The Housing Bubble and a New Approach to Accounting for Housing in a CPI
W. Erwin Diewert (University of British Columbia), Alice O. Nakamura (University of Alberta)and Leonard I. Nakamura (Philadelphia Federal Reserve Bank) 1(July 17, 2009)
Abstract
Over the course of the recent house price bubble in the United States, the price of homes rose rapidly
from 1999 Q4 to 2005 Q4 (11.3 % annually as measured by the Case-Shiller index, and 8.4 percent
annually as measured by the Federal Housing Financing Agency) but slowly as measured by ownerequivalent rents (3.4 percent), so measured core inflation remained relatively docile during this period
since only rents are used to measure inflation for housing services in the United States. Over the last
several decades, the U.S. Bureau of Labor Statistics (BLS) has experimented with both rental equivalence
and user cost approaches for accounting for owner occupied housing (OOH) services in the CPI. We
explain the basics of these approaches, and outline the BLS experiences with using them. This assessment
leads us to conclude that the time has come to try a new approach: the opportunity cost approach. We
argue this approach has advantages over both the conventional rental equivalence and user cost
approaches, though it embeds components of the measures for both those approaches and builds solidly
on the research of Verbrugge and others at the BLS. Also, we take up empirical issues that must be faced
regardless of which of the approaches discussed is adopted. We explain how the repeat-sales and various
hedonic regression methods can be placed in a common framework, thereby facilitating understanding of
the properties of and the tradeoffs between the methods. We also consider measurement complications
that arise because the land and structure components of properties depreciate at different rates.

1.

Introduction

How is the cost of housing services changing over time for those living in their own
homes? Good measures are needed by economic policy makers managing everything from themoney supply to benevolent income transfer programs, but are hard to come by since
homeowners do not actually pay themselves for the services of their owned homes.
Over the course of the recent house price bubble in the United States, the price of homesrose rapidly from 1999 Q4 to 2005 Q4 (11.3 % annually as measured by the Case-Shiller index,
and 8.4 percent annually as measured by the Federal Housing Financing Agency) but slowly as
measured by owner equivalent rents (3.4 percent). One consequence was that measured core
inflation remained relatively docile during this period since only rents are used to measure
inflation for housing services in the United States. Yet, as Gallin (2009) and Crone et al. (2009)1

Financial assistance from the Canadian SSHRC is gratefully acknowledged, as is the hospitality of the Centre forApplied Economic Research at the University of New South Wales. This paper reflects the opinions of the authorsand does not represent the views of the Federal Reserve Bank of Philadelphia or of the U.S. Federal Reserve System.
The authors thank the participants in a seminar and a workshop held at the Bank of Canada, a seminar at the
Department of Economics at Ottawa University, meetings at the Philadelphia Federal Reserve Bank, and a sessionof the 2009 annual meeting of the Canadian Economics Association for helpful comments. Special thanks are due to
Randal Verbrugge, Thesia Garner, Pierre Duguay, Andrew Baldwin, Marc Prud’homme, Emi Nakamura, Kam Yu,
Marion Steele, Mason Gaffney, Anne Laferrère, Dennis Capozza, Dennis Fixler, Johannes Hoffmann, and Paul
Schreyer for helpful comments on various versions of this paper and related discussions, but all remaining faults ofthe paper are our sole responsibility.

-1-have argued, rents and home prices are cointegrated over the long run. Moreover, Gallin haspresented evidence that home prices tend to overshoot while rents tend to lag.
If the housing bubble took the form of unsustainable increases in home prices, theexpectation that these increases could be sustained in the short run perhaps lead to a short run
divergence between rents and home prices. If there is information about inflation trends in both
rents and home prices, it might be useful to combine both types of measures. The opportunity
cost approach which we develop combines information on home price change, as part of afinancial user cost component, and information on rents as a rental equivalent component.
Over the last several decades, the U.S. Bureau of Labor Statistics (BLS) has
experimented with both rental equivalence and user cost approaches for accounting for owneroccupied housing (OOH) services in a Consumer Price Index (CPI). We explain the basics ofthese approaches in sections 2 and 3, respectively, and outline the BLS experiences with using
them in section 4. This assessment leads us to conclude that the time has come to try a new
approach to accounting for OOH services costs in measures of inflation: a new approach thatnevertheless builds on BLS expertise and research findings, especially including the work of
Randall Verbrugge and his collaborators. 2
The opportunity cost approach, introduced in section 5, was first suggested at a 2006
OECD Workshop by Erwin Diewert. 3 We argue this approach has advantages over both theconventional rental equivalence and user cost approaches, though it embeds components of themeasures for both those approaches. Also, in sections 6 and 7, we take up empirical issues thatmust be faced regardless of which of the approaches discussed is adopted. We explain how therepeat-sales and various hedonic regression methods can be placed in a common framework,
thereby facilitating understanding of the properties of and the tradeoffs between the methods. We
also consider measurement complications that arise because the land and structure components
of properties depreciate at different rates. Section 8 concludes.
2.

The Rental Equivalence Approach

The rental equivalence approach values the services yielded by an owned dwelling at thecorresponding market rental value for the same sort of dwelling for the same period of time. Thisis the approach used by the BLS at present for the CPI. 4 The price data needed for the CPI rental
equivalence component for OOH services are observations on rents paid by renters: the same
price data also used by the BLS to compile the rental component of the CPI.
The location of each rental unit for which rent and other data are collected is unique.
Empirical studies have shown location to be a key determinant not only of both rents andresidential real estate price levels, but also of the rates of change over time in the levels. Hence,
after choosing a sample of dwelling units to use for the collection of rent data, the BLS
repeatedly samples those units. It is assumed that the changes in owners’ equivalent rents withinsmall geographic areas (areas of 3-4 city blocks, sometimes called segments) will move similarly
to changes in actual rents. (The nature of this rent data, and some of the main data sets for2

See Verbrugge (2008) and Garner and Verbrugge (2009), and also Poole, Ptacek and Verbrugge (2005).
See Diewert (2006/2009).
4
This section draws on the U.S. Bureau of Labor Statistics (BLS) (2007).
3-2-

housing price data too, are why, in section 6, we explore the relationship between the repeat saleand hedonic estimation methods.) Each rental unit that is priced does double duty: it represents
the rents for renters within the segment, and it also separately represents the rent equivalents
implicitly paid by owners within the segment.
3.

The User Cost Approach

The only nations that use the user cost approach to account for the cost of OOH services
in their official measures of inflation omit the property appreciation term. 5 However, reports on
the treatment of OOH by official statistics agencies, including the BLS, make frequent reference
to the shared theoretical underpinnings for the user cost and the rental equivalency approaches,
and it is the user cost, including the property appreciation term, that is relevant in this regard.
The property appreciation term of the user cost formula also plays an important role in the
research of Verbrugge and his collaborators. Hence, in this section, we describe the user cost
approach and show why and how the property appreciation term enters into the user cost formula.
The user cost approach is routinely used in a variety of other measurement andaccounting contexts too, such as in the capital asset pricing literature, in production function
studies, in the measurement of total factor productivity growth, and in the analysis of tax
depreciation rules. The underlying theoretical framework is provided by the fundamental
equation of capital theory. According to this equation, in equilibrium, the price of a durable asset
equals the present discounted value of the future net income that is expected to be derived fromowning it. Thus, if the future income flow that an asset such as a machine can generate is known
or can be readily forecast, then this information can be used to infer what the asset would be
worth to a buyer. On the other hand, in the literature on inflation measurement for OOH services,
what is directly observed are the purchase prices for houses and there are no observabletransactions for the rent that owner occupiers implicitly charge themselves for use of their homes.
Instead, the fundamental equation of capital theory is used to try to back out the period by period
costs to the owner occupier of the OOH services they are using.
Diewert (1974, p. 504) sets out the user cost principles for consumer durables: 6“To form the rental price (or user cost) for the services of one unit of the nth good
during period t, we imagine that the consumer purchases the good during period t
and then sells it during the following period (possibly to himself). Then the
discounted expected rental price for the nth consumer good during period t isgiven by the discounted cost of the purchase of the nth good during period t minus
the discounted resale value of the depreciated good during period t + 1.”
And, Bajari, Benkard, and Krainer (2003, p. 3) outline the connection in the housing economics
literature of user costs to the rental equivalence approach:5

See Diewert and Nakamura (2009) for summary information and references regarding the use of the user cost
approach by Statistics Iceland and Statistics Canada.
6
Diewert (1974, 1980) followed Fisher (1897) and Hicks (1939) in deriving the user cost using a discrete time
approach rather than the continuous time approaches used by Jorgenson (1963, 1967), Griliches (1963), Jorgenson
and Griliches (1967, 1972) and Christensen and Jorgenson (1969, 1973). See also Schreyer (2009a, 2009b).

