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Why Do Home Prices Appreciate Faster in Center Cities? The Role of

Risk-Return Trade-Offs in Real Estate Markets

April 21, 2021

Maeve Maloney

Syracuse University

Email: mmalon01@syr.edu

Stuart S. Rosenthal

Maxwell Advisory Board Professor of Economics

Syracuse University

Email: ssrosent@syr.edu

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Abstract

Using Zillow data, this paper shows that central city housing appreciates faster than in suburban

markets, echoing evidence of higher rates of appreciation in superstar cities (e.g. Gyourko et al

(2013)). Additional results confirm that housing supply constraints within and across cities

contribute to these patterns by increasing volatility and investor exposure to risk. Allowing for

other mechanisms, zip code and CBSA-level CAPM betas explain up to half of cross-sectional

variation in appreciation rates. These patterns confirm that risk-return trade-offs are important

drivers of returns across markets, and that different rates of home price appreciation across

markets do not necessarily indicate opportunities for arbitrage.

JEL Codes: R0, G1

Key words: CAPM, Risk-Return, Home price appreciation rates, supply constraints

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1. Introduction

This paper establishes a new stylized fact that helps to highlight market mechanisms that

drive home price appreciation rates. Among large cities, year-over-year neighborhood level

single family home price appreciation rates decline almost monotonically both with distance to

the central city (Figure 1a) and as density declines (Figure 1b).1 In some respects, these patterns

echo more widely recognized tendencies for metropolitan level home price appreciation rates to

vary across urban areas, as is evident in Table 1. Recent work by Gyourko et al (2013) suggests

that for highly attractive supply constrained “superstar” cities, such cross-metro differences can

be sustained if a growing population of high-income households are drawn to such cities with

their scarce mix of amenities.

The superstar city argument is compelling when comparing higher appreciation rates for

amenity rich cities like San Jose to less sought after locations like Milwaukee or Cleveland, as in

Figure 2. It does not, however, explain differences in appreciation rates among cities with similar

appeal (e.g. New York and Los Angeles based on quality of life estimates in Chen and

Rosenthal, 2008). A different explanation is also needed for within-city variation in appreciation

rates. This is especially apparent if central city and suburban locations are viewed as close

substitutes on the demand side of the market, consistent with growing tendencies for central city

gentrification (Couture and Handbury, 2020). It will also be true because of equilibrating forces

from the supply side as developers direct investment to higher yielding locations, creating

pressure for similar rates of within-city home price appreciation across neighborhoods (e.g. Liu

et al, 2016).

1Estimates in Figures 1a and 1b are based on all CBSAs in the United States with population over 100,000 using

monthly zip code-level home price indexes from Zillow Inc. for the period 1996-2019. Additional detail will be

provided later in the paper.

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We draw on standard finance principles to provide a simple but unified explanation for

both the macro (cross-city) and micro (within city) patterns just described. As with previous

literature in the urban area (e.g. Glaeser, Gyourko and Saiz (2008)), we argue that cross-sectional

differences in housing supply elasticities and related supply constraints contribute to differences

in market volatility across locations. To the extent that supply constraints are known to investors

and perceived as enduring, systematic differences in investor exposure to risk should then be

offset by higher rates of home price appreciation in more volatile locations. Controlling for other

factors, our empirical work provides support for this idea both across and within metropolitan

areas.

To establish these and related results, we begin by estimating a CAPM model that

measures local housing market risk relative to the broader housing market to which a community

belongs, as in Case, Cotter, and Gabriel (2011). When we examine within-CBSA patterns of

home price appreciation we adopt two different approaches to measuring local markets. In some

models we divide the CBSA into just two locations, central city and suburb, and estimate

separate betas for each treating the entire CBSA as the broader market. In other models we

estimate separate betas for each zip code in the CBSA, again treating the entire CBSA as the

broader market. When we examine cross-CBSA patterns, we estimate separate betas for each

CBSA treating the entire set of CBSAs as a broader national market to which a CBSA belongs.

Our analysis of the relationship between the CAPM betas and cross-sectional indicators

of housing supply constraints yields striking patterns at both the cross-metro and within-metro

levels of geography. For the cross-metro analysis, we proxy for CBSA level housing supply

elasticities using the Wharton Land Use Regulatory Index (WLURI) developed by Gyourko,

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Saiz and Summers (2007).2 Estimates based on the entire population of CBSAs indicate that a

one standard deviation increase in the level of supply restrictiveness is associated with a roughly

20% to 30% increase in CBSA level risk relative to the national market. Moreover, the most

supply constrained quartile CBSAs are exposed to 45% more risk while the least constrained

quartile exhibit 15% lower risk.

