Understanding Housing Market Volatility

Joseph Fairchild∗ Jun Ma† Shu Wu‡

January 7, 2014

∗Bank of America, 600 Peachtree Street, Atlanta, GA 30308, phone: 949-422-0968,email: joseph.fairchild@bankofamerica.com.†Department of Economics, Finance and Legal Studies, Culverhouse College of Com-

merce and Business Administration, University of Alabama, Auscalosa, AL 35487, phone:205-348-8985, email: jma@cba,ua.edu.‡Department of Economics, University of Kansas, Lawrence, KS 66045, phone: 785-

864-2868, email: shuwu@ku.edu.

Abstract

The Campbell-Shiller present value formula implies a factor structure

for the price-rent ratio of the housing market. Using a dynamic factor

model, we decompose the price-rent ratios of 17 major housing markets

into a national factor and independent local factors, and we link these

factors to the economic fundamentals of the housing markets. We find

that a large fraction of housing market volatility is local. And the local

volatilities mostly are due to time-variations of idiosyncratic housing

market risk premiums, not local growth. At the aggregate level, the

growth and interest rate factors jointly account for up to 47% of the

total variations in the price-rent ratio. The rest is due to the aggregate

housing market risk premium and a pricing error. We find evidence

that the pricing error is related to money illusion, especially at the

onset of the recent housing market bubble. The rapid rise in housing

prices prior to the 2008 financial crisis was accompanied by both a

large increase in the pricing error and a large decrease in the housing

market risk premium.

Key words: housing market, dynamic factor model, price-rent ratio,

risk premium, money illusion.

JEL Classifications: G10, R31, C32

1

1 Introduction

Housing markets are segmented. There does not exist a centralized market

for housing assets. Demographic changes, household preferences for geo-

graphic locations and climate plus inelastic land supply can lead to hetero-

geneous regional price dynamics. Some existing studies, such as Gyourko,

Mayer and Sinai (2006) and Del Negro and Otrok (2007), have already ob-

served the price level and growth rate vary drastically across major U.S.

housing markets in the past few decades. As we can see in Table 1, for

example, the average annual nominal price change was 6.8% in New York

City but was only 3.5% in Kansas City during the period between 1979 to

2009. Moreover, the volatility of house price also varies greatly across dif-

ferent cities. Table 1 shows the standard deviation of annual nominal price

changes for the same period. It was 7.5% in New York City but was 2.7% in

Kansas City. Table 2 reports similar statistics for the log price-rent ratios

of the same cities. For example, in New York City the log price-rent ratio

had an annual standard deviation of 22% while in Kansas City it was only

8.7% between 1979 to 2009.

On the other hand, all housing markets are obviously affected by a few

aggregate variables such as the monetary policy, mortgage market innova-

tions and national income. When the central bank lowers the key interest

rate, it could stimulate the demand for houses in all markets and have a

positive effect on housing prices. In fact Table 1 and Table 2 show that the

correlation among some housing markets can be very high (e.g. New York

City and Boston, Los Angles and Philadelphia).

In this study we use a dynamic factor model to decompose housing prices

into a common national factor and idiosyncratic local factors in order to

better understand the sources of housing market volatility. We treat a resi-

dential house as a dividend-paying asset and base our dynamic factor model

on the Campbell-Shiller log-linear approximate present value formula for

the price-dividend ratio (Campbell and Shiller, 1988). Such an approach

allows us to link the unobservable factors to the economic fundamentals of

2

the housing markets such as interest rates and expected rent growth.

Quantitatively distinguishing the national factor from local factors in the

housing markets is important. From the perspective of policy makers, for

instance, it is crucial to know if monetary policy was responsible for creating

a national housing market bubble by keeping the short-term interest rate

too low for too long, or if the increase in housing prices prior to the 2008

financial crisis instead reflected a collection of local bubbles. On the other

hand, identifying a market “bubble” is intrinsically difficult. An increase in

asset prices could be due to improved economic fundamentals as perceived by

investors or due to purely speculative activities. By linking the unobserved

price factors to economic fundamentals, our paper also seeks to distinguish

between the part of housing market volatility attributable to changes in

expected rent growth and the discount rate and the part that could be due

to speculations or pricing errors.

Given the importance of the housing sector in the aggregate economy,

there have been many studies on housing markets in recent years. For ex-

ample, Fratantoni and Schuh (2003) uses a heterogeneous-agent VAR to

examine the effect of monetary policy on regional housing markets. Davis

and Heathcote (2005) points out that residential investment is more than

twice as volatile as business investment and leads the business cycle. Ia-

coviello (2005) develops and estimates a monetary business cycle model

with housing sector. Brunnermeier and Julliard (2007) finds evidence that

money illusion can play an important role in fueling run-ups in housing

prices. Stock and Watson (2009) estimated a dynamic factor model with

stochastic volatility for the building permits of the U.S. states from 1969-

2007. Mian and Sufi (2009) uses detailed zip-code level data to examine

the role of subprime mortgage credit expansion in fueling the house price

appreciation prior to the recent financial crisis. Kishor and Morley (2010)

uses an unobserved component model to estimate the expectations of the

housing market fundamentals and investigate the sources of the aggregate

housing market volatility. Ng and Moench (2011) estimates a hierarchical

factor model of the housing market and examines the dynamic effects of

3

housing market shocks on consumption. Favilukis et al (2011) argues that

international capital flow had played a small role in driving the last house

market bubble, and the key causal factor was instead the cheap supply of

credits due to financial market liberalization. Our paper contributes to this

growing literature that seeks to understand the fundamental driving forces

of the housing market and its relationship to the aggregate economy.

Perhaps the closest studies to ours are Del Negro and Otrok (2007) and

Campbell et al (2009). Del Negro and Otrok (2007) was among the first to

apply dynamic factor models to housing markets.1 In their study state-level

house price movements are decomposed into a common national component

and local shocks via Bayesian methods. They find that historically the lo-

cal factors have played a dominating role in driving the movement in house

prices in different states. But a substantial fraction of the recent increases in

house prices is due to the national factor. They further use a VAR to inves-

tigate the effect of monetary policy on housing markets. The key difference

between our study and Del Negro and Otrok (2007) is that we treat a house

as a dividend-paying asset and infer a factor structure for the price-rent ratio

based on the Campbell-Shiller log-linear present value formula. As a result,

we can explicitly link the unobserved factors to economic fundamentals of

the housing markets. The Campbell-Shiller formula has been widely used

to analyze the volatility of bond and equity markets. In an intriguing study,

Campbell et al (2009) applied the same method to price-rent ratio in housing

markets.2 The ratio is split into the expected present values of rent growth,

the real interest rate and a housing risk premium. The study found that

the housing risk premium accounts for a significant fraction of the price-

rent volatility. An important difference between our paper and Campbell et

al (2009) is that we are able to disentangle the relative importance of the

common component in the price-rent ratios across individual markets from

1Recent applications of dynamic factor models include Cicarelli and Mojon (2010) onglobal inflation, Ludvigson and Ng (2009) on bond risk premiums and Kose et al. (2003,2008) on global business cycles among many others. Forni et al. (2000) provides a thoroughanalysis of the identification and estimation of generalized dynamic factor models.

2 Brunnermeier and Julliard (2007) also uses the same approach to isolate the pricingerror in the aggregate housing market due to money illusion.