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“Dougherty and Van Order (1982) were among the first to recognize that the user
cost ... should be equal to the rental price of a single unit of housing services
charged by a profit-maximizing landlord. Thus, the inherently difficult task ofmeasuring an unobservable marginal rate of substitution is replaced by the mucheasier task of measuring rents.”
3.1

The Fundamental Equation of Capital Theory

A clear understanding of the basics of the financial theory justification conventionallycited as the basis for both the rental equivalence and the user cost approaches is helpful to have
before we discuss the opportunity cost approach. We review this financial theory basis here.
Attention to timing matters for understanding user costs. Realized prices are determinedat points in time. Rates of interest are regarded as fixed at points in time. In contrast, rates ofinflation are defined for time intervals. If there is inflation, money is less valuable when received
at the end versus the beginning of a period. An end of period t value can be converted to its
equivalent at the beginning of that same (not the next) period by discounting by the term 1+ r t ,where r t is the period t nominal interest rate.
As Arnold Katz (2009) of the U.S. Bureau of Economic Analysis (BEA) 7 explains, the“user cost of capital” is based on the fundamental equation of capital theory. In box 1, thederivation of the user cost by Katz (2009, appendix A) is shown, recast using the notation for ourpaper. We denote the value of a home that is v periods old at the start of period t by Vvt . Given
only the information available at the start of t, the expected price a home could be sold for at theend of period t, which is the start of period t + 1 , is denoted by Vvt ++11 . O tv denotes the anticipatedoperating costs including maintenance expenses.

Katz explains that the user cost measure is typically derived by assuming that flowtransactions within a period occur at the end of the period.8 Following this convention leads tothe end of period user cost, shown in box 1 as equation (3-4): 9
u vt ≡ r t Vvt + O tv − (Vvt ++11 − Vvt ) .
The symbol m in box 1 (above expression (3-1)) denotes the remaining service life of the home.
The expected market value of a home at the end of period t ( Vvt ++11 ) is conditional on the homehaving a remaining service life of m periods.

7

The BEA makes use of the OOH component of the CPI to supplement the other information they use foraccounting for OOH inflation in compiling the National Income and Product Accounts for the United States.
8
Diewert (2005a, 2005bb) also carefully distinguishes between beginning and end of period user costs andrecommends the use of end of period user costs since they are more consistent with financial accounting conventions.
9
Unlike the home value variable where we need to refer to both the beginning and the end of period values, we only
need to refer to the end of period values for the other anticipated variables and denote them simply using t as thesuperscript, as Katz does. We also forego using a special designation for expected values.

-4-Box 1. Derivation of the User Cost Measure from Katz (2008, Appendix A)
The user cost of capital measure provides an estimate of the market rental price based on costs of owners. It
is directly derived from the principle that, in equilibrium, the purchase price of a durable good will equal the
discounted present value of its expected net benefits; i.e., it will equal the discounted present value of its expectedtfuture services less the discounted present value of its expected future operating costs. To see this, let Vv denote thet +1purchase price of a v periods old durable at the beginning of period t; Vv +1 denote its expected purchase price at thetbeginning of period t+1 when the durable is one period older; u v denote the expected end of period value of thetperiod t services of this durable; O v denote the expected period t operating expenses to be paid at the end of periodtt, for this v periods old durable in period t; and r denote the expected nominal discount rate (i.e., the rate of returnon the best alternative investment) in period t. Expected variables are measured as of the beginning of period t.
Assume that the entire value of the durable’s services in any period will be received at the end of theperiod, and that the durable is expected to have a service life of m periods. From the definition of discounted present
value,
u tv++11t + m − v −1
t + m − v −1
O tv
O tv++11
um
Om
−1
−1
K
K
+
+
−
−
−
−
1+ r
(1+ r t )(1+ r t +1 )
Π it=+tm − v −1 (1+ r i ) 1+ r t (1+ r t )(1+ r t +1 )
Π it=+tm − v −1 (1+ r i )When the durable is one period older, the services it renders in period t will have been received and theoperating expenses of period t already incurred. Thus, the expected price of the durable at the beginning of period
t+1 is:

u tv

t

(3-1)

Vv =

(3-2)

Vv +1 =t +1+
t

u tv++11
1+ r

+
t +1

u vt ++22(1+ r t +1 )(1+ r t + 2 )+K+t + m − v −1um−1Π it =+tm+1− v −1 (1+ r i )−O tv++11
1+ r

−K−
t +1

t + m − v −1
Om−1

Π it=+tm+1− v −1 (1+ r i )Dividing both sides of (3-2) by (1+ r t ) , subtracting the result from equation (3-1), and rearranging terms yields
(3-3)

Vvt

Vvt ++11

−

1+ r

t +1

=

u tv
1+ r

t

−

O tv
1+ r t.

Multiplying through equation (3-3) by (1+ r t ) and rearranging terms, then yields the end of period t user cost:
t

t

t

t

t +1

t

(3-4)

u v = r Vv + O v − ( Vv +1 − Vv ) .

3.2

The Verbrugge Variant (VV) of the User Cost Approach

Randall Verbrugge, in research of his own and with various colleagues, has sought todetermine whether rents and user costs move together, as the financial economics theory outlined
in box 1 seems to imply should be the case. We refer to the specification of the user cost
implemented in Poole, Ptacek and Verbrugge (2005) (PPV hereafter), and that is explained andinvestigated more fully in Verbrugge (2008) and Garner and Verbrugge (2008 and 2009), as the
Verbrugge variant user cost: the VV user cost for short. The VV user cost can be stated as:(3-5)

t
u t = [r t + γ H+ Eπ]V t ,tcollects
where V t is the beginning of period home value; r t is a nominal interest rate; γ Hongoing “housekeeping” operating expenses; and Eπ is an estimate of the expected home priceappreciation. Note that (3-5) is essentially the same as formula (3-4) in box 1.

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What distinguishes the VV user cost from the generic one is that Verbrugge uses
alternative forecasting equations for the Eπ term in (3-5). In his preferred equation, documented
most fully in Verbrugge (2008) and Garner and Verbrugge (2009), the term Eπ is specified to be
the expected value of the 4-quarter home price appreciation. With this setup, changes in homeprices have an immediate within-year impact on the user cost. Verbrugge shows empirically that,since 1998, the VV user cost tracks neither rents nor house prices when evaluated using his
preferred forecasting equation.
4.