Parallel analysis for within-CBSA patterns requires that we specify the center of each

CBSA which, to simplify discussion, we refer to as the CBD for a given urban area. For this we

use coordinates adopted by Holian and Kahn (2012) who code the latitude and longitude of

CBDs based on coordinates provided by Google Earth. Neighborhood level housing supply

elasticities are then proxied in two different ways including distance from the CBD and density.

Both proxies are motived by the idea that pre-existing high-density development makes land

assembly and new development more difficult, lowering housing supply elasticities in central

city and high-density locations (as in Baum-Snow and Han (2019)). Results indicate that CAPM

betas decline with distance from the CBD: zip codes 10 miles distant from the center exhibit 5%

less risk. Analogous estimates are also obtained when we distinguish zip codes by employment

density, and when we segment zip codes into central city versus suburban groups.3 These

patterns confirm that within cities, higher density, central locations tend to be associated with

greater housing market risk.

The final portion of this paper estimates the extent to which spatial variation in CAPM

betas helps to explain the spatial patterns of home price appreciation rates described above, both

2 See also Gyourko, Hartley, and Krimmel (2019) for related work. 3 For these exercises, high-density zip codes were defined as those above 95th percentile zip code employment

density and low-density zip codes were defined as those below the mean zip code employment density for the

national sample of CBSA zip codes. Suburban zip codes were classified as those 4 to 8 miles from the CBD, while

city center zip codes were classified as those within one mile of the city center.

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across and within CBSAs. This portion of the analysis controls for other plausibly important

drivers of price appreciation beyond risk in ways that we elaborate on later in the paper. At both

the metropolitan and zip code level we find a positive and significant relationship between

locally estimated betas and long-run home price appreciation. At the CBSA level a 50-

percentage point increase in beta (e.g. from 1.0 to 1.5) is associated with a 0.36 percentage point

increase in average year-over-year price growth. At the neighborhood level the relationship

between asset risk and home price appreciation is of the same order of magnitude but stronger; a

50-percentage point increase in local risk relative to the market is associated with a 0.58

percentage point increase in growth. We also find that spatial variation in betas accounts for

roughly half of the variation in home price appreciation rates across and within CBSAs.

Our paper contributes to literature on risk-return tradeoffs in housing markets. Crone and

Voith (1999) and Cannon, Miller and Pandher (2006) find positive risk-return patterns in housing

markets. However, Fan, Huszar and Zhang (2012) show that the risk-return relationship is

positive only when risk is low; when risk becomes high, the relationship becomes negative. Han

(2013) finds that while the positive relationship holds for some US metropolitan areas (San

Francisco and San Jose), there is a negative relationship in others (Chicago and Cincinnati).

Relative to these and other related studies, this paper introduces two ideas not previously

emphasized. First, we rely on a simple risk-return argument that allows for a unified explanation

for why long run home price appreciation rates differ both within and across CBSAs. Second, we

show that home price appreciation rates decline as density and related local supply elasticities

increase both across and within cities. We argue that these and other patterns are consistent with

market adjustments for spatial variation in investor exposure to risk.

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We proceed as follows. Section 2 presents a set of stylized facts on long-run price

appreciation and volatility within and across metropolitan areas. Section 3 reviews the capital

asset pricing model and estimates CAPM betas within and across CBSAs. Section 4 establishes a

positive relationship between indicators of housing supply elasticity and asset risk at both levels

of geography. Section 5 examines the extent to which differences in risk contribute to long-run

differences in price appreciation. Section 6 concludes.

2. Spatial Variation in House Price Appreciation and Volatility

2.1 Data

The primary data source is the monthly Zillow Home Value Index (ZHVI) for single

family-homes. We work with both the zipcode and CBSA level versions of the ZHVI, both of

which are seasonally adjusted. For both levels of geography, the index is designed to measure

quality adjusted home price appreciation in the target area. Index values are further scaled so that

the index value for December 2019 is equal to the average home value in the target area in that

month. In this way, the ZHVI captures home price appreciation while facilitating comparison of

home price levels across locations.

More precisely, for a given target location i and period t, ZHVI is computed as follows.