4

idiosyncratic local factors using a dynamic factor model. We show that this

factor structure is an implication of the Campbell-Shiller present formula

and both factors have similar Campbell-Shiller representations. Moreover,

we show that a pricing error associated with money illusion is also important

in driving the housing market dynamics.

To implement the Campbell-Shiller formula, we need to estimate the

expected future rent growth and real interest rate. Another innovation of

our paper is that the forecasting vector auto-regression model (VAR) for

future rent growth and real interest rate is embedded in a dynamic factor

model, and the two models are estimated jointly. The macro variables in the

VAR are correlated with the national factor of rent growth but are indepen-

dent of the local factors. Such a specification is important for appropriate

identifications of the national and the local factors.

The rest of the paper is organized as follows. Section 2 describes our

model. Section 3 discusses the data and estimation strategy. Section 4

presents the main empirical results. Section 5 concludes.

2 Model

We treat a house as a dividend-paying asset and equate the house price

to the present value of the expected future rental income under rational

expectations.3 Following Campbell and Shiller (1988), we can write the

price-rent ratio as the sum of expected growth rate of rental income minus

the expected rate of return on the housing asset.

In particular, if Pi,t denotes the ex-dividend price of a housing asset in

market i at time t, Di,t+1 the rental income of the housing asset between t

and t+1, let xi,t = log(Pi,t

Di,t

), di,t = logDi,t and ri,t+1 = log

(Pi,t+1+Di,t+1

Pi,t

).

3Using rent as an approximation of the dividend income of a housing asset, we implicitlyassume that individuals are indifferent between owning and renting. Glaeser and Gyourko(2007) points out that the rental units in the housing markets tend to be very differentfrom owner-occupied units.

5

Under log-linear approximation, we have (ignoring constant terms):

xi,t = Et

∞∑τ=0

ρτ [∆di,t+1+τ − ri,t+1+τ ] (1)

where ρ = 1/(1+e−x), and x is the steady state price/rent ratio. The house

price today should equal the present value of expected future rent growth

minus the weighted average of expected future rates of return.

We assume in this study that the growth rate of rent in one market

consists of two components, a national factor that is common to all markets

and an independent local factor that is specific to market i. We can now

rewrite the standard Campbell-Shiller decomposition as

xi,t = Et

∞∑τ=0

ρτ∆dt+1+τ+Et

∞∑τ=0

ρτ∆di,t+1+τ−Et∞∑τ=0

ρτrf,t+1+τ−Et∞∑τ=0

ρτeri,t+1+τ

(2)

where ∆dt is the national factor of rent growth rate,4 ∆di,t is the idiosyn-

cratic rent growth rate in market i, rf,t is the real interest rate and eri,t is

the excess rate of return in market i, eri,t = ri,t − rf,t.

The last term in Equation (2) corresponds to the risk premium for in-

vesting in the housing market, which also can be written as the sum of two

components

Et

∞∑τ=0

ρτeri,t+1+τ = Et

∞∑τ=0

ρτ ert+1+τ + Et

∞∑τ=0

ρτ eri,t+1+τ (3)

The first part on the right side of the equation above can be thought of as the

national housing market risk premium and the second part an idiosyncratic

risk premium component that is specific to market i. This decomposition

can be justified as follows: if housing markets were fully integrated without

4For the purpose of exposition we have assumed that the factor loading is 1 for allmarkets. We also estimate a more general case where the factor loadings vary acrossdifferent markets. The results under the two specifications are very similar. See morediscussions below.

6

transaction cost and other frictions, it would follow from the standard asset

pricing theory that Et(eri,t+1) = βiEt(ert+1), where ert+1 is the excess

return on a portfolio of housing assets that is perfectly negatively correlated

with the pricing kernel (or the stochastic discount factor).5 Of course much

evidence shows that housing markets are far from integrated and there are

many kinds of frictions within each market as well such as transaction costs,

liquidity constraints and etc. The second part on the right side of Equation

(3) therefore captures the expected excess rate return that is orthogonal to

the aggregate housing market risk premium. Notice that we again assume

in Equation (3) the factor loading is 1 for all i. Relaxing this assumption

doesn’t change our main empirical results.6

In summary, the log-linear Campbell-Shiller present value formula im-

plies a factor structure for the price-rent ratios of the housing markets as

follows:

xi,t = xt + xi,t, i = 1, 2, ..., N (4)

where

xt = Et

∞∑τ=0

ρτ∆dt+1+τ − Et∞∑τ=0

ρτrf,t+1+τ − Et∞∑τ=0

ρτ ert+1+τ = yt − lt − ηt

(5)

and

xi,t = Et

∞∑τ=0

ρτ∆di,t+1+τ − Et∞∑τ=0

ρτ eri,t+1+τ = µi,t − εi,t (6)

As we will further show in Section 4.3, there could be an additional

pricing error term in Equation (5) if investors are not able to form rational

expectations of future real rent growth or the real interest rate. For example,

they may suffer from money illusion and mistaken a decline in the nominal

interest rate due to a change in inflation for a decrease in the real interest

5Since we will estimate the risk premium as the residual term in the Campbell-Shilleridentity, the coefficient βi is not important as long as it is constant. If βi is time-varying,our decomposition in (3) will not be valid because βi can depend on local state variables.

6Another implicit assumption is that investors in different housing markets share thesame information set.

7

rate (or equivalently they extrapolate the historical nominal rent growth rate

without taking into account changes in inflation). Under money illusion, the

observed price-rent ratio, xt, will include an extra term (a pricing error) as

follows,

xt =Et

∞∑τ=0

ρτ∆dt+1+τ − Et∞∑τ=0

ρτrf,t+1+τ − Et∞∑τ=0

ρτ ert+1+τ

+ (Et − Et)∞∑τ=0

ρτrf,t+1+τ

(7)

where Et denotes people’s subjective expectation under money illusion, Et

denotes the rational expectation. Investors perceive the real interest rate to

be Et∑∞

τ=0 ρτrf,t+1+τ , while the actual real interest rate is Et

∑∞τ=0 ρ

τrf,t+1+τ .

The first part of Equation (7) corresponds to the “correct” or “true” house

value under rational expectations. If households underestimate the real in-

terest rate, for example, the observed house price xt will exceed its true

value by (Et − Et)∑∞

τ=0 ρτrf,t+1+τ . This pricing error will disappear if in-

vestors are able to correctly form rational expectations of future interest

rates (Et = Et). In our decomposition exercise, we will need to distinguish

empirically the pricing error from the risk premium term in the Campbell-

Shiller formula.7

3 Data and Estimation

In our model specification there are two types of unobserved factors: the

unobserved national and local factors for both rent growth and price-rent

ratio, and the unobserved agent’s expectations of future rent growth, future

interest rates and future excess returns or risk premiums. Typically the un-

observed national and local factors can be extracted from the observed series

by applying the type of Dynamic Factor Model (DFM) proposed in Stock

7We treat money illusion as a national factor because there is no appealing reason toassume only households in one or some particular markets make this mistake while othersdon’t.

8

and Watson (1991), and the unobserved expected future variables can be es-

timated by a VAR model that was first implemented in Campbell and Shiller

(1988) to study sources of price-dividend variations. We combine these two

lines of work and propose a novel VAR augmented DFM that allows us to

simultaneously decompose the observed series into the national and local

factors and obtain estimates of these expectations of future variables. Using

these estimated expectations along with the Campbell-Shiller present-value

accounting identity, we can further decompose the housing price variations

into movements in different economic fundamentals including time-varying

risk premiums.

