedonic equilibrium. In particular the results presented
in Table 5 point to a serious concern for the results. One of the severe biases that arise
is the parameter estimate of the equilibrium marginal value of square meters that turn
43
7.1 Hedonic Prices
7 RESULTS
Amenity
Estimate
Std. Dev.
Constant
10,341.138
391.406
322.123
10.739
6.316
0.161
Plot Size
-0.493
0.037
House Age
10.904
0.159
5,777.894
84.028
Basement Size
7.614
0.332
Business Size
-6.389
0.678
-115.863
9.565
-13,757.511
373.401
2,656.710
24.579
Sexual Offences
-995.697
19.166
Violent Crime
-530.878
6.928
357.199
3.968
5.184
0.020
WCs
House Size
Floors
Floors Squared
Ethnic Danes
School FE
Property Crime
Population Density
Year Dummies
Included
Table 4: Results from Rosen’s First Step
44
7.1 Hedonic Prices
7 RESULTS
negative. In particular it is the inclusion of the population density of municipalities that
facilitate the positive estimate in the full model above. Thinking about it, this is a result
of an omitted urban measure as house size is negatively correlated with the distance to the
nearest city center. Another result that follow the same arguments is that the marginal
value of toilets are cut by more than half.
Amenity
Estimate
Std. Dev.
Constant
-428.360
14.306
125.827
1.241
House Size
-0.686
0.019
Plot Size
-0.097
0.005
1.690
0.018
1024.728
14.442
Basement Size
0.902
0.041
Business Area
-0.141
0.071
Floors Squared
-21.535
1.846
Year Dummies
Included
WCs
House Age
Floors
Table 5: Hedonic Regression Omitting Neighborhood Amenities
The two regressions above both have very low explanatory power against the house
prices, as the Adjusted-R2 is only 0.16 and 0.12 for the full model and the reduced model
respectively. Furthermore, there are still some unreasonable estimates of the full model
that could be correlated with the distance to the nearest city center. Therefore I presume
that also the estimates in the full model suffer from omitted variables bias, as the population density of the municipality might not be a sufficient statistic for the preference of
cities.
45
7.2 Moving Costs
7.2
7 RESULTS
Moving Costs
Here the results from Stage 2 are presented in Table 6. As argued above the parameter
estimates for financial moving costs can be interpreted as the marginal value of wealth.
Therefore the first set of parameters presented in Table 6 will be interpreted as such.
The households have on average a positive marginal value of wealth, which is decreasing
in the income of the household, which is also expected as this correspond to a value
function that is strictly increasing in wealth but with diminishing marginal value. This
holds up until the income of a household is greater than DKK 15.3 million after which
the marginal value of wealth, with this specification, becomes negative. The marginal
value of wealth is decreasing if the household include children living at home, which can
be interpreted as households not caring as much about wealth as they care about their
children.
Financial
Estimate
Std. Dev.
Constant
58.34
0.00
Income
-0.38
0.00
-10.22
0.00
-0.42
0.00
Psychological Estimate
Std. Dev.
Children
Trend
Constant
12.48
0.00
0.16
0.00
-17.93
0.00
29.56
0.00
Income
Children
Trend
Table 6: Estimates of Psychological and Financial Moving Costs
However, as the financial moving costs have been parametrized to household characteristics, a constant, and a trend, this conversion will provide a new set of marginal
46
7.3 Marginal Willingness to Pay
7 RESULTS
willingness to pay estimates for each point in time when adjusting the marginal utilities
in Stage 4. This differs from Bayer, McMillan, Murphy, and Timmins (2011) that does
not parametrize using a trend for the financial moving costs. As the data used contain
nominal prices, it is clear that the marginal effect from house prices on the household’s
move-stay decision should be decreasing, as house values increase a lot over the sample
period and therefore the marginal value of wealth should decrease. This is in part due to
these nominal prices in the data sources.
Secondly, the psychological moving costs are estimated simultaneously in Stage 2. The
psychological moving costs have a positive constant and they are increasing in income.
This result probably stem from the correlation between high income and long working
hours, such that if you work a lot, it is more difficult to find the time to search for a
new house and to build up a new home after buying. The psychological moving costs are
decreasing if the family also include children living at home. This effect might be due to
parents’ desire to raise their children in a good and safe neighborhood and this is clearly
outweighing the negative social impact on moving the children from one school to another.