The Rental Equivalence versus the User Cost Approach

The BLS has experimented over the years with both the user cost and rental equivalence
approaches. 10 Until the early 1950s, homeowners’ costs to rent were imputed by the BLS: arental equivalence approach. Dissatisfaction with this approach developed due to widespread rent
controls, and this led the BLS to switch to what came to be called the “Asset Price” approach.
With this simplified user cost approach, which Greenlees (2003) terms an “ad-hoc user cost”
approach, OOH services costs within the CPI were built up from five elements: (1) homepurchase prices, (2) mortgage interest costs, (3) property taxes, (4) homeowner insurance charges,and (5) maintenance and repair costs. Over time, problems arose with this BLS approach too.
By the early 1980s, the quality of the data available to the BLS on house prices andmortgage interest rates was deteriorating. 11 The source of the house price and mortgage datautilized by the BLS then was the Federal Housing Administration (FHA) administrative databasefor FHA-insured houses: a small, atypical and shrinking segment of the housing market. Also,
the influential Stigler Report (Stigler 1961, p. 53) had come out strongly two decades earlier infavor of rental equivalency. 12 Thus, in 1983 the BLS switched back to a rental equivalence
procedure for dealing with OOH services in the CPI. 13 Rents, of course, were being collected allalong anyway for the rental component of the CPI.
The rental equivalence approach is still being used. However, PPV note that, by 2005, therapid rise in housing prices in the post 1999 years coupled with slow increases in the OOH
component of the CPI had led to concern among many economic analysts about the use of rental
equivalence for accounting for OOH services costs. Younger people, many of whom faced
deteriorating employment and earnings outcomes along with rising home prices, were among
those who argued that rising housing prices were pushing up their cost of living sharply. Yet, the10

The Poole, Ptacek and Verbrugge (2005) paper was prepared for presentation to the U.S. Federal Economic
Statistics Advisory Committee (FESAC) on December 9, 2005. The FESAC is a Federal Advisory Committeesponsored jointly by the Bureau of Labor Statistics of the U.S. Department of Labor and by the Bureau of Economic
Analysis and the Bureau of the Census of the Department of Commerce.
11
See Poole, Ptacek and Verbrugge (2005). Also, Katz (1982, 1983) at the BEA explored the sensitivity of user costestimates to alternative assumptions about expected rates of inflation and patterns of depreciation.
12
The Stigler Report (Stigler 1961, p. 53) states that: “The welfare of consumers depends on the flow of services
from houses and not upon the stocks acquired in any given period.” The report concluded that (p. 48) “If asatisfactory rent index for units comparable to those that are owner-occupied can be developed, this committee
recommends its substitution in the CP1 for the present series for the prices of new houses and related expenses.”13
See Gillingham and Lane (1982). The rental equivalence approach was implemented for the CPI-U in January
1983 and for the CPI for Urban Wage Earners and Clerical Workers (CPI-W) in January 1985.

-6-

rental equivalent component of the CPI rose little for the nation and indeed fell for some urban
areas.
Bauer, Haltom, and Peterman (2004) with the Federal Reserve Bank of Atlanta argue that,in the United States, the general softening of rents in 2002-2003 is causally related to increases
over the same period in the demand for owned homes. They argue that many of those who hadplanned to purchase homes over the coming years instead rushed to buy out of fear they would
be shut out of the owned housing market by the price increases. Many home buyers entered into
mortgage contracts with exceptionally little equity down based on expectations, on their part andheld as well by lenders, of rising home values.
Yet, concerning possible alternatives to rental equivalence that the BLS might consider,
PPV write that the user cost approach is “the only serious alternative to rental equivalency.” Weagree, at least, that neither of the other main approaches currently in use by major official
statistics agencies -- namely, the acquisitions and the payments approaches -- would be a suitable
alternative.14
We agree too with some of the problems that PPV raise concerning a user cost approach.
They note, for example, that the capital theory used to derive the user cost approach only holds
under equilibrium conditions. Yet, a housing bubble is a sign that the housing market is not in
equilibrium. Furthermore, as conventionally specified, in a period of rapid house price inflation,
the appreciation component of the user cost expression15 can grow large enough so that the user
cost turns negative. Yet it makes no sense to have a negative figure for the value owner occupiers
place on living in their homes!
We conclude that the time has come to find a new approach to accounting for OOH
services in a CPI. This is the task we turn our attention to in the following section. The newopportunity cost approach that we recommend builds directly on suggestions made by PPV in
their seminal 2005 paper. On an operational level, the proposed approach also builds on theexpertise and data collection instruments amassed over the previous decades at the BLS.
5.

The Opportunity Cost Approach

As noted in the introduction, in a 2006 address, Diewert recommended an opportunity
cost approach for dealing with OOH in a CPI: 16“[P]erhaps the correct opportunity cost of housing for an owner occupier is not hisor her internal user cost but the maximum of the internal user cost, which is thefinancial opportunity cost of housing, and what the property could rent for on therental market. After all, the concept of opportunity cost is supposed to represent
the maximum sacrifice that one makes in order to consume or use some object.”Diewert and Nakamura (2009, p. 20) followed up with a two-part suggestion for how an
opportunity cost approach might be used to compile a price index for OOH services:14

See Diewert (2003/2010) for more on this issue.
The appreciation component is the final term in parentheses on the right hand side of (3-4) in box 1.
16
See p. 113 in Diewert (2006/2009), which is the published version of Diewert’s presentation at the OECD-IMF
Workshop on Real Estate Price Indexes held in Paris, November 6-7, 2006..
15-7-

For each household living in owner occupied housing (OOH), the opportunity
cost is the maximum of what the dwelling could have been rented out for, whichis the rental equivalent, and the financial user cost of the funds tied up by owning
the property.
The OOHOC index for a nation can be defined as an expenditure share weighted
sum of a rental equivalency index and a financial user cost index, with theexpenditure share weights depending on the proportion of owner occupiers forwhom the financial user cost is estimated to exceed the rental equivalent cost.
Our purpose in this section is to further develop and explore the properties of two
components of the proposed OOHOC index. The term “opportunity cost” refers to the net valueof the best of the alternatives given up in taking the option chosen. Before proceeding further
with the derivation of the household level components of an OOHOC index, in the following
section, we briefly discuss the concept of an opportunity cost.
5.1

Opportunity Cost Basics

Consider the example provided in the Wikipedia article on opportunity costs. 17 The
example given is for a city that has decided to build a hospital on vacant land it owns. In opting
to build a hospital, the city passed up two competing development proposals: (A) a sports arena,and (B) another revenue generating commercial proposal. The Wikipedia write-up states that:“If the city decides to build a hospital on the vacant land it owns, the opportunity
cost is the value of the benefits forgone of the next best thing that might havebeen done with the land.”Thus, the opportunity cost is the present value of the greater of the two development
opportunities that were passed up in choosing to use the vacant land for the hospital.
In the Wikipedia article, it is explained that the reason that the opportunity cost is thegreater -- that is, the maximum -- rather than the sum of the expected outcomes for the two
proposals passed up is because those other opportunities and the one chosen were mutually
exclusive. Note too that the opportunity cost is being evaluated after the choice to use the land
for the hospital has been made. Equivalently, the opportunity cost could have been assessed prior
to the decision to go forward with the hospital, but from the perspective that this decision should
be adopted. To compute the opportunity cost of some course of action, the analysis must be
carried out from the perspective that the designated action was or will be chosen and that theother alternatives were rejected.
In the Wikipedia example, the construction cost of the hospital and an estimate of themarket value of the vacant land could be calculated, just as the full purchase price of a home canbe observed or estimated. However, another feature of the Wikipedia example that is relevant isthat the two options that the city passed up in choosing to build a hospital would both havegenerated net revenue flows that could be forecasted, and that furthermore should be considered
in determining the full economic cost of the decision to use the land for a hospital. This is a17

http://en.wikipedia.org/wiki/Opportunity_cost.