Our data are the annual real rent growth and price-rent ratio of 17

metropolitan areas. The data on rent growth cover a long span, from 1936 to

2009. The real rent growth are obtained through deflating the nominal rent

by the CPI. When applying the DFM to the real rent growth we augment the

DFM with a multivariate VAR that includes several important macroeco-

nomic variables (such as the interest rate) to allow for potential interactions

between the national factor of real rent growth and observed macroeconomic

activities. This is very important for several reasons. First it ensures the

appropriate identification of the local factor of real rent growth in (6) which

is supposed to be independent of the national factor of rent growth and the

real interest rate in (5). Second, as pointed out by Engsted et al. (2012),

a critical requirement for proper Campbell-Shiller VAR decompositions is

that the forecasting state variables should include the current asset price.

We address this issue by including the Case-Shiller home price index as one

of the macro variables in our model. Third, the extra information contained

in the macro variables can in principle improve the forecasts of the national

real rent growth as well as the future interest rate. More information on the

data used in this paper can be found in the appendix.

Denote the annual real rent growth in the 17 metropolitan areas by ∆di,t.

Assume a common national factor represented by ∆dt and the idiosyncratic

local factors denoted by ∆di,t. We use 2 lags for all dynamic factors since all

data are annual and 2 lags appear sufficient to capture potential dynamics.

9

Notice that all variables are demeaned before being used to estimate the

model. Specifically, the DFM part is set up as below:

∆di,t = βd,i∆dt + ∆di,t, i = 1, 2, ..., 17 (8)

∆dt = φ1∆dt−1 + φ2∆dt−2 + ωt (9)

∆di,t = ψ1∆di,t−1 + ψ2∆di,t−2 + νi,t, i = 1, 2, ..., 17 (10)

where ωt and νi,t are independent Gaussian shocks.

We augment the above DFM with a VAR to allow the latent national

factor ∆dt to interact with four macroeconomic variables that include the

real interest rate, rf,t, real GDP growth, gt, log changes in the Case-Shiller

home price index, st, and CPI inflation rate, πt:

Zt = Φ(L)Zt−1 + ξt (11)

where Zt = (∆dt, rf,t, gt, st, πt)′ and ξt = (ωt, εr,t, εg,t, εs,t, επ,t)

′. The vari-

ance matrix of the innovations to the VAR is given by Σ. We also use 2 lags

in the VAR specification.

To estimate this VAR-DFM model we cast it in a state-space framework.

The Kalman filter then can be conveniently employed to estimate such state-

space model. The resulting state-space model consists of the measurement

and transition equations as detailed below.

Measurement equation:

Ut = MYt (12)

10

where

Ut =

∆d1,t

...

∆d17,t

rf,t

gt

st

πt

, M =

1 0 · · · 0 0 βd,1 0 0 0 0 0 0 0 0 0...

......

...

0 0 · · · 1 0 βd,17 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 1 0 0 0 0 0 0 0

0 0 · · · 0 0 0 0 0 0 1 0 0 0 0 0

0 0 · · · 0 0 0 0 0 0 0 0 1 0 0 0

0 0 · · · 0 0 0 0 0 0 0 0 0 0 1 0

Transition equation:

Yt = ΛYt−1 + Θt (13)

where

Yt =

∆d1,t

∆d1,t−1

...

∆d17,t

∆d17,t−1

∆dt

∆dt−1

rf,t

rf,t−1

gt

gt−1

st

st−1

πt

πt−1

, Θt =

ν1,t

0...

ν17,t

0

ωt

0

εr,t

0

εg,t

0

εs,t

0

επ,t

0

11

and

Λ =

ψ1,1 ψ1,2

1 0...

· · · ψ17,1 ψ17,2

1 0

φ1,1 · · · φ1,10

1 0 0

φ2,1 · · · φ2,10

1 0

φ3,1 · · · φ3,10

1 0

φ4,1 · · · φ4,10

1 0

φ5,1 · · · φ5,10

1 0

We follow Kim and Nelson (1999) to maximize the log-likelihood func-

tion written via the Kalman filter to obtain the estimates of the hyper-

parameters. Once the hyper-parameter estimates are found the smoothing

algorithm is invoked to calculate the smoothed estimates of the national

and local factors: E[∆dt|IT ] and E[∆di,t|IT ], i = 1, 2, . . . , 17. The smoothed

estimates are based on the most available information in the sample and

thus provide the best estimates of these factors.

Iterating forward the dynamics of the unobserved national factor and

observed macroeconomic variables, we can derive the growth component

and interest rate component in the Campbell-Shiller decomposition (5) as

12

follows:

Et

∞∑τ=0

ρτ∆dt+1+τ = e′1 · F · (I − ρF )−1 ·Wt (14)

Et

∞∑τ=0

ρτrf,t+1+τ = e′3 · F · (I − ρF )−1 ·Wt (15)

where Wt = (∆dt,∆dt−1, rf,t, rf,t−1, gt, gt−1, st, st−1, πt, πt−1)′, i.e. the sec-

ond half of the state variable in the transition equation (13); F is the cor-

responding companion matrix in the VAR model. ej is a selection column

vector which has 1 as the j-th element and zero elsewhere. In the same

way, the idiosyncratic local growth component Et∑∞

τ=0 ρτ∆di,t+1+τ can be

computed relatively easily since it is by construction independent of the

macroeconomic variables.

The aggregate and local risk premium components are obtained as the

residual terms in the Campbell-Shiller accounting identity (5) and (6), re-

spectively. We first apply the DFM to the price-rent ratio and extract the

national and local factors from this series.8 Assume each price-rent ratio is

the sum of the unobserved national factor and local factor:

xi,t = βx,ixt + xi,t, i = 1, 2, . . . , 17 (16)

and the national and local price-rent ratios both follow the stationary AR(2)

processes:

xt = α1xt−1 + α2xt−2 + et, et ∼ i.i.d. N(0, σ2e) (17)

xi,t = γi,1xi,t−1 + γi,2xi,t−2 + ςi,t, ςi,t ∼ i.i.d. N(0, σ2ς,i) (18)

Again, the national and local factors are orthogonal to each other for

8Before applying the DFM to the log price-rent ratio data, we ran a panel unit roottest and rejected the unit root hypothesis. This is consistent with the finding in Ambroset al (2011) that house price and rent are cointegrated.

13

identification purposes following Stock and Watson (1991). This model can

be put into its state-space form and the estimation is done by following Kim

and Nelson (1999). The risk premium term is then obtained by subtracting

the rent growth and interest rate components from the price-rent ratio.

Also notice that in the dynamic factor models, the scale of the common

factor and the factor loading are not identified independently. We can either

normalize the factor loading to be 1 (i.e. βx,i = 1) or normalize the standard

deviation of the shocks to the common factor to be 1 (i.e. σ2e = 1). We

estimated both versions of the model and obtained similar results.