This effect might be promoted by couples staying in a smaller house or apartment until
they get children and first then they choose to move to something bigger.
For all of the household types that are considered in this thesis, the mean characteristics
are such that the marginal utility of wealth and the total moving costs are positive.
7.3
Marginal Willingness to Pay
The marginal utilities from the decomposition outlined in stage 4 of the estimation procedure are converted into marginal willingness to pay estimates. To simplify the results,
only marginal willingness to pay estimates for 2005 is presented below, however, there
exists a set of such estimates for each point in time as the marginal utility of wealth is
allowed to evolve over time.
In all of the decompositions of the marginal willingness to pay presented below, there
has been included dummies to control for both year and type. The year dummies are
included to clean the mean utilities from the exogenous utility shocks to the Danish
47
7.3 Marginal Willingness to Pay
7 RESULTS
population over time. The type dummies are then included to average the level of utility
across types, to make sure that only the marginal willingness’s to pay are estimated.
Amenity
Estimate
Std. Dev.
Ethnic Danes
42.727
10.044
School FE
15.646
1.463
Sexual Offences
2.309
1.075
Violent Crime
1.121
0.297
-0.789
0.108
0.013
0.001
Property Crime
Population Density
Year Dummies
Included
Type Dummies
Included
Table 7: Overall Marginal Willingness to Pay for Local Amenities
Table 7 presents the benchmark results for the estimation of the stages described above.
All of the estimates are significant at a 5% significance level, however, in these results and
many of the following the marginal willingness to pay for sexual crimes is in the range
of not being significant. This is probably because it is hard to observe neighborhoods in
which it is more likely to be confronted with sexual criminals.
On average households are willing to pay nearly DKK 43, 000 to increase the fraction of
ethnic Danes in the neighborhood by one percentage point. This is not necessarily related
to racism, but more to the idea that households would prefer to have neighbors with the
same cultural background as themselves. Secondly, the households are willing pay to a
significant amount to live in a neighborhood with a public school that can increase the
probability that their children will attend high school at some later point in time.
The estimated marginal willingness to pay for the crime amenities is interesting.
Households are willing to pay DKK 800 to live in a neighborhood that has an average
amount of convicts per 1000 inhabitant convicted for property crimes that is one convict
lower. However, the situation is the opposite for the case of both sexual offenders and vio48
7.3 Marginal Willingness to Pay
7 RESULTS
lent criminals. Considering violent crime, this leads to the thought that these crime rates
are correlated with something else, which is a positive amenity. This will be discussed in
grater detail below. Lastly, the mean single period utility is increasing in the population
density, which is in line with the expected outcome that houses in municipalities with
larger cities are more valued.
From here, the locally non-traded amenities will be divided into two groups based
on their time series properties. The first group consists of ethnic Danes and population
density as both of these are positively persistent, potentially following some underlying
autoregressive process. The second group consists of the crime rates and the school fixed
effect, as these amenities are mean reverting. For crime rates it is clear that the arrival
rate of convictions will be Poison distributed and for the school fixed effect the unobserved
ability of students in a small sample like a neighborhood can vary significantly over time,
however, the teachers’ abilities and the schools physical quality will be persistent over time.
This subdivision will be handy as the underlying time series properties of the variables
seem to be able to explain some of the biases identified below.
In the following, two different decompositions of the marginal willingness to pay estimates will be presented. The two are based on the type characteristics wealth and
children. Here, some significant differences across these types are identified.