First, for each individual home within i, Zillow computes the home’s value in each month (the

so-called “Zestimate”) based on a comparison to sales of comparable homes in the nearby

community although Zillow does not provide full detail on how this is done. Next, Zillow

measures home price appreciation in the target area based on a weighted average of appreciation

across all homes in the target location, 𝐴𝑖,𝑡 = ∑ 𝑤ℎ,𝑡𝑧ℎ,𝑡𝐻ℎ=1 , where wh,t is the value of home h in

period t divided by the value of all homes in the target area in that period and zh,t is the rate at

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which home h is estimated to have appreciated in the last month. As a final step, Zillow forms

𝑍𝐻𝑉𝐼𝑖,𝑇 =1

𝐻∑ 𝑍𝑒𝑠𝑡𝑖𝑚𝑎𝑡𝑒ℎ,𝑇

𝐻ℎ=1 , where ZHVIi,T is the average home value in the target

community in the final period of the sample horizon (December, 2019 in our case). Working

backwards from that end point, the ZHVI index for each period prior to T in target location i is

constructed as follows,

𝑍𝐻𝑉𝐼𝑖,𝑡−1 = 𝑍𝐻𝑉𝐼𝑖,𝑡

1+𝐴𝑡, for 𝑡 = 0, … , 𝑇 − 1 (1)

Using this measure, in all of the analysis to follow, house price appreciation in location i is

calculated based on the growth of ZHVIi between periods.

ZHVI is the only publicly available index we are aware of that has extensive coverage at

the zip code level. The index includes all single-family residences outside of condominiums and

co-ops, and in that sense includes both single family attached and single family detached homes.

We use the monthly index from April 1996 to December 2019 at both the zip code and CBSA

level. Throughout the paper, index values are always measured in nominal terms without

adjusting for macro or local general rates of inflation. In adopting this approach, we are

implicitly assuming that month-to-month changes in local cost of living are driven primarily by

home price appreciation and not non-housing goods and services.4

4 Additional details on the Zillow methodology is provided by Zillow at: Zillow Home Value Index Methodology,

2019 Revision: Getting Under the Hood - Zillow Research . The ZHVI is designed to measure the change in

aggregate home values within a given location, holding constant the stock of homes between adjacent periods. Noise

embedded in the ZHVI estimates for individual homes that underlie the index for a given location appear likely to

average away. Also, addition of new possibly more highly valued homes to the local stock of housing is

incorporated into time varying measures of a location’s price appreciation index in a manner that is unlikely to

introduce biases into our estimates. There are two reasons for this. Looking back in time, homes constructed in

period t are not used to measure appreciation in prior to period t, helping to ensure that a common stock of homes is

used to measure price appreciation. Looking forward in time, Liu et al (2016) argue that active new construction in a

given market helps to ensure similar rates of home price appreciation across homes of different size and quality

because of equilibrating effects on the supply side. Both features help to ensure that construction of new homes will

have little effect on the rate at which Zillow’s ZHVI index appreciates in a given location. Any remaining concerns

that might arise from changes in the composition of homes in a given geographic area are further mitigated by our

focus on monthly changes in ZHVI index values since the stock of homes changes little on a month to month basis.

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At the time our data was downloaded the core-based statistical area (CBSA) definition

used by Zillow was the September 2018 US Census definition of CBSAs in the United States.

We use this same definition when constructing CBSA population counts for 1990, 2000, and

2010, ensuring that these measures correspond to the geography used to measure home price

appreciation. We limit our sample to the 364 CBSAs with population greater than a hundred

thousand in 2000.

2.2 Stylized Facts

Here we present three sets of stylized facts about single-family home prices that hold

across and within CBSAs. First, there is significant spatial variation in home price appreciation.

Second, home prices tend to appreciate more quickly in supply constrained locations. Finally,

homes price volatility is higher in supply constrained locations. We describe these patterns at

both the CBSA and within CBSA levels of geography.

2.2.1 Across CBSAs

The urban literature is well aware that long-run home price appreciation rates differ

across CBSAs (Glaeser, Gyourko & Saiz, 2008; Gyourko & Saiz, 2008). Here we reconfirm that

pattern and place it in context within our framework. Table 1 shows the average year-over-year

SF ZHVI growth rates over the 1996-2019 period for the top and bottom fastest growing CBSAs

with population greater than 500,000. Growth rates range from 1.27% in Youngstown, OH to

7.23% in San Jose, CA. There is also considerable variation in growth rates even among CBSAs

with high rates of price appreciation. Washington, DC has average year-over-year growth of

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4.36% while San Francisco, CA experiences price growth almost 3 percentage points higher at

7.22%.