4 Results

4.1 Factor Decomposition

We first estimate a dynamic factor model of the price-rent ratios of the 17

cities in our sample. The model decomposes each price-rent ratio into a

common national factor and a local factor. We estimated two versions of

the model. In one model, we restricted the loadings on the national factor

to be 1 across the 17 cities in our sample. In the other model, the factor

loading is allowed to change but the standard deviation of the national factor

is normalized to be 1 in order to achieve identification. The results from

the two models are very similar and are summarized in Table 3. We find

that across the 17 cities local factors drive a significant portion of the total

volatility in the housing markets. We measure the volatility of a housing

market by the standard deviation of the annual price-rent ratio. As Table

3 shows, in the model with restricted factor loadings, an average of 47% of

the total volatility of the housing markets is attributable to local factors. In

some cities, the local factor shares are more than 60%. If we allow the factor

loadings to vary, the average local factor share is slightly lower, but is still

more than 40%. This is consistent with the results of Del Negro and Otrok

(2007), which finds that historically movements in house prices were mainly

driven by the local components. Figure 1 plots the estimated national factor

14

of the price-rent ratio together with the log Case-Shiller house price index

(normalized by the CPI index). We can see that these two series track each

other closely with a correlation coefficient of 0.86. Our series of the national

factor of the price-rent ratio is also very similar to the one estimated by

Davis et al. (2008). This confirms that the dynamic factor model provides

a good summary of housing market movements.

Table 3 also shows that the local factor shares vary greatly from city to

city. For example, while local factors contribute more than 60% of the total

volatility of the price-rent ratio in New York City and Los Angeles, the lo-

cal factor share is only 24% in Chicago. The log-linearized Campbell-Shiller

present value formula provides insights about what drives these local volatil-

ities. The Campbell-Shiller formula is an accounting identity that expresses

the (log) price-rent as a sum of two components: the present value of the

expected future rent growth rates and the present value of expected future

discount rates. House prices increase today either because people expect

higher future growth or a lower discount rate or both. As we have demon-

strated in Section 2, the growth component can be further decomposed into

a common national growth factor and an independent local growth factor.

The discount rate components can be thought of as consisting of three fac-

tors, a risk-free interest rate, an aggregate or national risk premium which

are common to all cities, and an idiosyncratic local risk premium. Therefore,

in cites where the local factors contribute a large share to the housing mar-

ket volatility there must be either volatile local growth or volatile local risk

premiums or both. In Table 4 we report the standard deviations of the local

growth rate and local risk premiums (see below for more on the estimation

of different components in the Campbell-Shiller accounting idendity). We

can see that local risk premiums are about 5 times more volatile than local

growth rates. There are also greater variations in the standard deviations

of local risk premiums. In Figure 2 we plot the scatter graph of the local

factor shares of the price-rent ratio against the standard deviations of the

local risk premiums. In Figure 3 we plot a similar graph with the standard

deviations of the local growth rate instead. We can clearly see from these

15

two figures that the local factor shares are mostly due to the volatility of

local risk premiums. Variations in the local growth rate have some, but very

limited, explanatory power for the local factor shares.

4.2 Economic Fundamentals of Housing Markets

We next examine the economic fundamentals that underlie the national and

local factors of the house price-rent ratios. Using the Campbell-Shiller log-

linear present-value formula, we are able to equate the national factor of the

log price-rent ratio to the sum of the present value of the expected national

growth rate in rent, the present value of the expected real interest rate

and an aggregate risk-premium term. Similarly, the local factor of a price-

rent ratio can be written as the sum of the present value of the expected

local growth rate in rent and a local risk premium term. To get reliable

estimates of the expected future rent growth rates and interest rates, we

use a long duration of historical data from 1936-2009 on rent and interest

rate. We embed a vector regression model into a dynamic factor model of

rent growth rates. This allows us to obtain joint estimates of the national

and local factors of rent growth as well as the expected future interest rates.

The national and local risk premium terms are then obtained as residuals

in the Campbell-Shiller accounting identity using our previous estimates of

the national and local factors of the log price-rent ratios. The results are

summarized in Table 4 and 5.

Table 5 shows that the long-run growth rate in rent is estimated to be

around 1.40% at the aggregate level. Our model also yields an estimate

of the expected long-run real interest rate of 2.27%. Since we can only

obtain index data on rent, the log price-rent ratio used in our study is

different from the true log price-rent ratio by a constant. Therefore we can’t

obtain the correct estimates of the mean of the national and local factors of

the log price-rent ratios, as well as that of the national house market risk

premium. Nonetheless this scale problem doesn’t affect our estimate of the

standard deviations of the log price-rent ratios and the underlying economic

16

variables as well as their correlations. Among the three economic variables

underlying the national factor of log price-rent ratio, the interest rate term

has the biggest standard deviation of 17.25%. The risk premium term has a

standard deviation of 11.95%, indicating strong evidence of time-varying risk

premium in housing markets. The growth term is the least volatile variable

with a standard deviation of 6.91%. Moreover, these three variables are

also highly correlated. The growth term and real interest rate are positively

correlated. The risk premium is negatively correlated with both the growth

term and real interest rate.9

To assess the impact of the economic fundamentals on housing markets,

we report in Table 6 the results from simple regressions of log price-rent

ratios on the growth and interest rate variables. Notice that our estimates of

the national growth factor and the real interest rate term are obtained from a

separate dynamic factor model than the one for price-rent ratio, and the risk

premium is obtained as the residual term in the Campbell-Shiller accounting

identity. Therefore a meaningful regression is a one that only includes the

growth and interest rate variables. Table 6 shows that the growth and

interest rate variables jointly explain up to 46.47% of the total variation

in the aggregate log price-rent ratios. The interest rate alone accounts for

about 17% of the variation in the aggregate log price-rent ratios. Moreover,

consistent with standard economic theory, a higher expected real interest has

a significant negative effect on house price while a higher growth expectation

has a significant positive effect. The regression result also indicates that a

large portion (more than 50%) of the variation in the national house market

is due to changes in the aggregate risk premium term. Table 6 also reports

the results from regressing local price-rent ratios on local growth variables.

Consistent with the result on the local factor shares in the previous section,

we find that local growth explains very little of the variation in local price-

rent ratios for most cities. The idiosyncratic volatilities in local housing

markets seem mostly due to time-variations in local risk premiums. Given

9The negative correlation between housing market risk premium and growth is consis-tent with the counter-cyclical risk premiums in stock market documented in many studiessuch as Campbell and Cochrane (1999).

17

that local price-rent ratios contribute to more than 40% of the total volatility

of house markets, it seems safe to conclude that variations in risk premiums

are the most important factor that drives housing market volatility. Changes

in the interest rate have the expected, but limited, direct effect on housing

market volatility.10

4.3 Housing Market Risk Premiums and the Pricing Error

The residual term from the Campbell-Shiller present value formula (5) is the

expected excess return or risk premium in the housing market, Et∑∞

τ=0 ρτ ert+1+τ .

This is a valid decomposition if investors have rational expectations and the

transversality condition holds, i.e., limT→∞ ρTEtxt+T = 0, where xt is the

national factor of log price-rent ratio. In general, however, the residual term

from the Campbell-Shiller formula may include a pricing error. This pricing

error can arise because either investors hold irrational expectations or there

is a speculative bubble that violates the transversality condition. We now

rewrite the Campbell-Shiller present value formula as

xt = Et

∞∑τ=0

ρτ∆dt+1+τ−Et∞∑τ=0

ρτrf,t+1+τ−Et∞∑τ=0

ρτ ert+1+τ+ limT→∞

ρTEtxt+T

(19)

or

xt = yt − lt − ηt + νt (20)

where yt, lt and ηt are, as before, the expected rent growth, the real interest

rate and the risk premium respectively, and νt denotes a possible pricing

error in the housing market. Our dynamic factor models produce estimates

of xt, yt and lt, and the residual term from the account identity (20) now

contains two components, the risk premium and the pricing error, ηt and νt.