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potential reason for assessing the opportunity cost even if an appropriate direct cost measure
could be developed.
Note though that it would make no sense to treat the acquisition costs of either a hospitalor a home as the costs of using the assets for a period of time such as a year; rather, some way
must be found to allocate the acquisition costs for these assets over their useful lives. The user
cost approach is one possible means of achieving this cost allocation. However, housing markets
are often not in equilibrium: a requirement for the derivation of the rental equivalent measure of
OOH services using a user cost approach as in section 3.1. Thus, interest in determining theopportunity cost of OOH services, arises, in part at least, because of the lack of some other valid
measure of the true economic cost involved.
5.2

The PPV Opportunity Cost Justification of the Rental Equivalence Approach

PPV state that the measurement question that must be addressed is:“How much richer would the homeowner be if he or she did not consume thehousing services provided by a dwelling?”
They go on to provide the following guidance on this opportunity cost calculation. They explainthat a homeowner always has the option of moving out of his/her house in order to rent it to atenant for rent r1, and of then moving into another rental unit whose rent is r2, with r2 < r1.
Doing so would free up (r1 – r2) income for other uses. This demonstrates, they note, that ahomeowner is, in effect, giving up income equal to r1 (not r2, and not r1-r2) if he/she occupies
his/her own house. They go on to note that: “The fact that shelter services are consideredessential for survival (‘one has to live somewhere’) is irrelevant to the argument.”
PPV argue that every owner occupier passed up the opportunity to rent out their home bychoosing to occupy the home themselves. Owning and occupying, and owning and renting, adwelling out are mutually exclusive alternatives. Treating this pair of options as the only relevantalternatives a homeowner faced provides an opportunity cost justification for the conventional
rental equivalence approach: a justification with the advantage that it pertains to the use of theservices of the durable asset -- the home -- for some period t. Also, this justification does not reston an assumption that real estate markets are in equilibrium, unlike the usual theoreticaljustification of the user cost approach (outlined in section 3.1 above). The development of an
opportunity cost justification for the rental equivalence approach by PPV is, in our view, an
important step forward. The arguments they level against the user cost approach would otherwise
apply as well to the rental equivalence approach if both are viewed as based on the same
theoretical arguments.
A theoretical justification of the rental equivalent that does not entail imposing a housing
market equilibrium condition means too that the rental equivalent is not invalidated by empiricalevidence that rents and housing prices often seem to move quite differently. This is helpful since
there are many reasons why house prices and rents may have differing developments over time.
Transactions costs can be substantial for real estate. Verbrugge (2008) suggests that “thelarge costs associated with real estate transactions would have prevented risk neutral investors
from earning expected profits by using the transaction sequence buy, earn rent on property, sell,
and would have prevented risk neutral homeowners from earning expected profits by using thetransaction sequence sell, rent for one year, repurchase.”-9-

Owners and renters are subject to differing sorts of uncertainty regarding changes overtime in housing related expenses. 18 And, because of agency problems, a landlord may not wish
to customize a rental unit to the same extent as a homeowner. Also rental units may be subject togreater depreciation (Crone et al., 2009). Landlords seem to be bound too by rental market
conventions to change rents infrequently. Rental rate stickiness has been shown empirically to beparticularly important for continuing tenants.19 In addition, the tax treatment of owner occupiers
and renters differs in many countries including the United States. 20
Also, the rental market for luxury homes is thin. Sometimes the owners of luxury homes
want or need to rent out their homes. Luxury homes tend to be offered for rent mostly underconditions that limit the options of a renter. To find renters, the owners of luxury homes often
must compete on price for tenants who would normally rent lower quality housing units andcannot afford to pay much more than what they normally would pay. 21
Yet another reason may be that a landlord’s horizon m may be longer than for an owneroccupier. When the landlord builds or buys a rental property, the landlord will want to set rents
at least equal to the user cost. However, once the property is built or bought, the cost is sunk andsupply and demand factors for rental properties could cause the market rents to diverge from theexpected user costs. Evidence of this factor at work includes cycles in the construction of rental
units; when the landlord’s user cost exceeds market rent, building of new rental properties
slumps and vice versa when the landlord’s user cost is below current market rent.
5.3

The Alternative Options Passed Up by Owner Occupiers

In the Wikipedia example, the set of alternative options the city had to consider whenchoosing the hospital was known: the options were the proposals submitted before a fixeddeadline (which included the option chosen). Now consider the problem of trying to measure theopportunity cost for an owner occupier. We can observe when a home owner has decided tocontinue to own and occupy their dwelling for the period (and perhaps for many periods tocome). But what are the options that the owner occupier had, but passed up in choosing tocontinue to own and occupy their home?
PPV argue that each owner occupier in each period gave up the alternative of renting out
their dwelling for that period. What we add to PPV’s contribution is the insight that most owneroccupiers also gave up alternative financial investment opportunities. The opportunity cost of achoice taken is the value of the next best available alternative. Thus, to properly determine thefinancial investment component of the opportunity cost of a choice to own and occupy in period t,

18

See Sinai and Souleles (2005).
See, for example, Gordon and Goethem (2004) and the findings of Genesove (2003). Also, Hoffmann and KurzKim (2006, p. 5) report the following: “In our sample, prices last on average more than two years... but then changeby nearly 10 %. The longest price durations are found for housing rents, which, on average, are for more than four
years.” Also, Hoffmann and Kurz-Kim (2006, p.5) report that German rents change only every 4 years on average.
20
See, for example, Poterba and Sinai (2008).
21
It also seems likely to us that, moving up the value scale, an increasing percentage of homes offered for rent are,in fact, offered with the terms of payment including house sitting duties along with the monetary rent obligations.
Situations like this should, of course, be caught by the questions asked as part of the collection of the rent data, but it
seems likely that not all the cases like this are properly identified.
19- 10 -

it is not necessary to know the entire alternative financial investment choice set that the owneroccupier faced. Rather, it is only necessary to consider the highest valued of those alternatives. In
addition, the alternatives considered as foregone opportunities must be mutually exclusive witheach other and with the option chosen of owning and occupying in period t. 22
5.4

The Financial User Cost of an Owner Occupier

The user cost approach is an appropriate way of allocating over time initial financial
investments made in assets that yield earnings over multiple subsequent time periods. Unlike realestate markets, financial markets are generally believed to satisfy the assumptions on which theuser cost derivation is based. Thus, we derive the financial user cost for an owner occupier with
non-negative home equity at the start of period t in sub-section 5.4.1. We explore the nature of
the financial user cost component of the OOH opportunity cost for a homeowner with nonnegative home equity in various special circumstances in sub-section 5.4.2. We then take up thenegative equity case in sub-section 5.4.3. Throughout section 5.4, we abstract from taxes, we
assume there are no defaults, and we do not explicitly deal with home rental or purchase or
financing or refinancing choices. Rather, we take the perspective of devising ways of accounting
for OOH services in a CPI conditional on the product choices people have made, including theirhousing services consumption choices. This is the usual practice for price index construction.
5.4.1

Homeowners with Non-Negative Equity

In what follows, we proceed under the conjecture that the highest value of the forgone
financial investment alternatives for an owner occupier with non-negative home equity is the
expected earnings that would have resulted from investment of the home equity funds that thesale of the home at the start of period t would have freed up: a mutually exclusive alternative tothe choice actually made of continuing to own and occupy and to the other alternative considered
and rejected of owning and renting out the dwelling. A homeowner is viewed as having made thechoice to continue owning their home with some level of debt ( D t ≥ 0 ) and equity ( V t − D t )and with a known, required mortgage payment (It) due at the end of each time period, 23 which
can be expressed as I t = (I t / D t )D t = rDt D t with rDt being defined by (I t / D t ) ≡ rDt .
The financial user cost for owning the home in period t and living in it, discounted to thestart of the period, is:22

There are multiple ways in which a homeowner might have withdrawn equity from their home. For example,
refinancing lets a homeowner sell (or buy back) a fraction of an owned home. However, for the purposes ofdetermining the opportunity cost of a choice taken, alternatives foregone that could have been jointly selected, suchas renting the home out for period t and withdrawing some home equity during that time period too, must be
considered as single, combined items in the set of mutually exclusive choices.
23
Mortgage contracts often contain fixed monthly payment requirements and also a fixed rate (or rates) of intereston the outstanding principal of the mortgage. The monthly payment typically represents a combination of interest
and principal, but all that matters in our analysis is that the amount is fixed at the start of each period for the durationof m subsequent periods.