The superstar city framework attributes differences in home price appreciation rates

between metropolitan areas to growing demand for select “superstar” cities that are in scarce

supply. These are cities that exhibit unusual high quality amenities not found elsewhere and

which also often have very inelastic local housing supply. Growing aggregate numbers of high-

income households increase demand for such locations over time pushing home prices higher.

While this may explain differences in growth rates between Cleveland and San Jose, it does not

explain differences in appreciation rates between similarly attractive, metropolitan areas. For

example, non-super star cities like Indianapolis and Minneapolis have average year-over-year

growth rates over the 1996-2019 period of 2.05 and 4.11 respectively.5

The next stylized fact is that price appreciation tends to be higher in more supply

constrained locations. At the CBSA level, supply restrictions are proxied using the Wharton

Land Use Regulation Index (WLURI) (Gyourko, Saiz, and Summers, 2008; Gyourko, Hartley,

and Krimmel; 2019). This index is based on the local regulatory environment including caps on

permitting and construction, density restrictions such as minimum lot size restrictions, affordable

housing requirements, and the number of re-zoning permits required. We match the index to 298

of the 364 CBSAs in our sample. Panel A of Table 2 shows the results from regressing average

CBSA year-over-year SF ZHVI growth on the WLURI. The coefficient on WRLURI is positive

and significant; long-run house price appreciation increases with the level of housing

restrictiveness. We repeat this exercise with the Siaz (2008) housing supply elasticities and find a

5 Amenity comparisons are based on the Chen and Rosenthal (2008) Quality of Life Index. Then mean Index value

for our sample is 231 with standard deviation 1880. The Index values for Indianapolis and Minneapolis are -1839

and -2054.

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negative relationship: CBSAs with more inelastic housing supply experience greater price

appreciation. Only 82 of the CBSAs within our sample can be matched to the Saiz measures,

however, and for this reason, we use the WRLURI as an indicator of CBSA level supply

elasticity in most of the analysis to follow.

Finally, not only do prices appreciate more quickly in supply constrained locations, they

are also more volatile. We measure local price volatility as the variance in the single family

ZHVI over our sample period. Regressing that measure on the WLURI we find that supply

constraints increase price volatility (Panel B of Table 2). Repeating the exercise using the Saiz

proxy for supply restrictiveness confirms this pattern.

2.2.2 Within CBSAs

A further stylized fact documented here is the positive relationship between supply

constraints and price appreciation within CBSAs.6 As noted earlier, we use two proxies for

within-CBSA supply constraints, distance to the CBD, whose coordinates are provided by Holian

and Kahn (2015), and zip code log employment density.

Panel A of Figure 1 plots the estimates from a nonparametric regression of zip code level

single-family ZHVI year-over-year percent growth on miles to the CBD. Observations are

restricted to within 30 miles of the CBD to reduce the tendency to encounter important

population subcenters and other cities situated within the CBSA. Observe that there is a roughly

monotonic negative relationship between distance to the CBD and percentage growth of ZHVI

with growth rates at the center roughly 3.9% versus 3.4% ten miles from the city center. Growth

6 Gleaser (2012) observe that city centers tend to experience more frequent housing bubbles than their suburb

counter parts. However, to our knowledge, we are the first to observe higher price appreciation in city centers/ high

density locations over a significant time horizon.

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rates stabilize thereafter, falling only slightly further to 3.3% by 20 miles distance from the CBD.

An analogous pattern holds for log employment density as well (Figure 1, Panel B). Price growth

is close to 3% when employment is largely absent (1 worker per square mile, or zero log

employment) but increases to over 5% percent for log density of 9, roughly 8,000 workers per

square mile.

Table 3 confirms that the patterns in Figure 1 persist event after controlling for other

factors. Column 1 adds no other controls and is analogous to the non-parametric plots in Figure

1. Column 2 includes CBSA fixed effects, and column 3 includes both CBSA and month fixed

effects. The coefficient on distance to the CBD is significant and similar in magnitude across all

three columns: moving ten miles away from the city center decreases price appreciation by

0.19% per year (Panel A). This is similar to the relationship plotted in Panel A of Figure 1. The

coefficient on log employment density is also significant and positive in all three specifications

but declines somewhat in magnitude with the addition of CBSA and month fixed effects (Panel

B).