For the case of having children the estimates can be found in Table 8. It is found that
families with children living at home are willing to pay over DKK 20,000 to increase the
share of ethnic Danes by one percentage point. Not surprisingly, these families are also
willing to pay significantly more to live in a school district that increase the probability
of the children to attend high school. Regarding crimes, the effect of having children is
dependent on the type of crime. Having children reduces the marginal willingness to pay
for living an a neighborhood with sexual offenders, such that it is insignificant, and almost
the same holds for violent criminals. Property crimes are however not as discredited for
these families as for the rest of the population. This result might either be because that
e.g. elderly people are very afraid of property crimes or because families with children do
49
7.3 Marginal Willingness to Pay
Type:
7 RESULTS
Children in Family
No Children in Family
55.468
32.237
(13.751)
(13.484)
15.393
10.080
(1.964)
(1.904)
1.776
2.885
(1.474)
(1.440)
0.925
1.549
(0.411)
(0.401)
-0.529
-1.157
(0.147)
(0.143)
0.013
0.006
(0.001)
(0.001)
Ethnic Danes
School FE
Sexual Offences
Violent Crime
Property Crime
Population Density
Year Dummies
Included
Type Dummies
Included
Table 8: Marginal Willingness to Pay for Local Amenities Dependent on whether any
Children of the Family Live at Home.
50
7.3 Marginal Willingness to Pay
7 RESULTS
not perceive property crimes as a large threat to their children, but are simply relatively
more concerned with other types of criminals that expose their children more directly to
danger.
Secondly, the two wealth types, high and low wealth, which are separated by the mean
wealth in each year, will be considered separately. The results for this segregation are
presented in Table 9.
Type:
Low Wealth
High Wealth
76.322
9.132
(11.385)
(11.640)
12.396
18.897
(1.794)
(1.858)
1.729
2.889
(1.181)
(1.217)
0.538
1.705
(0.322)
(0.332)
-0.521
-1.057
(0.136)
(0.136)
0.011
0.016
(0.001)
(0.001)
Ethnic Danes
School FE
Sexual Offences
Violent Crime
Property Crime
Population Density
Year Dummies
Included
Type Dummies
Included
Table 9: Marginal Willingness to Pay for Local Amenities by Wealth Type
The result first result to consider is the marginal willingness to pay estimates for ethnic
Danes. The low wealth type is willing to pay over DKK 75,000 to increase the fraction
of ethnic Danes by one percentage point whereas the model can not reject that the high
wealth type does not value ethnic Danes over others at all. This significant difference can
be interpreted in many ways, however I think that it is fair to say, that this indicates
51
8 EVALUATION
that ethnic preferences seem to be a socially skewed phenomena. The estimates of the
school fixed effect is significantly higher for the high wealth type. This is probably a
symptom that high wealth families are more focused on the need for education in order
to do financially better.
It is also interesting that it is only for the high wealth type that all types of crime
are significant, but for the low wealth type it is only property crimes that matter for the
demand function. But the low wealth type has a lower marginal willingness to pay for
not living in a neighborhood with an additional convict, such that lower wealth implies a
higher tolerance for this type of criminals.
8
Evaluation
This section will evaluate the performance of the model by considering the robustness of
the model with respect to the parametric assumptions imposed. Then it will reflect upon
the specification of the model and the data used to estimate the model. Lastly it will
examine the justification for building and estimating a dynamic discrete choice model of
demand for the Danish housing market.
8.1
Robustness
Very few assumption about parameter values have been imposed in this model, but some
have been needed. Therefore, this subsection will examine the model’s robustness with
regards to these assumption.
8.1.1
The Impact of the Rate of Time Preference
One of the only model primitives that is assumed exogenously given is the subjective
rate of time preference, β. The following will examine the models sensitivity towards this
parameter. As the baseline subjective rate of time preference is set to be 0.95, roughly
corresponding to a discounting of 5.3% annually, two alternative estimations will be run
with β equal to 0.90 and 0.98, which corresponds to 11.1% and 2.0% annual discounting
52
8.1 Robustness
8 EVALUATION
respectively. As a third comparison, I will estimate a model that sets the subjective rate
of time preference equal to null. This is of particular interest as it is comparable to the
case in which the households are a sort of hand-to-mouth consumers, that only care about
their instantaneous utility and not about tomorrow. This part is essential to justify the
use of a dynamic model of demand for neighborhoods as it is equivalent to estimating a
comparable static model. The households will assume that the neighborhoods will posses
the characteristics of today forever. Hence, the decision today is based on an implicit
assumption that the state variables are Markovian and therefore only based on todays’
observed states. Intuitively this means that the optimal decision today is assumed to be
optimal the optimal decision in the future.
Table 10 presents the results for the three alternative specifications along with the
benchmark results, for β = 0.95.
