ECN741: Urban Economics
Estimating Housing
Demand
Estimating Housing Demand
Class Outline
 Constant Elasticity Case
 Linear Expenditure System
 Housing price variable
 Submarkets and the Demand for Structures
 Tenure Choice
 Housing Durability
 Endogeneity of neighborhood choice
 Endogeneity of household formation
Estimating Housing Demand
Constant Elasticity Demand
 Let’s begin with a standard formulation of the demand
for housing services, H.
 The notation
▫
▫
▫
▫
▫
▫
Y = income
P = price of housing services
u = distance from worksite
t = round-trip commuting costs per mile
R = apartment rent = PH (≠ land rent!!)
V = house value = R/r
Estimating Housing Demand
Constant Elasticity Demand, 2
 The demand function is


H  C (Y  tu ) P{u}
where C is a constant
 Multiplying both sides by P, we obtain

1 
R{u}  P{u}H {u}  C (Y  tu) P{u}
Estimating Housing Demand
Constant Elasticity Demand, 3
 Taking logs, we obtain the estimating equation:
ln{R{u}}  ln{C}   ln{Y  tu}  (1   ) ln{P{u}}
 In practice, empirical work ignores theory!
 Everybody uses Y instead of Y- tu;
 P is usually measured with a metropolitan area
construction index; some studies divide R by P to get H;
 Some studies use V instead of R.
Estimating Housing Demand
Constant Elasticity Demand, 4
 Example: Zabel, Journal of Housing Economics,
March 2004.
 Uses data from AHS.
 The 2001 AHS data, including a few control variables,
imply that
▫ θ = 0.362
▫ μ = -0.052
Estimating Housing Demand
Linear Expenditure System
 Recall this problem with a Stone-Geary utility function:
Maximize:
U   Z  SZ 
1

(H  SH )
Subject to : Y  Z  P{u}H  tu
 Recall as well that the resulting demand for H is
 Y  tu  S Z  P{u}S H 
H  SH   

P{u}


Estimating Housing Demand
Linear Expenditure System, 2
 Now if we multiply both sides by P{u}, we have
R{u}  P{u}H   SZ  (1   )S H P{u}   (Y  tu)
 This is called a linear expenditure system.
 The survival quantities are coefficients to be estimated.
 If the price of Z varies in the data, it needs to be included,
too.
Estimating Housing Demand
Linear Expenditure System, 3
 Note that this functional form is quite different from the
constant elasticity form.
 The linear expenditure system has been widely used in other
contexts, but not so much in housing.
 An idea for a study: Use standard specification tests to
determine which of these specifications is appropriate.
 The Davidson/MacKinnon test, for example, calls for including
the predicted value from regression A as an explanatory variable
in regression B. If it is significant, the specification in
regression A adds explanatory power.
Estimating Housing Demand
The Housing Price Variable
 The standard approach allows prices to vary across
urban areas, but the basic urban model indicates that P
varies within an urban area, too.
 Some studies (e.g. Goodman, JUE, May 1988) first
regress V on u, A (neighborhood amenities that appear
in P), and X (housing characteristics that appear in H):
P{u , A}H { X }
V
r
Estimating Housing Demand
The Housing Price Variable
 These studies then predict P based on u and A and use
the predicted P in housing demand estimation.
 Allowing intra-area variation in P appears to make a
big difference:
 His price elasticities for owners and renters are -0.502
and -0.786, respectively, much larger (in absolute value)
than Zabel’s.
Estimating Housing Demand
Submarkets and the Demand for Structures
 Zabel’s study with AHS data pools across metropolitan
areas, treating each as a submarket.
 He allows the coefficients of the Xs to vary across
metropolitan areas.
 Then he defines a “structure price” to be the price of a
given housing bundle (set of Xs) in each metropolitan
area.
Estimating Housing Demand
Submarkets and the Demand for Structures, 2
 As an aside, some early studies applied the same type of logic
to data for a single urban area.
 The allow the coefficients of the X’s to vary across exogenously
determined “submarkets” within an urban area.
 In my judgment, this approach adds a lot of complexity without
much insight.
 But the interactions across submarkets are sometimes significant,
and some scholars think submarkets are important.
Estimating Housing Demand
Submarkets and the Demand for Structures, 3
 Back to Zabel: Let N be neighborhood traits (indexed
by n) and X be structural housing traits (indexed by m).
Then Zabel estimates:
R  PH    N n 
n
n
 X 
m
m
m
 The estimated coefficients are allowed to vary across
urban areas.
Estimating Housing Demand
Submarkets and the Demand for Structures, 4
 Zabel’s “structure price index,” PS, is defined by holding
the Xs at their mean and the λs at their estimated
values.
 The amount of “structure,” H*, is R/PS .
 With these terms, he can estimate the constant
elasticity demand function using H* as the housing
variable and PS (and the comparably defined
neighborhood price index, PN ) as explanatory variables.
Estimating Housing Demand
Submarkets and the Demand for Structures, 5
 Zabel’s logic is fine, but I find his terminology to be
misleading.
 In an urban model, H stands for “structure” already, and
P{u} is its price.
 It is fine to allow the weights that define H to vary across
urban areas, but it is confusing to create a new term for the
“price of structures.”
 I also prefer estimates based on a single urban area—
with intra-area variation in P.
Estimating Housing Demand
Adding Tenure Choice
 One of the most important behavioral issues in the
study of housing markets is tenure choice.
 Why do some households decide to buy a house while
others choose to rent?
 This topic has been widely studied. We will return to it
when we study race and ethnicity.
 But today it is important because it is a major source of
selection bias in estimating housing demand.
Estimating Housing Demand
Sample Selection Bias in Estimating
Housing Demand
 Owner-occupied houses tend to be larger than
apartments.
 Thus the amount of H (the dependent variable) is
correlated—highly—with tenure choice.
 This violates the principle that a sample selection rule (a
study of just owners or just renters) should not be
correlated with the dependent variable.
Estimating Housing Demand
Selection Bias in Housing Demand, 2
Error Distribution
Estimated Line for Owners
H
Owners
H*
True Relationship
Renters
Estimated Line for Renters
Y
Estimating Housing Demand
Selection Bias in Housing Demand, 3
 One way to handle this is to pool owners and renters.
 Use R, apartment rent, as the dependent variable for
owners.
 Use rV = annualized value as the dependent variable for
owners.
 But this can be complicated because of different tax
treatment, depreciation, mortgage interest, etc.
 And owners and renters may have different elasticities.
Estimating Housing Demand
Selection Bias in Housing Demand, 4
 Another approach is provided by Goodman (1988 and
several more recent articles).
 He estimates three equations:
 Housing demand for owners
 Housing demand for renters
 Tenure choice
Estimating Housing Demand
Selection Bias in Housing Demand, 5
 In his model ψ is the ratio of value to rent, a measure of
expected appreciation in owner occupied housing
(expected appreciation lowers the real discount rate),
which identifies investment incentives.
 In addition, λ is the ratio of owner to renter P, a
measure of relative price; f is the probability that a
household is an owner; C is a constant, and A is age.
 The resulting equations with owner, O, and renter, R,
subscripts are:
Estimating Housing Demand
Selection Bias in Housing Demand, 6
H O  COY
O
 PO {u} 
O
H R  C RY
R
R
 PR {u} 
R
f
f
f
O
f
f  CfY   A
O
A
R
A
Estimating Housing Demand
Selection Bias in Housing Demand, 7
 Now overall housing demand is
H  f HO  (1  f ) H R
 Differentiating with respect to income yields
the overall income elasticity, θ:
f O H O  (1  f ) R H R