We allow for further heterogeneity by estimating the distance and log employment

density models separately for each CBSA. This also addresses the possibility that the patterns in

Table 3 (and Figure 1) could be driven by a small number of large CBSAs with many zipcodes.

Because of the large number of CBSAs, Table 4 presents summary measures of the distribution

of estimates across urban areas.

For the distance model, notice that for 236 of the 352 CBSAs price growth varies

significantly with distance from the CBD. Of those, 58% are negative and 48% display a positive

growth-distance relationship. On the surface, this one measure casts doubt on whether there is a

systematic tendency for home price growth to decline with distance from city centers. Grouping

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CBSAs by population bins, however, reveals a more compelling pattern. For CBSAs with year-

2000 population between 100,000 and 500,000, home price growth does not appear to vary with

distance to the CBD. For CBSAs with population between 500,000 and 1 million, 57% of the

coefficients are negative. Between 1 million and 2.5 million, 86% of the coefficients are

negative, and for CBSAs with population over 2.5 million, all exhibit a negative price growth-

distance relationship. Similar patterns are also evident in Panel B of Table 4 for the log

employment density model.

The flat pattern between distance to the CBD and home price appreciation in small

CBSAs is suggestive of elastic housing supply and similarly stable rates of home price

appreciation throughout such areas. Conversely, the strong negative relationship between home

price appreciation and distance in large CBSAs suggests the presence of within-CBSA

heterogeneity. In these urban areas, it is possible that central city housing supply constraints may

amplify price volatility and investor exposure to risk, triggering risk-return tradeoffs, while more

outlying areas may exhibit more elastic supply, less price risk, and lesser appreciation for that

reason.

Table 5 reinforces this line of thinking. Estimates are presented of zip code level

regressions of the variance in ZHVI over the sample period on miles to the CBD (Panel A) and

log employment density (Panel B). The coefficient on miles to the CBD is negative and

significant, confirming that zip codes closer to the city center experience greater volatility in the

price index. These estimates are robust with and without controls for CBSA and month fixed

effects. Similar estimates are obtained for the density model as well, although here the

relationship is of smaller magnitude upon controlling for CBSA fixed effects.

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3. CAPM Betas

This section briefly reviews the theory behind the capital asset pricing model (CAPM)

and then estimates and presents summary measures of the betas for different locations as

described earlier in the paper. As emphasized earlier, beta is our primary measure of investor

exposure to risk.

3.1 Theory

Under the CAPM assumptions (Fama and French, 2004; Bodie, Kane, and Mohanty,

2009) all investments should offer the same reward-to-risk ratio in equilibrium:

𝐸(𝑟𝑖)−𝑟𝑓

𝐶𝑜𝑣(𝑟𝑖,𝑟𝑀)=

𝐸(𝑟𝑀)−𝑟𝑓

𝜎𝑀2 (2)

where 𝐸(𝑟𝑖) − 𝑟𝑓 is the expected return of asset i in excess of the risk-free asset return, 𝑟𝑓 , and

𝜎𝑀2 is the variance of the market portfolio – the measure of market risk. The standard CAPM

equation is obtained by rearranging the equilibrium condition:

𝐸(𝑟𝑖) − 𝑟𝑓 = 𝛽 ∗ [𝐸(𝑟𝑀) − 𝑟𝑓] (3)

Equation (3) states that the excess return or risk premium for asset i is the product of the

risk premium for the market and beta, where 𝛽 =𝐶𝑜𝑣(𝑟𝑖,𝑟𝑀)

𝜎𝑀2 measures the extent to which the

return on the asset and the return on the market move together. A beta equal to 1 indicates that

the respective returns for asset i and the market move together, and for that reason, asset i is risk

neutral relative to the market portfolio. A beta greater than 1 signals that asset i exposes investors

to greater risk relative to the market portfolio, where beta equal to 1.2, for example, indicates that

the asset’s return is 20% more volatile than that of the market.

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It is important to recognize that beta captures how the asset’s returns fluctuate in response

to systematic risk that cannot be diversified away. This includes broad market shocks as with

changes in interest rates, demand and other macroeconomic conditions, the effect of which

differs across housing markets with the local supply elasticity. Beta does not reflect idiosyncratic

risk that is specific to investment in a local housing market, as with discovery of contaminated

soil from past but forgotten development, or some other unanticipated catastrophic event.