To start with consider the comparable static model. This model seem to overestimate
all the marginal willingness to pay results of the model, except for the case of property
crimes that is underestimated compared to the benchmark dynamic model. In particular
note that the estimates are all significantly different under the two regimes.
As previously argued, the parameter estimates regarding sexual and violent criminals
might be correlated with an unobserved attribute, which is why the estimates are also
overvalued in the static model, as the time series properties of the variables in itself suggest
that the opposite should be expected. Regarding the school fixed effect, then there might
be some short run persistence in the outcomes, which is the only part that will be collected
in Stage 3 as the number of lags in the autoregression that specify expectations is limited
to two. However, the school fixed effect might also be correlated with other unobserved
neighborhood characteristics that are of a more persistent nature. This is supported by
the result in Table 8 that show that even households with no children living at home are
wiling to pay a significant amount to live in neighborhoods with a high school quality.
The overall conclusion is still clear though: There is a need for considering the dynamic
character of a market when estimating the demand function as the alternative would
provide biased estimates as is here identified for the Danish housing market.
53
8.1 Robustness
Subjective Discount Factor
Ethnic Danes
School FE
Sexual Offences
Violent Crime
Property Crime
Population Density
8 EVALUATION
β = 0.00 β = 0.90
β = 0.95 β = 0.98
105.718
46.043
42.727
40.738
(10.237)
(10.045)
(10.044)
(10.043)
26.945
16.241
15.646
15.290
(1.523)
(1.465)
(1.463)
(1.461)
-2.044
2.080
2.309
2.446
(1.115)
(1.077)
(1.075)
(1.075)
-2.078
0.953
1.121
1.222
(0.308)
(0.298)
(0.297)
(0.297)
-0.192
-0.758
-0.789
-0.808
(0.111)
(0.108)
(0.108)
(0.108)
0.022
0.014
0.013
0.013
(0.001)
(0.001)
(0.001)
(0.001)
Year Dummies
Included
Type Dummies
Included
Table 10: Marginal Willingness to Pay for Amenities given Different Rates of Time Preference
54
8.1 Robustness
8 EVALUATION
The results that stem from the estimation with the two alternative parameter assumption for the subjective rate of time preference, β, indicate that there are no significant
changes in the marginal willingness to pay estimates, which suggest that the model is
relatively robust to smaller changes in the subjective rate of time preference.
8.1.2
The Formation of Expectations to House Prices
There has been devoted a lot of focus on modeling house prices and forming expectations
about these correctly in this paper. To evaluate whether this has been valuable in terms
of correcting for potential biases, this subsection will identify the impact of cleaning the
house prices for house specific characteristics. This is what is done in estimation Stage 0
and then utilized in Stage 3. Hence, Table 11 presents the estimated marginal willingness
to pay for the local amenities, first by using an autoregression on the raw price series to
form expectations about the evolution of these in Stage 3, which is presented in column
1, and then using the cleaned prices in column 2.
The problem by not correctly cleaning the house prices is that there is captured some
endogenous features due to private investments in the attributes of homeowners’ houses
beyond the 5% user costs. In particular it seems as if the largest problem with this
endogeneity is that the houses with the highest level of the attributes used to clean house
prices, they are owned primarily by ethnic Danes. This point is underlined by the marginal
willingness to pay estimate that more than double if the house prices are not cleaned.
The only variable that is not significantly different is the marginal willingness to pay
for avoiding property criminals. Hence, this leads to the conclusion that the effort put in
cleaning the house prices for house specific attributes is justifiable.