It is well documented that the excess return in equity market can be

predicted by some state variables such as yield spread and dividend yield

10Kishor and Morley (2010) reports a similar finding that variations in risk premiums ex-plain a large fraction of housing market volatility. Cochrane (2011) argues that most assetmarket puzzles and anomalies are related to large discount-rate/risk premium variations.

18

etc. To distinguish between the risk premium and the pricing error in the

housing market, we project via OLS regression the residual term from the

Campbell-Shiller formula (20) onto the same state variables that are known

to predict stock market returns. We interpret the fitted value as the housing

market risk premium and the OLS residual as the pricing error. The results

are reported in upper panel of Table 7. We can see that both the yield

spread (the difference between the yield on Treasury bonds and that on

Treasury bills) and S&P 500 dividend yield are significant predictors of

the housing market return. We also used the S&P earning-price ratio in

the linear projection and obtained similar results. The R2 from the OLS

regression is high, indicating a large part of the Campbell-Shiller residual

is the housing market risk premium that varies over time. Nonetheless the

pricing error is also significant. The estimated housing market risk premium

and pricing error are plotted in Figure 4. We can clearly see that during the

housing market frenzy between 2000 and 2006, there was a large increase in

the pricing error, νt, and an even larger decrease in the risk premium, ηt,

both contributing to the sharp increase the housing market price before the

2007-2008 financial crisis (Figure 1). In contrast, during the early sample

period (1979 to 1985), the pricing error was decreasing and the risk premium

was increasing. As a result, the housing price declined during that period.

It is interesting to notice that in the risk premium regression (the upper

panel of Table 7), the coefficient on the yield spread is positive while the

coefficient on the stock dividend yield is negative. A large positive yield

spread indicates that interest rates are likely to rise in the future, and rising

interest rates decrease values of long-term assets such as houses. Therefore

a large yield spread increases the risk to participate in the housing market,

resulting in a higher risk premiums. On the other hand, it is well known that

the dividend yield has strong forecasting power for future stock returns. For

example Cochrane (2011) shows that a one percentage point increase in the

dividend yield forecasts a nearly four percentage point higher excess return

in the stock market. In states where the dividend yield is high, investors

perceive a larger risk in the stock market and demand a higher expected

19

return. As a result they are more willing to accept a lower expected return

in the housing market. A higher dividend yield in the stock market predicts

lower returns in the housing market. Housing assets provide a hedge against

stock market risk. Of course, such interpretations are subject to the caveat

that part of the estimated risk premium may actually be the projection of

the pricing error on the state variables or that the two state variables fail

to capture all the variations of the housing market risk premium.

The orthogonal residual term from the above OLS regression can be

interpreted as an estimate of housing market pricing error. One possible

source of the pricing error is money illusion. For example, as pointed out by

Modigliani and Cohn (1979), investors may fail to distinguish between the

real interest rate and nominal interest rate. They may interpret a decline in

the nominal interest rate due to changes in inflation as a decline in the real

interest rate, and therefore bid up the real housing price. As pointed out by

Brunnermeier and Julliard (2008), in the simplest case with constant real

rents and real interest rates, the price-rent ratio will be simply determined

asP

D=∞∑τ=1

1

(1 + rf )τ=

1

rf(21)

where rf is the real interest rate. Under money illusion, however, investors

would value the housing asset as

P

D=∞∑τ=1

1

(1 + i)τ=

1

i(22)

where i is the nominal interest rate. And if i declines due to a reduction

in inflation, the price-rent ratio will increase even if the real interest rate

remains constant.

To see if the estimated pricing error is indeed related to money illusion,

we run an OLS regression of the pricing error on the inverse of the nominal

interest rate as well as the inverse of inflation. The results are reported in the

second panel of Table 7. We can see that the estimated pricing error is indeed

20

positively related to the inverse of the nominal interest rate and inflation.

Since a change in inflation may reflect a change in inflation volatility rather

than its level, we also include the square of inflation in the regression to

control the effects of inflation volatility and obtain the same result. Notice

that the R2 of the regressions are not very high, suggesting that money

illusion may not be the only source of the pricing error.11 The pricing error

and its fitted value are plotted in Figure 5. It is interesting to note that

while the fitted value of the pricing error remains close to 0 and is very

different from the actual value in most periods (which explains the low R2),

it increases sharply in between 2000 and 2003 and tracks the actual value of

the pricing error closely. The short-term nominal interest rate was pushed

down to a very low level in a short period of time by the monetary policy in

the aftermath of the tech bubble and the subsequent economic recession. It

was also during that period the increase in housing prices accelerated (see

Figure 1). We can also see from Figure 5 that the money illusion effect

explains most of the pricing errors during the early 1980s as well. These

results suggest that money illusion is most severe during periods of drastic

changes in the nominal interest rate such as the early 2000s and the early

1980s. And the monetary policy may have contributed to the initial stage of

the last housing market bubble indirectly through the money illusion effect.

5 Conclusions

This paper is an empirical analysis of housing market dynamics. The hous-

ing asset is probably the single most important component of an average

household’s financial portfolio. Housing market movements also have great

impacts on macroeconomic fluctuations. Compared to equity and bond mar-

kets, however, there have relatively fewer studies on the nature and sources

of housing market volatility. We contribute to a growing literature on hous-

ing market by estimating a dynamic factor models of the price-rent ratios

11This is in contrast to the result in Brunnermeier and Julliard (2007) where the pricingerror is almost entirely explained by money illusion.

21

of 17 major metropolitan areas of the U.S. The model allows us to disen-

tangle the common and idiosyncratic local components of housing market

volatility. We find that a large fraction of housing market volatility is lo-

cal. Our dynamic factor model is based on a present-value representation

of the price-rent ratio of the housing asset and is jointly estimated with

a forecasting VAR that includes important macroeconomic variables. This

approach enables us to relate otherwise unobservable latent factors to eco-

nomic fundamentals of the housing markets. Our results indicate that at

both the local and the aggregate levels time-variation in risk premiums is

the most important source of housing market volatility. Interest rates play a

smaller role in driving the movements of the housing markets. Nonetheless,

changes in the interest rate can have a direct and indirect impact on the

housing market. A decrease in the interest rate directly lowers the discount

rate, and therefore pushes up home prices. Moreover, a sharp decline in

the interest rate can also fuel housing market speculations through a money

illusion effect. We find evidence that the housing market bubble leading to

the 2008 financial crisis was indeed accompanied by both a large decrease in

the housing market risk premium and a large increase in the pricing error

associated with money illusion.

22

References

[1] Ambrose, B.W., P. Eichholtz and T. Linderthal, 2011, “House Prices

and Fundamentals: 355 Years of Evidence”, working paper, Pennsylva-

nia State University.

[2] Brunnermeier, M. K. and C. Julliard, 2008, “Money Illusion and Hous-

ing Frenzies”, The Review of Financial Studies, 21, 135-180.

[3] Campbell, J. and R. Shiller, 1988, “The Dividend-Price Ratio and Ex-

pectations of Future Dividends and Discount Factors”, Review of Fi-

nancial Studies, 1, 195-228.

[4] Campbell, J. and J. Cochrane, 1999, “By force of habit: A

consumption-based explanation of aggregate stock market behavior”,

Journal of Political Economy 107, 205-251.