The market in CAPM applications typically refers to a portfolio of all possible

investments. For that reason, CAPM applications often use a composite index to represent the

market, as with the S&P 500, which is based on many different financial assets. In this context,

previous work has found that the correlation between real estate returns and broad financial

market indexes is close to zero (e.g. Geltner, 1989). One reason for this may be that much of the

variation in homes prices is local, while variation in stock prices is not (Goetzmann, 1993). For

these reasons, Case, Cotter, and Gabriel (2009) develop what they refer to as a Housing-Capital

Asset Pricing Model or H-CAPM in which local housing market risk is compared to returns from

a broader real estate market but not to returns from a portfolio of financial assets. Specified in

this fashion, the H-CAPM measures beta for a local housing market relative to volatility of real

estate returns for the broader real estate market to which the local area belongs. We adopt the

same approach in this paper.

3.2 Estimating CAPM betas across and within CBSAs

In this section we apply the CAPM framework outlined above to estimate betas for

individual CBSAs and zip codes. The estimating equation is specified in equation 4 and is

standard in the CAPM literature.

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𝑅𝑖,𝑡 = 𝛽𝑅𝑀,𝑡 + 휀𝑖,𝑡 (4)

When we estimate (4) at the CBSA level, there is one beta for each CBSA. The market here is

defined to be the national single family real estate market and market value is measured as the

average of CBSA level home values for CBSAs within our sample. When we estimate (4) at the

zip code level, there is one beta for each zip code in our sample and the CBSA to which a zip

code belongs is specified as the relevant market.

Figure 4 plots the beta distributions for both levels of geography. The mean beta across

CBSAs is 0.963 with a standard deviation 0.781. The distribution is also centered close to one

indicating that the modal CBSA return follows the market quite closely. The distribution also

displays an elongated right tail, with several CBSAs exposing housing market investors to

considerably more systematic risk relative to the national market. Table 6 lists the CBSAs with

the highest and lowest CAPM betas. CBSAs with especially low betas are often in smaller, less

sought after metropolitan areas that are not growing rapidly, an example of which is Syracuse,

NY. CBSAs with especially high betas mostly include high-amenity locations that have been

growing sharply in recent years, as with Phoenix, San Francisco, and Tampa.

In Figure 4, we also plot the distribution of betas for 17,837 zip codes. These are drawn

from all CBSAs in our sample with population greater than 100,000 in year 2000. Once again the

mean beta is close to one, equalling 0.990 in this instance. The standard deviation of the betas is

0.491. The distribution of zip code level betas are more tightly centered around 1 than the CBSA

levels betas. However, the zip code level beta distribution also has long, thin tails reflecting that

there are a handful of zip code level betas with extreme values.

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4. Supply elasticities, volatility and spatial variation in risk premia

This section examines the relationship between the betas described above and differences

in home price volatility and housing supply restrictions across locations. As before, we first

consider differences across CBSAs and then focus on variation at the zipcode level within a

given urban area.

4.1 Variation across CBSAs

We begin by considering the relationship between a CBSAs local housing market risk

premium and the volatility of its price index. Treating each CBSA as a separate observation,

Panel A of Table 7 reports regressions of the CBSA beta on the log variance of its house price

index over the sample period.7 As anticipated, there is a positive and significant relationship

between volatility and asset risk: a one percent increase in variance is associated with a 0.32

increase in beta. This confirms that at the CBSA level, increased volatility contributes to greater

market allowance for risk.

Next, we regress βi on a proxy for local housing supply restrictions and other CBSA level

attributes that capture potential for demand shocks. The regression equation is specified as:

𝛽𝑖 = 𝛼 + 𝜇 𝑊𝐿𝑈𝑅𝐼𝑖 + 𝛿𝑋𝑖 + 휀𝑖 , 𝑖 = 1, … , 𝑛 (5)

where n is the number of CBSAs.

The term WLURI in expression (4) denotes the Wharton Land Use Regulatory Index and

is used to proxy for local housing supply restrictions. The index has mean zero with standard

7 The variance of price levels is not mechanically related to beta. Beta is equal to the covariance between the growth

rate of the asset and market normalized by the variance in the market growth rate. Both the asset and market growth

rates are measured in excess of the risk free asset.