8.1.3
Realtor fees
Lastly, I will consider the assumption about realtor fees. In this thesis the realtor fee is
set equal to the average fee in Denmark when the realtor is hired to handle everything
concerning the sale, which is approximately 4%. However, it is possible for households to
sell at a much lower cost if they choose to handle part of the promotion and advertisement
55
8.1 Robustness
8 EVALUATION
Prices
Non-Cleaned
Cleaned
104.541
42.727
(10.231)
(10.044)
19.278
15.646
(1.522)
(1.463)
0.586
2.309
(1.115)
(1.075)
-2.088
1.121
(0.308)
(0.297)
-0.626
-0.789
(0.111)
(0.108)
0.019
0.013
(0.001)
(0.001)
Ethnic Danes
School FE
Sexual Offences
Violent Crime
Property Crime
Population Density
Year Dummies
Included
Type Dummies
Included
Table 11: Marginal Willingness to Pay and the house Price Dynamics
56
8.1 Robustness
8 EVALUATION
themselves. Therefore, the model is estimated with two alternative realtor fees, 3% and
2% and he results are presented in Table 12. Realizing that the lowered realtor fee will
only spill over to the estimated parameter vector of marginal value of wealth in Stage 2,
it is obvious that cutting the cost by half for the entire sample, will simply double the
estimated marginal value of wealth. This higher marginal value of wealth will then cut the
estimated marginal willingness’s to pay estimates by half in the decomposition of stage 4.
Hence, the results presented in Table 12 under House Prices are linear in the realtor fee.
It is found that even a 1% change in the realtor fee implies significant changes to
the marginal willingness to pay estimates of the school fixed effect, property crime and
population density, and all except for sexual offences are significantly different, when
considering a 2% decrease in the realtor fee. These results indicate that the model is
not particularly robust to changes in the realtor fee which is of some concern as it is
not possible to obtain data on the actual fee corresponding to individual sells. This is
worrying, as there might be significant cross sectional variations in realtor fees as a faction
of house prices.
Bayer, McMillan, Murphy, and Timmins (2011) models the realtor fee as a linear
function of the mean prices in the neighborhood that is left. This is unreasonable, as there
can be large differences in the value of houses within the same neighborhood. Especially
on the countryside there will be great differences as the realtor fee of selling a large farm
can easily be greater than the value of the neighboring country house. This implies that
if there are differences in which type of houses that sell more often, then the estimates
of the marginal utility of wealth will be biased. Imagine that small houses sell more
frequently, then the moving costs are lower for the majority of houses sold in reality than
in the model and hence, the model will estimate that the marginal utility of wealth will
be underestimated, as small houses will sell despite the high moving costs. This bias will
spill over to the estimates of marginal willingness to pay for neighborhood amenities which
will then be overestimated.
Table 12 also include a model estimated with the assumption in Bayer, McMillan,
Murphy, and Timmins (2011). These results identify large differences in almost all of the
57
8.1 Robustness
8 EVALUATION
House Prices
Fee
Ethnic Danes
School FE
Sexual Offences
Violent Crime
Property Crime
Population Density
Neighborhood Prices
4%
3%
2%
4%
42.727
28.485
21.364
93.629
(10.044)
(6.696)
(5.022)
(1.290)
15.646
10.431
7.823
0.989
(1.463)
(0.975)
(0.731)
(0.134)
2.309
1.539
1.155
-1.930
(1.075)
(0.717)
(0.538)
(0.119)
1.121
0.748
0.561
-1.545
(0.297)
(0.198)
(0.149)
(0.035)
-0.789
-0.526
-0.395
-0.742
(0.108)
(0.072)
(0.054)
(0.011)
0.013
0.009
0.007
-0.008
(0.001)
(0.000)
(0.000)
(0.000)
Table 12: Marginal Willingness to Pay and the house Price Dynamics
58
8.2 Discussions of Model Limitations
8 EVALUATION
variables except for property crime, hence, in the case of Denmark it will be important to
use the actual house prices in order to improve the models estimates.
8.2
Discussions of Model Limitations
The model estimated in this thesis is build around an infinite horizon optimization problem
of the household. The advantage of this theoretical approach is that no assumption is
needed on how to handle differences in expected remaining lifetime. But, it is potentially
unsatisfying that the families are assumed to maximize under an infinite horizon as the
life-cycle will most probably affect the housing decision of families. In particular, when
people become older, they can not manage as large a house and might move to a smaller
place or choose to rent, if they do not leave the sample in a more natural way.
The life-cycle is not captured by the current discretization of the type-space, but one
simple way to include effects from the life cycle could be to segregate households on the
basis of e.g. the age of the oldest household member.
As the families are identified by the social security number of one of the family members, the life-cycle of the family in the data also have this more technical side to it.