- 11 -

(5-1)⎡ − r t D t − O t + (V t +1 − D t ) ⎤⎥,≡ [V t − D t ] − ⎢ D
⎢⎣
⎥⎦
1+ rt
1+ rt
ut

where V t +1 is defined as the value of the home at the beginning of the period plus the expected
per period average appreciation of the home over the m subsequent time periods that thedwelling is assumed to be able to provide housing services. We recommend that m should be set
at a value at least as large as the median number of years that the relevant population of
homeowners report having lived in their present homes. 24 This implies that a very long term rateshould be used for the expected home value appreciation. Now, if we multiply expression (5-1)
through by the discount factor, 1+ r t , we obtain the following equivalent expressions for the endof period user cost:(5-2)

u t = rDt D t + r t (V t − D t ) + O t − (V t +1 − V t )(5-3)= O t + (rDt − r t )D t − [V t +1 − (1 + r t )V t ] .

5.4.2

Properties of the Financial User Cost for Homeowners with Non-Negative Equity

The user cost expression given in (5-3) can be better understood by considering somespecific types of situations. Consider first of all a homeowner with no mortgage debt. For them,
expression (5-3) reduces to(5-4)

u t = O t − [V t +1 − (1 + r t )V t ] .

This expression is essentially the same as the customary user cost derived by Katz (2009) andshown in (3-4) in box 1 above, and specified by Verbrugge and shown in (3-5).
We next consider the extreme case of owner occupiers with positive equity whose
mortgage payment rate equals their expected rate of return on financial holdings (i.e., rDt = r t ).
In this case too, (5-3) reduces to (5-4) above. Thus, the conventional user cost expression
implicitly assumes that homeowners with mortgages make payments such that rDt equals their
expected rate of return on alternative financial investments.
Among owner occupiers with positive home equity, well off households often can get
mortgages with interest rates that are less than prime and moreover so that rDt < r t . In this case,the user cost expression (5-3) can be written as:(5-6)

u t = O t − (r t − rDt )D t − [V t +1 − (1 + r t )V t ] ,24

Most people own their homes for 7 years or more. In 2004, for example, there were 72 million owner-occupied
homes and existing home sales were 6.8 million. Indeed, owner occupiers typically roll forward the equity
accumulated in one owned home into another owned home when their housing needs change. Few return to being
renters, even for brief periods. Many people move into their own owned homes as soon as they can afford to after
reaching adulthood and die still owning their own homes.

- 12 -

where the term (r t − rDt ) is now positive. Thus, all else equal, for these homeowners, higher
mortgage debt reduces the financial user cost of OOH services.
On the other hand, most subprime loans are high cost. For positive equity owneroccupiers with rDt > r t , the user cost expression (5-3) reduces to:(5-7)

u t = O t + (rDt − r t )D t − [V t +1 − (1 + r t )V t ],where (rDt − r t ) is positive. All else equal, for these homeowners, higher mortgage debt means ahigher financial user cost for OOH services.
5.4.3

Homeowners with Negative Equity

We now turn our attention to the negative equity case: a case of special relevance in thewake of the recent burst of a housing market bubble. Right now, many households are living inhomes that could not be sold for enough to cover their debts. Ashcraft and Schuermann (2008)
cite data showing that second mortgages were quite rare in the 1990s, but that by 2006, manyfirst mortgages were accompanied by second mortgages. In a comprehensive survey report of the
Bank for International Settlements (a central bank forum), for the post-2000 period in the UnitedStates, Ellis (2008) calls attention to an increased use of second mortgages, and observes thatmany U.S. households were able to obtain 100 percent financing in this way. Ellis notes that, in
addition to initial debt-to-value ratios being higher than before, the debt-to-value ratios for a
growing proportion of home owners failed to decline over time as expected based on previous
patterns, or even increased because of recent declines in home values, moving increasing
numbers of home owners into negative home equity situations. Cagan (2007) estimates thataround 5 percent of loans made in the early 2000s were already in negative equity at the end of
2006, though the figure for older loans was lower. Cagan estimates that around 18 percent ofmortgages originated in 2006 were in negative equity by the end of that year.
An owner occupier who has borrowed using their home as collateral basically is renting
part of their home from themselves and part from their creditor, with the rent that the creditor ischarging (i.e., the interest and the required partial repayment of principal that must be made at
the end of each time period for the owner to be able to either occupy or rent out the home) being
specified in the mortgage contract. As before, the mortgage debt in nominal terms is denoted by
D t , the required payment on the mortgage that is due at the end of period t is rDt D t , thebeginning of the period t market value of the home is V t , and the expected end of period tmarket value is V t +1 . Mortgage debt in the amount of D t can consist of some combination of aconventional first mortgage and subprime or second mortgage funds. Negative equity at the
beginning of period t means that debt D t is larger than beginning of period t equity V t so that
(5-8)

V t − D t < 0 , and hence D t − V t > 0 .
If the homeowner had let go of their home at the beginning of period t by either selling it

or “giving it back to the bank” for the current market value of V t , and if they either could not orchose not to consider simply walking away from their residual debt, then household net worth- 13 -

would have had to decrease at the start of period t by the amount of the homeowner’s negative
equity position at the start of t, which is D t − V t > 0 . Of course, there are differences in pricelevels at the beginning and end of the period and so the end of the period costs must be
discounted relative to the beginning of the period costs. But what is the appropriate discountfactor in this case?
If the homeowner has investments that are earning more than the mortgage payment rate
rDt , it would not make sense to liquidate these investments to pay off the negative equity. Nor
would it make sense for the homeowner to borrow at a rate higher than rDt to pay off thenegative equity. 25 Rather, ignoring possible institutional or moral constraints, it would only makesense for the homeowner to borrow or to liquidate investments with an associated rate of lessthan rDt (say the rate of ρ t ≤ rDt ). We let ρ t denote a suitable discount rate for the homeowner.
The cheapest funds for paying off negative equity at the start of t, and hence the best financial
alternative, are funds in a conventional savings or checking account that are earning no interest,or household savings achieved by cutting back on other beginning of period t household
expenses, in which case the appropriate discount rate might be the anticipated Consumer Price
Index Inflation rate over period t, since discounting by one plus this rate will make asset values
at the end of the period comparable to asset values at the beginning of the period in terms of
consumption equivalents. To know the end of period value of the beginning of period negative
equity, we need to know where those funds would have come from. In computing the net valueof the most attractive way in which the homeowner could have let go of the home at the start ofperiod t, we must also allow for the fact that the end of period negative equity position, given
continued ownership and occupation of the dwelling, is expected to be D t − V t+1 , with the costs
as well of rDt D t and O t due at the end of the period.
In what follows, we will assume that the appropriate period t discount rate, ρ t , is a rate atleast as high as the anticipated CPI inflation rate and less than the mortgage payment rate of rDt .
Thus assuming that mortgage interest and operating costs are paid at the end of the period, we
find that the beginning of the period user cost u t /(1 + ρ t ) is defined as follows:⎡ − r t D t − O t − (D t − V t +1) ⎤⎥≡ −[D t − V t ] − ⎢ D⎢⎣⎥⎦
1 + ρt1 + ρtut(5-9)⎡ r t D t + O t + (D t − V t +1) ⎤⎥.
= −[D t − V t ] + ⎢ D⎢⎣⎥⎦
1 + ρtMultiplying both sides of the above expression through by (1 + ρ t ) leads to the following
expression for the end of period t user cost for owner occupied housing if beginning of the period
equity is negative:
25

Some negative equity homeowners will have low interest mortgage options. However, it is typically the case thatnegative equity homeowners have relatively high interest mortgage debt.

- 14 -

(5-10) u t = −(1 + ρ t )(D t − V t ) − [−rDt D t − O t − (D t − V t +1 )](5-11)= O t + (rDt − ρ t )D t − [V t +1 − (1 + ρ t )V t ] .