[5] Campbell, S. D., M. A. Davis, J. Gallin, and R. F. Martin, 2009, “What

Moves Housing Markets: A Variance Decomposition of the Rent-Price

Ratio”, Journal of Urban Economics, 66, 90-102.

[6] Ciccarelli, M. and B. Mojon, 2010, “Global Inflation”, Review of Eco-

nomics and Statistics, 92, 524-535.

[7] Cochrane, J.H., 2011, “Discount Rates”, The Journal of Finance, 66,

1047-1108.

[8] Davis, M.A. and J. Heathcote, 2005, “Housing and the Business Cycle”,

International Economic Review, 46, 751-784.

[9] Davis, M.A., A. Lehnert and R.F. Martin, 2008, “The Rent-Price Ra-

tio for the Aggregate Stock of Owner-Occupied Housing”, Review of

Income and Wealth, 54, 279-284.

[10] Del Negro, M. and C. Otrok, 2007, “99 Luftballons: Monetary Policy

and the House Price Boom Across U.S. States”, Journal of Monetary

Economics, 54, 1962-1985.

23

[11] Engsted, T., P.Q. Thomas and C. Tanggaard, 2012, “Pitfalls in VAR

based Return Decompistions: A Clarification”, Journal of Banking and

Finance, 36, 1255-1265.

[12] Favilukis, J., D. Kohn, S.C. Ludvigson and S. van Nieuwerburgh, 2011,

“International Capital Flows and House Prices: Theory and Evidence”,

working paper, New York Univesity.

[13] Forni, M., M. Hallin, M. Lippi and L. Reichlin, 2000, “The General-

ized Dynamic Factor Model: Identification and Estimation”, Review of

Economics and Statistics, 82, 540-554.

[14] Fratantoni, M. and S. Schuh, 2003, “Monetary Policy, Housing, and

Heterogeneous Regional Markets”, Journal of Money, Credit and Bank-

ing, 35, 557-589.

[15] Glaeser, E.L and J. Gyourko 2007, “Arbitrage in Housing Markets”,

NBER working paper 13704.

[16] Gyourko, J., C. Mayer, and T. Sinai, 2006,“Superstar Cities”, NBER

Working Paper 12355.

[17] Iacoviello, M., 2005, “House Prices, Borrowing Constraints, and Mon-

etary Policy in the Business Cycle”, American Economic Review, 95,

739-764.

[18] Kim, C.J. and C. R. Nelson, 1999, State-Space Models with Regime

Switching: Classical and Gibbs-Sampling Approaches with Applications,

The MIT Press.

[19] Kishor, N.K. and J. Morley, 2010, “What Moves the Price-Rent Ratio:

A Latent Variable Approach”, working paper, University of Wisconsin-

Milwaukee.

[20] Kose, A., C. Otrok and C. Whiteman, 2003, “International Business

Cycles: World, Region and Country Specific Factors”, The American

Economic Review, 93, 1216-1239.

24

[21] Kose, A., C. Otrok and C. Whiteman, 2008, “Understanding the Evolu-

ation of World Business Cycles”, Journal of International Economics,

75, 110-130.

[22] Ludvigson, S.C. and S. Ng, 2009, “Macro Factors in Bond Risk Premia”,

Review of Financial Studies, 22, 5027-5067.

[23] Mian, A. and A. Sufi, 2009, “The Consequences of Mortgage Credit

Expansion: Evidence from the U.S. Mortgage Default Crisis”, Quarerly

Journal of Economics 124, 1449-1196.

[24] Modigliani, F. and R. Cohn, 1979, “Inflation, Rational Valuation and

the Market”, Financial Analysts Journal, 37, 24-44.

[25] Ng, S. and E. Moench, 2011, “A hierarchical factor analysis of U.S.

housing market dynamics,” Econometrics Journal, 14, 1-24.

[26] Stock, J. H. and M. W. Watson, 1991, “A Probability Model of the Co-

incident Economic Indicators”, in Leading Economic Indicators: New

Approaches and Forecasting Records, ed. K. Lahiri and G. H. Moore.

Cambridge University Press, 63-89.

[27] Stock, J. H. and M. W. Watson, 2009, “The Evolution of National and

Regional Factors in U.S. Housing Construction”, in Volatility and Time

Series Econometrics: Essays in Honour of Robert F. Engle, T. Boller-

slev, J. Russell and M. Watson (eds), 2009, Oxford: Oxford University

Press.

[28] Shiller, R., 2005, Irrational Exuberance, second ed. Princeton University

Press, Princeton, NJ.

25

A Data

Rent growth. Nominal rent indexes are the rent of primary residence for

major U.S. metropolitan areas from Bureau of Labor Statistics (BLS).12

The beginning periods of these indexes vary for different cities: for some

cities such as Boston or San Francisco we start to have observations from as

early as 1915 but for some other cities such as Miami or Denver we do not

have observations until as late as 1978 and 1971, respectively. To explore

as much information as possible in the long-run time series data and at the

same time to include as many cities as possible to span wide geographic re-

gions, we choose to start our sample period from 1935 and we have to drop

many cities such as Dallas, Miami, Tampa, Anchorage, Denver, Honolulu,

San Diego. Furthermore we have to drop St. Louis because of the lack of

house price data. This leaves us with a set of observations of the annual

rent indexes from 1935 to 2009 for 17 cities: New York City, Philadelphia,

Boston, Pittsburgh, Chicago, Cincinnati, Cleveland, Detroit, Kansas City,

Milwaukee, Minnesota, Atlanta, Houston, Los Angeles, Portland, San Fran-

cisco, and Seattle. Evidently the data covers most major cities ranging from

the east coast to the midwest and west coast. The nominal rent then is

deflated by the consumer price index (CPI). The real rent growth is the

difference of the logarithms of the real rent.

Log price-rent. Nominal housing price indexes are the repeat-transaction

house price indexes from the Federal Housing Finance Agency (FHFA). The

original data start from 1975 for most cities. But we do not have observations

for some cities, such as St, Louis, until as late as 1983. Therefore, we choose

to start our sample period from 1979 and this, together with the considera-

tion of the rent data as described earlier, leaves us with a set of observations

of the annual housing price data from 1979 to 2009 for 17 cities: New York

City, Philadelphia, Boston, Pittsburgh, Chicago, Cincinnati, Cleveland, De-

12These indexes are the Consumer Price Index (CPI) rent components whose construc-tion has been criticized by a number of studies such as the Boskin Commission Report.While we are aware of this problem, these indexes are the only long-run rent series avail-able.

26

troit, Kansas City, Milwaukee, Minnesota, Atlanta, Houston, Los Angeles,

Portland, San Francisco, and Seattle. We then divide the nominal housing

price by the nominal rent and take the logarithm to get the log price-rent

ratio. Notice that since both price and rent are indexes, the log price-ratio

deviates from its true value by a constant. This caveat, however, won’t

affect our analysis of housing market volatility.

Other macroeconomic variables. Annual real GDP growth rate and the

consumer price index are from the FRED website. The Case-Shiller home

price index and the short and long rate are taken from Robert Shiller’s book

Irrational Exuberance, available on Shiller’s website. The real interest rate

used in our estimation is the short rate minus the CPI inflation rate. The

sample period for the macro variables is 1935 to 2009.