Therefore, the formation of families could be included to grasp the transition probabilities
between types that reflect changes in the family composition. This could be the result
of marriages, divorces, children growing up and leaving home or spouses that die. These
transitions could either be incorporated as endogenous choices when considering marital
changes or children moving, or it could be thought of as exogenous shocks that hit the
family at given rates, which would be more reasonable when thinking about deaths.
This leads to a second concern of the model. In general, the model does not capture the
potential transitional probabilities between types, that are not endogenously determined
by the move-stay decision of the household. Along with the infinitely lived households
this imply that the households are initially assigned an income and child type which they
will posses forever. The concern is that this distort the expectation to the future states as
the households should also have some probability of changing type in the two exogenous
type-dimensions, which clearly is in line with reality.
59
8.2 Discussions of Model Limitations
8 EVALUATION
With this model at hand, the user cost of housing is the only measure of financial
costs of owning a house. However, the loan type can have large effects on household’s
wealth over time. The uniform user costs arise from the implicit assumption of the model
that the capital markets are perfect. In this way the only cost of buying the house is
the financial moving costs paid to the realtor, and the rest of the costs can be neglected
as buying a house is only considered as a reallocation of wealth post this realtor cost.
The user cost of housing is 5 pct. of the house value at every year, which might be fair
enough if user costs are thought of as costs of keeping the quality of the house constant
and paying constant property and land taxes. However, there are significantly varying
costs for household over time depending of the financing of the part of the house that
is not privately owned. Especially the possibility to include interests-only loans in the
model could be interesting as the impact on the dynamics of he model could be large.
Furthermore, it could be very interesting to consider the effect of changing property and
land taxes that both influence the user costs of housing.
One of the key problems with the data used in this model is that it does not seem
to include a sufficient statistic to handle the effects from living near a city center. The
problem is that there is a potential for omitted variables bias in all of the results of this
model as the population density at municipal level does not proxy good enough for this.
In some of the results it is possible to find indications that such a measure is indeed distorting the estimates of marginal willingness to pay. Given the size of the neighborhoods,
municipalities cover on average nine neighborhoods and therefore an appropriate measure
could be nine times better.
The model estimation uses nominal data and does not handle the impact of inflation
explicitly. The way that this is handled is through the year-dummies in Stage 0 and Stage
4 and the time trends in Stage 2 and Stage 3 of the estimation procedure. However, the
data might be better handled if prices were indexed to a specific year. This could also
improve the conversion of marginal utilities to marginal willingness to pay estimates as
the time trend in the marginal utilities of wealth would then only include the trend in
60
8.3 Hedonic Equilibrium vs. Dynamic Demand
8 EVALUATION
the marginal utility of wealth and no nominal noise to this parameter, which could be an
interesting statistic.
The robustness checks above find that the model is very sensitive to the assumption
about realtor fees. Unfortunately these data does not seem to be available as realtors are
not willing to publish such numbers, which is obvious, but bad for the estimation of the
model.
Looking further, beyond this model it is somewhat obvious that the link between
the physical location of the workplace and the location decision of the household of an
employee should be very strong. Hence, the model will most probably be significantly
improved if such a link is included. However, the causality is definitely not clear in this
context. Do workers move to minimize the transportation cost or do they choose only
among jobs that are within a certain range of their home? Reality most probably lie
somewhere in between as unemployed workers will look for a job nearby their home, and
households will move not too far away from their current workplace. In this way it might
be reasonable to model the location decision of a household as a function of the location
of the workplace(s) if the aim is only to explain housing demand.
A link between workplace and employees would probably help reduce the omitted
variables bias that might underlie the marginal willingness to pay, as it will play a crucial
role in urbanization.
8.3
Hedonic Equilibrium vs. Dynamic Demand
Rosen’s hedonic equilibrium model rely on two assumptions which are both unreasonable.
The first assumption is that the housing market is a perfect market, this has long been
known to be false as there are significant moving costs associated with finding a new house
and paying realtor fees. this assumption has been handled by the use of discrete choice
models. However, these models are still static in their simple form, which implicitly is the
second assumption of Rosen’s model. This thesis has estimated a dynamic discrete choice
model of demand, that can cope with both of these assumption.