The expression for the user cost in (5-11) says that user cost is equal to operating expenses O tplus real mortgage interest (rDt − ρ t )D t less the anticipated real capital gain in the value of thehouse [V t +1 − (1 + ρ t )V t ] . This expression makes sense intuitively. 26
We will not develop a user cost formula for a homeowner with negative equity who canwalk away from paying off their debt with no consequences. The problem of transactions andadjustment costs becomes critical in this case. If there were no transactions and no otheradjustment costs of buying and selling and building homes and there were no other consequencesof defaulting, then in the negative equity case, it would make sense for the homeowner to default
and immediately repurchase their home at the new lower price (or buy another home ofequivalent value). However, with transactions and adjustment costs, it can make sense for thenegative equity homeowner to stay in the home without defaulting. Thus the size of thetransactions and adjustment costs becomes critical.
6.

Inflation Measurement Given Dwelling UniquenessA decision by the BLS and other official statistics agencies to move to an opportunity
cost approach for accounting for OOH in a CPI would be a step forward, in our view. However,regardless of whether an opportunity cost or a user cost or a rental equivalent approach is used,statistical agencies will still face empirical methodology problems arising from dwelling
uniqueness. Each dwelling has a unique location and dwellings continually evolve via dwellingspecific depreciation and renovation. Dwelling uniqueness presents similar empirical challenges
regardless of whether the price data are sale prices or rental prices. Thus, in this section, we use
the term “price” to refer to either dwelling sale or rent observations, unless otherwise specified.
The depreciation rate for a dwelling can be defined and can hypothetically be measured
by the ratio of the same period prices for identical dwellings that have been used for different
lengths of time. The inflation rate for a dwelling can be defined and can hypothetically bemeasured by the ratio of the prices at different points in time for identical dwellings that wereused the same length of time. However, no two dwellings are identical. Among other differences,
they all have different physical locations. Cross sectional information on used asset sale prices orrents at any one point in time will not allow us to separate out the separate effects of depreciationand inflation for durable assets that must be viewed as unique for price measurement purposes. 27
26

We have not modeled various tax consequences associated with home ownership. This is left to further research.
Special cases of this fundamental identification problem have been noted in the context of various econometrichousing models: “For some purposes one might want to adjust the price index for depreciation. Unfortunately, a
depreciation adjustment cannot be readily estimated along with the price index using our regression method.… In
applying our method, therefore, additional information would be needed in order to adjust the price index for
depreciation,” Bailey, Muth and Nourse (1963, p. 936). “The price index and depreciation are perfectly collinear, so
if one cares about the price index, it is necessary to use external information on the geometric depreciation rate ofhouses,” Palmquist (2003, p. 43).
27

- 15 -

However, this separation can be based on empirical evidence if some way can be foundfor deciding when dwellings can be viewed as comparable for price measurement purposes. 28
Achieving this separation is especially important for any category of durables where
maintenance and renovation expenditures and consequences are substantial, and also where
inflation and deflation movements in prices can be substantial. The prospects for separating out
depreciation and inflation effects are much improved if dwellings can be viewed as the same forprice measurement purposes provided they have certain shared characteristics. This is true for thehedonic methods, as used for measuring inflation for both dwelling sale prices and dwelling rents,
but not for the repeat sales method, which only compares the same unit over time.. The repeat
sales and hedonic methods are two seemingly very different types of empirical methods which
we explain here can be placed within a common mathematical framework, thereby allowing
users to take advantage of insights from research into the properties of both these methods.
With a pure hedonic method, data collected over multiple periods are classified by value
determining characteristics that are not unique such as neighborhood or distance from the down
town core, type of dwelling unit (e.g., single detached or a unit in a multiple unit building), somemetric for size such as floor space, and the age of the property. To apply a hedonic method, data
are needed on the selected list of value determining characteristics, and there must be agreementthat this list is appropriate and adequate to control for differences in the value of the housing
services provided by the different purchased or rented dwellings. Thus the hedonic method datarequirements are usually very extensive. The inflation rate is then estimated using observations
over time while controlling econometrically for changes in the value determining characteristics.
The repeat sales method compares the price observations for housing properties that were
sold, or that were rented, multiple times over the time interval spanned by the available data. 29
This method is popular with U.S. real estate researchers and practitioners because it uses only the
information readily available in all localities of the United States: sale or rental prices and theunique legal property descriptions. A key underlying assumption is that, with similar
maintenance expenditures, owners of residential properties that they occupy or rent out usually
manage to maintain their properties in unchanged condition over the dwelling service lives. Thismethod does, however, control for the quality determining attributes of each dwelling that do notchange over time, and it does so without the need for having data on (or even knowing about) allof those value determining attributes.
We now turn our attention to the specifications of the repeat sales and hedonic methods
and how these methods are related. We begin with the repeat sales method which is due to
Bailey, Muth and Nourse (1963). Since hedonic regression models, as usually used in the pricemeasurement literature, have price levels (often in logarithmic form) as dependent variables,
rather than price ratios as is the case for the repeat sales method, it is helpful to introduce therepeat sales method as it arose historically: as a generalization of the chained matched model
methodology. 30 This is the motivation for how the repeat sales method is presented in box 2.

28

For CPI rent measurement, BLS age adjusts inflation rates using the hedonic method of Randolph (1988a, 1988b).
The repeat sales procedure, now in widespread use, dates back to Bailey, Muth, and Nourse (1963). See also
Dreiman and Pennington-Cross (2004) for the uses of this method and see Green and Malpezzi (2003, pp. 32-60) fora review of the repeat sales index literature.
30
See Wyngarden (1927) and Wenzlick (1952).
29- 16 -Box 2. An Exposition of the Repeat Sales Method
Let S(0,1) denote the set of housing units that are in scope for the index and were sold in both periods 0
and 1. Denote the price for property n sold in period t by Vnt . Here attention is confined to just two time
periods 0 and 1, so n∈S(0,1). Let P 0,1 be the real estate price index going from period 0 to 1. For housing
units in S(0,1), suppose the stochastic model relating the property sales price ratio, Vn1 / Vn0 , to P 0,1 is:(4-1)

Vn1 / Vn0 = P 0,1 exp ε 0n,1 ,where ε 0n,1 is assumed to be an independently distributed error term with mean 0 and constant variance.
Taking logarithms of both sides of (4-1) leads to the following linear regression model:(4-2)

ln[Vn1 / Vn0 ]= π 0,1 + ε 0n,1 ,where π0,1 ≡ ln P 0,1 . The least squares estimator for π 0,1 is the arithmetic average of the logarithms of thesales price ratios. Exponentiating this estimator yields a preliminary matched model property price index
going from period 0 to 1:(4-3)

P 0,1* ≡ ∏ n∈S( 0,1) [ Vn1 / Vn0 ]1 / N ( 0,1) ,where N(0,1) is the number of houses in the set S(0,1). This index is seen to be the equally weighted
geometric mean of sales price ratios Vn1 / Vn0 for all the properties that changed hands in both periods 0
and 1: a typical matched model estimator for an elementary price index.
Next let N(1,2) denote the number of sales of houses in set S(1,2) and consider the set S(1,2) ofhouses that sold in both periods 1 and 2. Now the preliminary matched model price index going fromperiod 1 to 2 can be shown to be:(4-4)

P1, 2* ≡ ∏ n∈S(1, 2 ) [ Vn2 / Vn1 ]1 / N (1, 2 ) .

Using the above results, the levels of the property price index, P t , for t = 0,1,2 can be defined as:(4-5)

P 0 ≡1; P1 ≡ P 0,1* ; P 2 ≡ P 0,1*P1, 2* .

Thus the price index P t is set equal to 1 in period 0; in period 1, it equals the matched model price index
going from period 0 to 1, and in period 2, it equals the product of the preliminary price indexes given in(4-3) and (4-4).
The Bailey, Muth and Nourse (1963) innovation was to reparameterize the model describedabove and to add an additional set of estimating equations for repeat sales pairs in periods 0 and 2: i.e., forhousing properties in S(0,2). Their estimating equations with three periods of data on repeat sales are:(4-6)

ln[Vn1 / Vn0 ]= π1 − π 0 + ε 0n,1

for n∈S(0,1),(4-7)

ln[Vn2 / Vn1 ]= π 2 − π1 + ε1n, 2for n∈S(1,2),(4-8)

ln[Vn2 / Vn0 ]= π 2 − π0 + ε 0n, 2

for n∈S(0,2),

where now we have π0 ≡ ln P 0 , π1 ≡ ln P1 , and π 2 ≡ ln P 2 , with the following normalization imposed (where
adding a constant to each πt leaves the regression unchanged): π0 = 0 or P 0 =1 . This leads to a model thatcan be estimated using least squares regression. Exponentiating the least squares estimates for theparameters π1 and π 2 , denoted here by π1* and π 2* , leads to estimates for the preliminary indexes P1*
and P 2* . The BMN estimates for the housing price levels in periods 1-3 are:(4-9)

P 0 ≡1 , P1* ≡ exp π1* , P 2* ≡ exp π 2* .