27

Ta

ble

1S

um

ma

ry S

tati

stic

s o

f H

ou

sin

g M

ark

ets

, P

rice

Ch

an

ge

s 1

97

9-2

00

9

NY

CP

HIL

BO

ST

PIT

TC

HIC

CIN

CC

LEV

DE

TK

CM

ILW

MIN

NA

TL

HO

US

LAP

OR

TL

SF

RS

EA

Log

Pri

ce C

ha

ng

e0

.06

77

0.0

55

90

.06

79

0.0

36

50

.04

08

0.0

35

80

.03

23

0.0

35

30

.03

50

0.0

40

40

.04

56

0.0

42

80

.03

08

0.0

58

00

.03

32

0.0

63

90

.05

78

Std

De

v0

.07

47

0.0

54

60

.08

11

0.0

22

90

.03

36

0.0

24

80

.02

93

0.0

63

40

.02

67

0.0

31

40

.04

31

0.0

31

60

.04

60

0.0

97

00

.10

15

0.0

85

40

.05

70

Co

rre

lati

on

1.0

00

00

.81

51

0.8

92

90

.10

81

0.3

99

40

.39

55

0.1

86

10

.38

18

0.5

75

10

.21

28

0.5

56

90

.71

29

-0.3

66

70

.58

52

0.5

67

00

.58

78

0.0

94

3

1.0

00

00

.54

53

0.2

90

00

.60

97

0.5

24

50

.27

63

0.4

10

60

.43

85

0.4

11

90

.52

80

0.5

67

6-0

.19

17

0.8

45

10

.71

94

0.7

58

30

.41

50

1.0

00

0-0

.00

99

0.2

60

40

.30

21

0.1

90

30

.35

37

0.5

16

90

.06

79

0.4

77

90

.70

35

-0.3

78

50

.39

98

0.3

80

40

.48

66

-0.0

79

9

1.0

00

00

.40

50

0.5

33

20

.59

51

0.2

83

50

.03

05

0.6

22

30

.27

95

-0.1

28

4-0

.12

07

0.2

39

40

.23

02

0.2

46

80

.31

62

1.0

00

00

.79

05

0.7

59

50

.71

66

0.3

15

80

.73

75

0.4

81

10

.30

47

-0.2

35

50

.68

00

0.6

48

80

.64

84

0.6

03

5

1.0

00

00

.87

94

0.9

13

50

.59

24

0.7

66

20

.64

23

0.4

26

2-0

.26

60

0.5

52

40

.52

23

0.6

69

00

.39

68

1.0

00

00

.84

20

0.3

19

70

.74

05

0.4

27

40

.16

38

-0.3

74

80

.36

85

0.4

03

20

.44

44

0.3

45

1

1.0

00

00

.66

58

0.6

10

90

.64

45

0.5

15

9-0

.26

84

0.5

04

70

.51

67

0.6

26

20

.34

59

1.0

00

00

.44

85

0.8

68

80

.76

39

0.0

69

00

.47

21

0.5

71

40

.59

46

0.2

10

5

1.0

00

00

.59

86

0.1

16

0-0

.08

49

0.5

16

20

.61

79

0.5

33

40

.59

39

1.0

00

00

.69

00

0.2

30

20

.64

80

0.7

83

60

.70

08

0.4

07

3

1.0

00

00

.08

42

0.5

39

90

.58

89

0.6

45

20

.24

13

1.0

00

00

.13

70

0.1

75

60

.10

56

0.3

15

0

1.0

00

00

.85

13

0.9

03

60

.62

00

1.0

00

00

.73

39

0.6

46

7

1.0

00

00

.58

69

1.0

00

0

Ta

ble

2S

um

ma

ry S

tati

stic

s o

f H

ou

sin

g M

ark

ets

, Lo

g P

rice

/Re

nt

Ra

tio

s 1

97

9-2

00

9

NY

CP

HIL

BO

ST

PIT

TC

HIC

CIN

CC

LEV

DE

TK

CM

ILW

MIN

NA

TL

HO

US

LAP

OR

TL

SF

RS

EA

log

(P/D

)-0

.41

05

-0.4

37

1-0

.35

78

-0.3

48

5-0

.51

28

-0.3

83

0-0

.41

85

-0.3

65

0-0

.27

00

-0.3

92

0-0

.23

66

-0.3

41

4-0

.07

83

-0.2

66

6-0

.27

48

-0.3

92

6-0

.42

53

Std

De

v0

.22

36

0.1

50

90

.21

00

0.1

12

40

.08

76

0.0

91

00

.08

71

0.2

00

10

.08

78

0.1

70

90

.19

21

0.0

93

40

.10

84

0.2

25

40

.21

93

0.2

27

20

.22

85

Co

rre

lati

on

1.0

00

00

.87

15

0.9

77

30

.54

29

0.4

69

60

.46

05

0.3

63

80

.46

65

0.2

27

40

.50

92

0.6

42

00

.65

04

-0.1

47

50

.75

50

0.6

07

40

.77

44

0.7

15

2

1.0

00

00

.79

27

0.7

71

10

.77

82

0.6

76

50

.58

35

0.5

61

40

.52

13

0.7

58

20

.81

90

0.8

34

60

.03

77

0.9

32

80

.70

83

0.9

48

60

.88

14

1.0

00

00

.54

54

0.4

18

30

.48

34

0.3

89

60

.52

40

0.1

97

90

.50

67

0.6

34

90

.60

67

-0.2

25

90

.68

35

0.5

04

20

.71

17

0.6

99

5

1.0

00

00

.91

04

0.9

40

50

.91

39

0.8

51

40

.68

02

0.9

40

80

.89

56

0.7

81

8-0

.09

59

0.8

03

90

.37

60

0.8

39

00

.90

67

1.0

00

00

.90

67

0.8

95

70

.77

68

0.7

77

10

.92

05

0.8

68

50

.79

79

0.0

48

70

.85

02

0.5

40

20

.86

37

0.8

50

8

1.0

00

00

.93

62

0.8

99

40

.82

31

0.9

68

60

.93

13

0.8

14

50

.02

76

0.7

75

50

.42

90

0.7

94

90

.83

97

1.0

00

00

.94

36

0.6

54

10

.85

94

0.7

85

40

.61

05

-0.2

35

20

.68

47

0.2

36

60

.71

43

0.7

68

2

1.0

00

00

.55

28

0.7

98

50

.77

57

0.5

88

4-0

.33

69

0.6

75

90

.22

26

0.7

02

40

.75

48

1.0

00

00

.85

73

0.8

25

30

.83

72

0.5

51

60

.65

04

0.5

92

80

.63

94

0.5

76

4

1.0

00

00

.97

25

0.9

01

40

.15

79

0.8

26

10

.54

25

0.8

44

50

.89

36

1.0

00

00

.95

43

0.1

80

30

.88

19

0.6

66

00

.88

91

0.9

12

5

1.0

00

00

.39

94

0.8

97

30

.81

09

0.8

86

50

.83

49

1.0

00

00

.12

48

0.4

96

10

.07

51

-0.0

92

9

1.0

00

00

.77

83

0.9

91

30

.88

34

1.0

00

00

.72

22

0.5

60

3

1.0

00

00

.90

91

1.0

00

0

Table 3 Factor Decomposition of Log Price/Rent Ratios 1979-2009

Restricted Model Unrestricted Model

Nat'l Factor Local Factor Local Share Nat'l Factor Local Factor Local Share

NYC 0.0943 0.1885 0.6666 0.1579 0.1838 0.5379

PHIL 0.0943 0.0959 0.5043 0.1637 0.0922 0.3604

BOST 0.0943 0.1773 0.6528 0.1292 0.1750 0.5754

PITT 0.0943 0.0355 0.2735 0.1147 0.0310 0.2127

CHIC 0.0943 0.0300 0.2412 0.1062 0.0301 0.2208

CINC 0.0943 0.0182 0.1614 0.0611 0.0389 0.3888

CLEV 0.0943 0.0397 0.2964 0.0778 0.0350 0.3099

DET 0.0943 0.1289 0.5775 0.0928 0.1302 0.5837

KC 0.0943 0.0540 0.3640 0.0813 0.0616 0.4311

MILW 0.0943 0.0792 0.4564 0.1198 0.0673 0.3597

MINN 0.0943 0.1042 0.5249 0.1099 0.1000 0.4764

ATL 0.0943 0.0460 0.3281 0.0684 0.0530 0.4367

HOUS 0.0943 0.1386 0.5952 0.0434 0.1187 0.7324

LA 0.0943 0.1517 0.6168 0.1999 0.1078 0.3504