61
8.3 Hedonic Equilibrium vs. Dynamic Demand
8 EVALUATION
To compare these two models, start by considering Rosen’s model. In the first step
Rosen estimates the marginal equilibrium cost of the amenities. Then in the second step,
Rosen assume a functional form of the utility function and use the first order condition
of the households’ problems to identify the marginal willingness to pay for the various
locally non-traded amenities. Then the model assumes that the cost and the willingness
to pay equal as in (5), which can also be written as
a1 + a2 xj = b1 + b2 xj + b3 zi + νi
(53)
Now, turning to the dynamic discrete choice model. This model recovers marginal
willingness to pay estimates from the single-period utilities estimated. If I allow the
utility function of the dynamic discrete choice model to be of the form in (3), then the
estimates of marginal willingness to pay will be equivalent to what is put on the right
hand side of (53) with two notes. First the term b3 zi is handled by the discretization of
the type space and the inclusion of type dummies in Stage 4, and as the right hand side
is estimated in the dynamic model, the error, νi , will be null.
To sum up, Stage 0 of the estimation procedure estimate the left hand side of (53) and
Stage 1 through Stage 4 estimate the right hand side of (53). In this way it is possible back
test Rosen’s model by considering how far from perfect market-equilibrium the estimates
are. Whether the two misspecifications of Rosen’ model provide estimates of marginal
equilibrium cost that are significantly different from the equilibrium willingness to pay
estimates can be used to identify the implications of the assumption.
Clearly, the right hand side and the left hand side of (53) are far apart as only two
out of six amenities share the same sign on the parameter estimates. Furthermore, for
these two amenities, population density and school fixed effect, the estimates are more
than a factor 100 larger in the hedonic equilibrium model contra in the dynamic discrete
choice model. Thereby it is demonstrated that the assumptions in the hedonic equilibrium
model stemming back from Rosen (1974) are too strict to give reasonable estimates of the
marginal willingness to pay for locally non-traded amenities.
62
9 CONCLUDING REMARKS
9
Concluding Remarks
There has been a long literature that strives to estimate the demand function of housing
including locally non-traded amenities. This thesis point out some of the seminal models
in the literature that seems to have established broad support and wide applications. The
models presented covers both a simple hedonic equilibrium model, a static demand model
of durable products, and finally a dynamic model of demand. The two first models suffer
from misspecification due to unrealistic assumptions and are both expected theoretically
to yield biased estimates of the demand function.
All three models are estimated using a combination of least squares estimation and the
estimation procedure proposed by Bayer, McMillan, Murphy, and Timmins (2011). This
estimation procedure consist of four stages that identify the demand function by revealed
preferences in a discrete choice model that ultimately identify the single period utility of
different types of households. This procedure has been extended in the thesis by cleaning
house prices and using neighborhood prices to determine the endogenous evolution of
wealth.
The thesis have estimated all of the proposed models using a new and very rich Danish
dataset with a sample period from 1992 to 2005. These data allowed the thesis to relax
some of the strict assumptions of the original model, such as realtor fees being linear in
the average house prices of houses in each neighborhood.
The thesis finds that households are willing to pay significant amounts for improvements in several locally non-traded amenities, such as school quality, ethnicity and crime
rates. I particular it is supported that the time series properties of the locally non-traded
amenities does seem to have an impact on the direction of the bias from a comparable
static model of demand. This bias is such that amenities that are mean reverting in the
short run are found to be underestimated, whereas positively persistent amenities seem to
be overestimated in the static model. These willingness to pay estimates are decomposed
for different household types, based on wealth and children. The differences across these
types in the estimates are in many cases significantly different from each other, especially
in the case of wealth, whereas the differences are more moderate when segregating based
63
9 CONCLUDING REMARKS
on children in the family, however, the estimates still differ significantly across types.
The dynamic model of demand is passed through several robustness checks and it is
found that the estimates are very sensitive to changes in the realtor fee and whether the
prices used to form expectations are cleaned for house specific attributes or not. However,
the model is found to be very robust to marginal changes in the subjective rate of time
preferences.