The 3-period model generalizes easily to the T-period case of Bailey, Muth and Nourse (1963).

- 17 -

As originally proposed, the repeat sales method can only be used to measure price levelchange over time, and thus is not well suited for empirical studies that seek to exploit crosssectional variation by using the absolute dollar values of dwellings. 31 However, in the literatureon spatial price level comparisons, Summers (1973) proposed a hedonic regression model where
the only explanatory variables are dummy variables for the country and the product: the countryproduct-dummy or CPD method. In box 3, we show the formal steps for how the repeat sales and the
CPD methods are related for the case where complete matched model data are available.

Box 3. The CPD Model with Complete Matched Model Data
Consider a sample of N houses ( n =1,2,K, N ) each of which sold in each of the three periods ( t= 0,1,2 ):S(0,1,2) . A stochastic model for the house prices, Vnt , in each period t can be specified as follows:(4-12)

Vnt = α n P t exp ε nt ,n =1,K, N,

where P t is the housing price index level for period t, α n is a parameter that reflects the quality of
housing unit n relative to “average” quality and ε nt is an independently distributed, mean zero, constantvariance error term. Taking logarithms of both sides of (4-12) leads to the following system of estimating
equations for the N houses:(4-13)

ln Vnt =β n + π t + ε nt ,n =1,K, N, t = 0,1, 2 .

where β n ≡ ln α n and π t ≡ ln P t . The model defined by (4-13) and the normalization (4-14), the leastsquares (LS) estimators for the model parameters satisfy the following N+2 equations:(4-14)

1*
1*∑ nN=1ln Vn = ∑ nN=1β n + Nπ ,(4-15)

2*
2*∑ nN=1ln Vn = ∑ nN=1β n + Nπ , and(4-16)

ln Vn0 + ln Vn1 + ln Vn2 =3β*n + π1* + π 2* ,n =1,K, N .

Using equations (4-16) to eliminate the β*n from (4-14) and (4-15) yields solutions for the unknowns:(4-17)

π1* = (1/ N ) ∑ nN=1ln[ Vn1 / Vn0 ], π 2* = (1/ N ) ∑ nN=1ln[ Vn2 / Vn0 ] .

After exponentiating these estimates, this complete information CPD model leads to the following
geometric mean of the period 1 relative to the corresponding period 0 values as the estimate for the period
1 housing price level, P1* , and the geometric mean of the period 2 values relative to the corresponding
period 0 values as the estimate for P 2* :(4-18)

P1* = ∏ nN=1[ Vn1 / Vn0 ]1 / N , P 2* = ∏ nN=1[ Vn2 / Vn0 ]1 / N .

Finally, the details are shown in box 4 for how the repeat sales approach can be modified
to incorporate hedonic regression corrections for changes in observed dwelling characteristics
between price observations. The resulting linear regression model (equations (4-19)-(4-22)) is
the same as the two country CPD model (with incomplete information).32 Exponentiating (4-27)
reveals that this hedonic regression model leads to a period 0 to 1 price index that equals the31
32

See, for example, Capozza, Hendershott, Mack, and Mayer (2002).
It is also identical to the two period Aizcorbe-Corrado-Doms (2001) dummy product hedonic regression model.

- 18 -

equally weighted geometric mean of the selling prices in period 1 divided by the geometric meanof the selling prices of the matched models in period 0. 33
In its basic form, the general time dummy hedonic method involves regressing thelogarithm of the property sale price on the characteristics of the property and a time dummy
variable for each period spanned by the estimation data set (except the omitted base period).
Once the estimation has been completed, the time dummy coefficients can be exponentiated tocreate an index. Alternatively, using information on the characteristics of the properties sold, thedata can be stratified and a separate regression can be run for each time period for specifiedclasses of residential properties. Thus the hedonic regression method could be used to produce afamily of indexes. Diewert, Heravi and Silver (2009) outline alternative formulations andestablish the relationships among them. 34
7.

The Structures and Land Decomposition Problem 35

Usually the logarithm of the purchase price is taken as the dependent variable in realestate price models. While this specification accords with the directly observable property priceinformation, it is inconsistent with certain aspects of the structure and land components of theprice of a property. Residential real estate usually involves both a structure and the land that thestructure is built on (the site). To model this composite, consider a sample of dwelling units
purchased at the beginning of period 0.
Suppose the purchase price of property n is p nt . The value for property n can be regarded
as the sum of the (often unobserved) cost per square meter for the structure, times the floor spaceof the structure in square meters (denoted by A in box 5), plus the price per square meter of land
(often not directly observed) times the area of the site in square meters (denoted by B in box 5).
For period 0, the property value can be represented as in equation (6-1) in box 5, and for period t,
the value for this property can be represented as in equation (6-2).
The structure and land components are probably subject to different rates of inflation anddepreciation. Indeed, land is often viewed as depreciating little if at all (though there can be
depreciation of site infrastructure). The asset inflation and depreciation effects are embedded in
the coefficients of A and B. Estimating equations are given by (6-3) and (6-4). If data are alsoavailable for the characteristics of the structure and the land, then the pair of equations, (6-5) and(6-6), can be estimated instead. This model is flexible and provides a means of decomposing aproperty price index into structural and land components, though the model is nonlinear.
Moreover, the rate of depreciation may vary across rental and owner-occupied property.
33

In a series of papers, Diewert (2002, 2003a, 2003b, 2003c, 2004, 2005a, 2005b, 2006a, 2006b), Diewert, Heravi
and Silver (2007/2010), Silver (2003), and Silver and Heravi (2005) show how alternative specifications andweights can be used within the CPD framework to derive a number of known index number formulas. See also deHaan (2003), Silver (2003) and Silver and Hervari (2005). Diewert (2005b) shows that the unweighted indexes canbe far from their weighted counterparts. Thus it is important to run appropriately weighted regressions.
34
Rao (2008) notes that the CPD method -- the weighted version in particular -- is being increasingly used inderiving spatial comparisons due to its ability to handle price quotations. See, for example, Aten and Menezes(2002), Heston and Atten (2002), Rao (2003, 2005), and Deaton, Friedman and Alatas (2004).
35
See Diewert (2003/2010, 2006/2009). Discussions between Erwin Diewert and Anne Laferrère helped improve
the presentation of the model here.

- 19 -Box 4. The CPD Model with Incomplete Matched Model Data
Next a model is considered where not every house must trade in each period for information about thehouse to be included in the analysis data set. In order to minimize notational complexities, Diewert(2003a) provides the following details for the case of two periods. Let S(0,1) be the set of housing units
that sold in both periods 0 and 1. Taking into account the normalization (4-10), the estimating equations
corresponding to these houses are:
(4-19)

ln Vn0 =β n + u 0n ,for n∈S(0,1) ,(4-20)

ln Vn1 =β n + π1 + u1n

for n∈S(0,1) .