PORTL 0.0943 0.1862 0.6638 0.2004 0.2079 0.5092

SFR 0.0943 0.1509 0.6154 0.2209 0.0992 0.3098

SEA 0.0943 0.1493 0.6129 0.2138 0.0867 0.2886

Average 0.0943 0.1044 0.4795 0.1271 0.0952 0.4167

Note: this table reports the standard deviations of the national factor and local factors of the price-rent

ratios in 17 cities. In the restricted model, the loading on the national factor is restricted to be 1. In the

unrestricted model, the factor loading can change across different cities. Local share is the percentage of

the total standard derivation attributable to local factors.

30

Table 4 Volatility of Local Economy

Local Growth Rate Local Risk Premium Correlation

NYC 0.0588 0.1516 -0.7652

PHIL 0.0347 0.0838 -0.1945

BOST 0.0353 0.1743 0.0199

PITT 0.0066 0.0357 0.1280

CHIC 0.0078 0.0303 0.1421

CINC 0.0018 0.0185 0.2547

CLEV 0.0083 0.0378 -0.1972

DET 0.0110 0.1281 -0.1022

KC 0.0116 0.0546 0.1636

MILW 0.0174 0.0819 0.3492

MINN 0.0071 0.1057 0.2321

ATL 0.0145 0.0505 0.4200

HOUS 0.0292 0.1531 0.5383

LA 0.0385 0.1377 -0.1863

PORTL 0.0212 0.1903 0.2386

SFR 0.0293 0.1692 0.6132

SEA 0.0142 0.1482 -0.0134

Average 0.0204 0.1030 0.0965

Note: this table reports the standard deviations of the present value of the expected local rent growth

rates and the standard deviations of the local risk premiums (first two columns). The last column

includes the correlation coefficients between local growth rate and local risk premium.

31

Table 5 Macroeconomic Fundamentals of House Markets

Nat'l Growth Interest Rate Nat'l Risk

Mean 0.0140 0.0227 n/a

Std dev 0.0691 0.1725 0.1195

Correlation 1.0000

0.8342 1.0000

-0.8140 -0.7826 1.0000

Note: this table reports the summary statistics of the present value of the expected national rent growth

(2nd

column), the present value of the expected real interest rates (3rd

column) and the aggregate house

market risk premium (4th

column). The first two series are estimated on the annual data from 1936 to

2009 using a dynamic factor model. The last series is obtained as a residual term from the Campbell-

Shiller formula using data from 1979 to 2009.

32

Table 6 Price-Rent Ratio and Economic Fundamentals

Nat'l Price-Rent Real Interest Rate Nat'l Rent Growth R2

-0.8878** 2.2318** 0.4647

(0.1885) (0.5767)

-0.2076**

0.1783

(0.0827)

Local Price-Rent Local Rent Growth R2

NYC

3.5609** 0.7320

(0.4001)

PHIL

1.4930** 0.2650

(0.4617)

BOST

0.9026 0.033

(0.9066)

PITT

0.3475 0.0047

(0.9387)

CHIC

-0.1076 0.0002

(1.4329)

CINC

-1.8497 0.0284

(2.0093)

CLEV

2.1090** 0.1276

(1.0240)

DET

3.0455 0.0228

(3.6983)

KC

0.2930 0.0047

(0.7917)

MILW

-2.1878 0.0614

(1.5885)

MINN

-2.2109 0.0263

(2.4985)

ATL

-0.2518 0.0086

(0.5023)

HOUS

-1.5951** 0.13335

(0.7545)

LA

1.5961** 0.2050

(0.5836)

PORTL

-1.0367 0.0154

(1.5390)

SFR

-2.0719** 0.2152

(0.7348)

SEA

1.1211 0.0152

(1.6741)

Note: Coefficient estimates with ** indicate that they are significant at 5% level.

33

Table 7 Housing Market Risk Premium and Pricing Error

Risk Premium Yield Spread Dividend Yield Earning/Price R2

0.0542** -1.7722*

0.7248

(0.0070) (0.9351)

0.0459**

-1.3901** 0.7883

(0.0067)

(0.3845)

Pricing Error 1/Rate 1/Inflation Inflation2 R2

0.0843

0.0854

(0.0512)

0.1141**

0.0005 0.1556

(0.0538)

(0.0003)

0.0940* 0.0434

0.159

(0.0504) (0.0277)

Note: The first panel of this table contains the results of an OLS regression of the Campbell-Shiller

residuals on the yield spread, S&P 500 dividend yield and S&P 500 earning/price ratio. The second panel

reports the results of an OLS regression of the regression residual from the first OLS on inverse interest

rate and inflation. Numbers in parentheses are standard errors.

34

Figure 1: National Factor of Price-Rent Ratio and House Price Index

-.8

-.6

-.4

-.2

.0

.2

1980 1985 1990 1995 2000 2005

LOG(NP/CPI) PDN

This figure plots the estimated national factor of log price-rent ratio, PDN, and log Case-

Shiller house price index over the CPI index, LOG(NP/CPI).

35

Figure 2: Local Factor Share vs Local Market Risk Premium Volatility

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Vo

lati

lity

of

Idio

syn

cra

tic

Ris

k P

rem

ium

s

Price/Rent Volatility Due to Local Factors

This figure plots the fraction of the total volatility in the house price-rent ratio due to the

local factor (the horizontal axis) versus the volatility of local risk premium (the vertical

axis) across the 17 housing markets.

36

Figure 3: Local Factor Share vs Local Growth Volatility

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Vo

lati

lity

of

Loca

l R

en

t G

row

th

Price/Rent Volatility Due to Local Factors

This figure plots the fraction of the total volatility in the house price-rent ratio due to

the local factor (the horizontal axis) versus the volatility of local rent growth (the vertical

axis) across the 17 housing markets.

37

Figure 4: Housing Market Risk Premium and Pricing Error

-.6

-.4

-.2

.0

.2

-.10

-.05

.00

.05

.10

.15

1980 1985 1990 1995 2000 2005

RISK_PREMIUM MIS_PRICING

This figure plots the estimated aggregate housing market risk premium (left scale) and

the pricing error (right scale).

38

Figure 5: Housing Market Pricing Error

-.12

-.08

-.04

.00

.04

.08

.12

1980 1985 1990 1995 2000 2005

Pricing Error Fitted Value

This figure plots the estimated housing market pricing error and its fitted value from a

regression on the inverse of the nominal interest rate and the square of inflation.

39