Finally, the thesis identify a novel way to compare the simple hedonic equilibrium
model of Rosen (1974) and the dynamic demand model. This comparison indicates that
the two will provide estimates of marginal willingness to pay for locally non-traded goods,
that are significantly different, as the equilibrium condition, which s normally utilized in
Rosen’s second step, is very far from satisfied.
64
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Appendices
Appendix A0
Below can be found histograms for the six locally non-traded amenities considered in the
thesis. The histogram collect all time periods in one.
9000
8000
7000
6000
5000
4000
3000
2000
1000
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Figure 1: Histogram of Share of Ethnic Danes
68
1
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8000
7000
6000
5000
4000
3000
2000
1000
0
−8
−6
−4
−2
0
2
4
6
8
Figure 2: Histogram of School Fixed Effect
4
2.5
x 10
2
1.5
1
0.5
0
0
2
4
6
8
10
12
Figure 3: Histogram of Sexual Offense Convictions per 1000 Inhabitants
69
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7000
6000
5000
4000
3000
2000
1000
0
0
5
10
15
20
25
30
35
Figure 4: Histogram of Violent Crime Convictions per 1000 Inhabitants
8000
7000
6000
5000
4000
3000
2000
1000
0
−20
0
20
40
60
80
100
120
140
Figure 5: Histogram of Property Crime Convictions per 1000 Inhabitants
70
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14000
12000
10000
8000
6000
4000
2000
0
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
Figure 6: Histogram of Share of Municipal Populatiopn Density
71
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Appendix A1
The cumulative density (CDF) function of the Type 1 Extreme Value distribution is given
by the following expression.
Fx (x) = e−e
−x
Then the probability density function (PDF) of the Type 1 Extreme Value distribution is
calculated as the partial derivative of the CDF and equal to the expression,
−x
fx (x) = e−e e−x
1,5
Both the CDF and the PDF is illustrated in Figure 1
1
0,5
-1
0
1
2
3
Figure 7: The CDF and -0,5
the PDF of the Type 1 Extreme Value Distribution
-1
Appendix A2
This Appendix will outline the Delta Method, a simple numerical procedure to get esti-1,5
mates of the standard errors of the parameters from a likelihood function.
Assume that the parameter vector is k-dimensional
θ = (θ1 , θ2 , . . . , θk )
and that the sample consists of the following N observations,
x = (x1 , x2 , . . . , xN )
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Let f (θ, xi ) be the pdf. ∀i ∈ {1, . . . , N } and let the cumulated likelihood of the entire
sample be,
L = ΠN
i=1 f (θ, xi ).
Then, define the k-dimensional vector h(·) consisting partial derivatives of f (θ, xi ) with
respect to the parameter vector θ








≈
∂f (θ,xi )
 ∂θ1 




..
.
h(θ, xi ) ≡ 5f (θ, xi ) = 
∂f (θ,xi )
∂θk

f (θ1 ,xi )−f (θ,xi )


η




..
.
f (θk ,xi )−f (θ,xi )
η





where θj is identical to θ except for θjj = θj + η, where η is a small number.
With these definitions, the following three-step-procedure will generate the approximate standard error
(1) Compute the (k × k)-dimensional matrix, A,
A=
"N
X
0
#
[h(θ, xi )] · [h(θ, xi )]
i=1
(2) Form the (k × k)-dimensional matrix B
B = A−1
(3) Finally, compute the standard errors of the parameters as
s
σ(θi ) =
1
· B(i, i)
N
Appendix A3
This Appendix will the estimation of robust standard errors in an OLS routine18 .
Assume that the following model is estimated where y is a vector of length Y of
explained variables, that X is a matrix with dimensions (N × p) of p explanatory variables
and that ε is a vector of N i.i.d. N (0, σ 2 ) numbers,
Y = Xβ + ε
18
The procedure follows Wooldridge (2002)
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Then the OLS estimator of the coefficient vector is,
β̂ = (X 0 X)−1 X 0 Y
Let the residuals from the least squares estimation be = Y − Xβ then the Covariance
matrix of the estimates is calculate as
N
1 X
2i xi x0i (X 0 X)−1
N i=1
!
σ̂β2
0
= (X X)
−1
Then the standard errors reported for parameter j is
σ̂βj =
r
74
σ̂β2
j,j