Let S(0,∼1) denote the set of housing units in the target population that sold in period 0 but not inperiod 1. The estimating equations for these observations are:0
(4-21)ln Vm= γ m + u 0m ,for m∈S(0, ~1) ,where γ m is the logarithm of the quality adjustment factor for the mth housing unit that sold in period 0
but not in period 1. Similarly, let S(1,∼0) denote the set of housing units in the target population that soldin period 1 but not in period 0. The estimating equations for these observations are:
(4-22)
ln Vk1 = δ k + u 0k
for k∈S(1, ~ 0) ,where δ k is the logarithm of the quality adjustment factor for the kth dwelling for which priceinformation is available in period 1 but not 0.
Let π1* , β*n , γ *m and δ*k denote the least squares (LS) estimates of the parameters π1 , β n , γ mand δ k that appear in (4-19)-(4-22). The stacked vector of dependent variables for equations (4-19)-(422) can be written as the sum of the vectors of exogenous variables times their corresponding leastsquares estimates plus the vector of least squares residuals. As noted above, the inner product of eachexogenous vector with the vector of LS residuals is zero. Thus the LS estimators for the unknown
parameters in the regression model must satisfy the following equations:1*

1*(4-23)∑ n∈S( 0,1) ln Vn1 + ∑ k∈S(1, ~ 0 ) ln Vk1 = ∑ n∈S( 0,1) β*n + N (0,1) π + ∑ k∈S(1~ 0 ) δ*k + N (1, ~ 0) π ;(4-24)

ln Vn0 + ln Vn1 = 2β*n + π1* ,(4-25)

0ln Vm= γ *m ,for m∈S(0, ~1) ,(4-26)

ln Vk1= δ*k ,

for k∈S(1, ~ 0) ,for n∈S(0,1)

where N(0,1) is the number of dwellings that traded in both periods and N(1,∼0) is the number that sold in
1 but not 0. Equations (4-26) can be used to eliminate the δ*k in equation (4-23), and equations (4-24) canbe used to eliminate the β*n from equation (4-23). The resulting equation for π1* is:(4-27)

π1* =[1 / N (0,1)]∑ n∈S( 0,1) ln[ Vn1 / Vn0 ] ,which is the arithmetic average of the logarithms of the sales price ratios for the two periods.
For the housing units that sold (or were rented) in t, a more general hedonic regression model is:tt(4-28)
ln Vnt = π t + ∑ Kn∈S( t ) .
k =1 z nk β k + ε n ,t

ε nt is an independently distributed error term with mean 0 and constant variance, Vn is the observedtselling price or rent of dwelling n in period t, z nk
is the amount of characteristic k that dwelling n has,and π t equals the logarithm of the constant quality price index, P t ; i.e., π t = ln P t for t = 0,1,…,T. The
parameter β k transforms amounts of characteristic k, z k , into constant quality utility units for k = 1,…,K.

- 20 -Box 5. Structure and Land Decomposition
Suppose the total cost, p, of a property after the structure is completed will equal the floor space area ofthe structure, say A square meters, times the building cost per square meter, α say, plus the cost of theland, which will equal the cost per square meter, β say, times the area of the land site, B. Now think of asample of properties of the same general type, with prices, p 0n , in period 0 and structure areas A 0n andland areas B0n for n =1,K, N ( 0) , and where these prices are equal to costs of the above type times error
terms η0n which have mean 1. This leads to a hedonic regression model for period 0 where α and β are theparameters to be estimated in the regression:(6-1)
p 0n =[αA 0n +βB0n ]η0n .
Taking logarithms of both sides of (6-1) leads to the following traditional additive errors regression
model:(6-2)
ln p 0n = ln[αA 0n +β B0n ]+ ε 0n ,where the new error terms, ε 0n ≡ ln η0n for n =1,K, N ( 0) , are assumed to have 0 means and constantvariances.
For a subsequent period t, the price per square meter for the given type of structure will havechanged from α to αγ t and the land cost per square meter will have changed from β to βδ t where γ t is
the period 0 to t price index for the type of structure and δ t as the period 0 to t price index for the landthat is associated with this type of structure. For n =1,K, N ( t ) , the period t counterparts to (6-1) and (6-2)
are:
(6-3)

p nt =[αγ t A nt +βδ t Bnt ]ηnt and(6-4)

ln p nt = ln[αγ t A nt +βδ t B nt ]+ ε nt ,where ε nt ≡ ln ηnt , the period t property prices are p nt , and the structure and land areas are A nt and Bnt .
Diewert (2006a) suggests that equations (6-2) and (6-4) can be run as a system of nonlinear
hedonic regressions. The main parameters of interest are γ t and δ t , which can be interpreted as period tprice indexes (relative to the corresponding period 0 price levels of 1) for the price of a square meter ofthis type of structure and the price per meter squared of the underlying land.
This framework can be generalized to encompass the traditional array of characteristics used inreal estate hedonic regressions. Suppose that we can associate with each property n that is transacted in t atlist of K price determining characteristics X nt 1 , X nt 2 ,K, X nK
for the structure and a similar list of M pricetdetermining characteristics Ynt 1 , Ynt 2 ,K, YnM
for the type of land. The equations that generalize (6-2) and(6-4) are:000(6-5) ln p 0n = ln{[ α 0 + ∑ kK=1 X 0nk α k ]A 0n +[β 0 + ∑ M
n =1,K, N ( 0) andm =1 Ynm β m ]B n }+ ε n ,

(6-6)

t
t
t
t
t
t
M
ln p nt = ln{γ t [α 0 + ∑ K
k =1 X nk α k ]A n + δ [β 0 + ∑ m =1 Ynm β m ]B n }+ ε n , n =1,K, N ( t ) ,where the parameters to be estimated are now the K+1 quality of structure parameters, α 0 ,α1 ,K,α K , theM+1 quality of land parameters, β 0 ,β1 ,K,β M , the period t price index for structures parameter γ t and theperiod t price index for the land underlying the structures parameter δ t . Note that [α 0 + ∑ kK=1 X 0nk α k ] in (605) and (6-6) replaces the single structures quality parameter α in (6-2) and (6-4) and [β 0 + ∑ M
m =1 Ynm β m ] in(6-5) and (6-6) replaces the single land quality parameter β in (6-2) and (6-4).

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8.

Concluding Remarks

The recent housing bubble has raised several questions for the measurement of housing
service inflation. The most fundamental question is, are rents always the best measure of theopportunity cost of owner-occupied housing? Since, as Verbrugge has shown, rents often diverge
from conventional user cost measures, the possibility has arisen that the answer is no.
Also, the markets for rental and owner occupied housing units can be quite isolated fromone another. This is evident in Crone et al.’s hedonic measures of housing services for owneroccupied housing. Heston and Nakamura (2009) show too that for more expensive homes in the
United States, the estimated rent/price ratio is substantially lower, suggesting that the housing
services of more expensive homes are being underestimated in the U.S. national expenditures asthese are being evaluated now. The equilibrium conditions under which rents equal user costs
may not be fulfilled for sustained periods of time, either because the rental market is thin in someareas and price ranges where there are substantial proportions of owner occupied housing, or
because of disequilibria due to other factors including the time required for new housing
construction and government regulations governing land use and building that are believed tomake the housing market prone to bubbles.
We have explored one comprehensive measure of housing services that may remain valid
or may better approximate the trend rate of housing services cost inflation during conditions inwhich rents and user costs diverge. Our proposed measure of the opportunity cost of housing
services is the greater of rents and user costs. This measure not only arguably permits moreaccurate measurement of housing services in disequilibrium, but also avoids the difficult
problem of the possibility that user cost in the short run may be negative.
It is worth remarking that the difficulty of measuring housing services arises from threesources: (1) homes are unique, because of location, (2) homes are durable assets, and (3)homeowner transaction and housing supply adjustment costs are large. With large transaction
and adjustment costs, prices may remain out of equilibrium because arbitrage is too costly. As
durable assets, the rate of depreciation and the rate of interest, enter nontrivially. And because of
uniqueness, prices of different units are not easily comparable. We have explored the interrelationships of the repeat sales and hedonic methods for dealing with the comparability problem.
Owner-occupied housing services are in many countries the largest item in the consumer
basket. In the United States consumer price index, it accounts for 24 percent of the total weight.
How we measure inflation in this item is highly consequential for our understanding ofmacroeconomic dynamics, monetary policy, and growth.
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