 f
H
 HR 
1 

H 

Estimating Housing Demand
Selection Bias in Housing Demand, 7
 Goodman results, are as follows:
O  0.308;  R  0.134;   0.423
 Recall that Goodman also accounts for intraurban variation in P. If you are interested in
housing demand, his work is well worth
studying!
Housing Demand Theory
Accounting for the Long Lifetime of Housing
 The long lifetime of housing makes housing decisions
different from many other decisions.
 One implication is that housing demand is more closely
linked to permanent income than to temporary income.
 Households may not immediately adjust their housing
consumption in response to temporary income shocks.
 Thus, temporary shocks are like measurement error,
and the income elasticity of demand for housing is
higher when permanent income is used instead of
current income.
Housing Demand Theory
The Long Lifetime of Housing, 2
 Olsen (Handbook chapter 1988) emphasizes the long
time frame of housing decisions.
 Because housing decisions are forward-looking, models
of housing demand need to consider wealth, age, and
expectations.
 At the very least, studies should try to have controls for
wealth, education, and age (which help to predict
permanent income).
Housing Demand Theory
The Long Lifetime of Housing, 3
 The Olsen chapter is valuable because it presents all
these issues in fairly straightforward models.
 He shows, for example, that one might also want to
interact age with other parameters in the demand
model, because age changes expectations and time
horizons!
 Moreover, the formulations given above implicitly
assume static expectations, and a formal treatment of
expectations might alter the estimating equation.
Housing Demand Theory
The Endogeneity of Neighborhood Choice
 Households must decide where to live as well as how
much housing to consume.
 These decisions are related, so failure to consider
neighborhood choice might bias estimates of housing
demand.
 Papers that make this argument include Rapaport
(JUE, September 1997) and Iaonnides and Zabel (J. of
Applied Econometrics, 2003).
Housing Demand Theory
The Endogeneity of Neighborhood Choice, 2
 Rapaport’s methodology is complicated.
 She estimates a logit model of neighborhood choice and
then includes the predicted probabilities in a constantelasticity housing demand equation.
 In effect, she adds to the standard model an additional
variable for each neighborhood.
 She finds that accounting for neighborhood choice increases
the estimated (absolute value of) price elasticities
considerably.
Housing Demand Theory
Endogeneity of Neighborhood Choice, 3
 A similar endogeneity will show up later in the class when
we study “hedonics,” which is the name economists give to a
regression of house value or rent on neighborhood and
housing attributes.
 Because households compete for entry into desirable
neighborhoods, prices rise with neighborhood quality.
 As a result, households simultaneously select housing price
and neighborhood quality.
 As we will see, this form of endogeneity makes it difficult to
study the demand for neighborhood amenities.
Housing Demand Theory
The Endogeneity of Household Formation
 Individuals must decide how to form themselves into
households when they make housing decisions.
 Single adults may decide to move in with other single
adults or with their parents when the price of housing
is high.
 Elderly parents may move in with their adult children
when the price of housing is high.
Housing Demand Theory
Endogeneity of Household Formation, 2
 A few studies have shown that accounting for
household formation decisions is important in
estimating housing demand.
 See, for example, Boersch-Supan and Pitkin (JUE
September 1988).
 This study uses nested logit analysis; in the first level,
adults decide on household formation and in the second
level they decide on housing type.
Housing Demand Theory