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deviation one. It is also designed so that urban areas in the lowest quartile of the index are the

least supply constrained while those in the top quartile are the most supply constrained.

The term X in expression (5) includes proxies for potential demand shocks. This includes

log population in 2000, log median income in 2000, and the Chen and Rosenthal (2008) Quality

of Life Index that measures the amount of real wage workers forgo to live in the CBSA. As such,

the index is an indicator of the amenity appeal of the urban area, as in Rosen-Roback.8 Two

additional controls capture whether the urban area grew or lost population between 1990 and

2010, Growing and Shrinking. These are measured as the absolute value of the percent increase

(if growing) or decrease (if shrinking) in CBSA population between 1990 and 2010.

Table 9 presents estimates of equation (5).9 In column 1 beta is regressed just on the

Wharton Land Use Regulatory Index. The coefficient on WLURI is 0.26. This indicates that a

one standard deviation increase in the index corresponds to a 26% increase in local housing

market risk relative to the national market. In Columns 2 through 4 we progressively add in

additional controls for log population and median income, Growing and Shrinking, and the

Quality of Life Index. The coefficient on WLURI remains positive and significant in all

specifications except the last. With the addition of the quality of life index the coefficient on

WLURI loses significance but remains positive.

Table 10 provides further evidence that supply restrictions are associated with higher

values for beta. We group CBSAs into four clusters for the 1st through 4th quartile of the WLURI,

or from the least to the most restricted housing supply. The CAPM model is then estimated

8 The index measures nominal wage adjusted for the local cost of living and is scaled to reflect how many thousand

dollars (in year 2000$) a worker gives up to locate in a given urban area relative to an “average” urban area. The

QOL index from Chen and Rosenthal was based on metropolitan statistical areas (MSAs). We were able to match

that geography to the CBSA data in this study for 261 CBSAs. Matches were formed using the name of the primary

city for each defined area, MSA and CBSA. 9 Table 8 presents summary measures of the variables in expression (4).

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separately for each group of CBSAs, constraining the beta and the constant term to be alike

across CBSAs within a given group. The market is the national real estate market as before when

we estimated individual CAPM models for each CBSA.

In Table 10, there is a monotonic relationship between level of supply restrictiveness and

beta: moving from left to right in the table, more heavily regulated, supply constrained CBSAs

expose investors to a greater level of systematic risk. The magnitude of the pattern is also

noteworthy. The least restricted quartile experiences return volatility approximately 25% less

than the national market, while the most restricted quartile experiences volatility 45% more than

the national market, a 70 percentage point difference.

Summarizing, at the CBSA level, local housing market risk premia increase with the

volatility of local house prices, confirming that volatility exposes housing market investors to

increased risk. Evidence above also confirms that risk premia increase with housing supply

restrictions and demand shocks that drive volatility in local markets.

4.2 Variation within CBSAs

As in the previous section, we first examine the relationship between asset risk and

supply restrictiveness within CBSAs by pooling observations across all zip codes. We also

compare betas estimated for city centers and suburbs as well as high- and low-density areas. In

all exercises we find that locations with greater supply restrictiveness are associated with higher

asset risk. Finally, we look at the relationship between zip code level price volatility and asset

risk finding a positive and significant relationship. However, the magnitude of this relationship is

smaller than it is at the CBSA level.

18

Figure 5A shows the results of a local constant regression of zip code level betas on

distance to the CBD. There is a distinct negative relationship between distance and risk. Zip

codes at the CBD experience 15% more systematic risk than those ten miles out. Running the

analogous regression with log employment density on the right-hand side we observe a strong

positive relationship between employment density and beta (Figure 5B). As log employment

density increases betas increases, rising sharply from 0.95 at natural log 4 (around 60 workers

per square mile) to 1.1 at natural log 8 (around 3000 workers per square mile) where the

relationship flattens.

Running the linear versions of these regressions with CBSA fixed effects confirms that

the patterns are not driven by differences across CBSAs (Table 11). Additionally, coefficients

from the linear specifications are of the same order of magnitude as the results from the

nonparametric model. Moving ten miles from the city center decreases beta by 0.058 (columns 1

and 2 of Table 11). A one percent increase in employment density increases beta by 0.013

(columns 3 and 4). The specifications in columns 5 and 6 include both distance to the CBD and

log employment density. The coefficient on distance to the city center remains significant and

consistent in terms of sign and magnitude to the previous results. However, the coefficient on

log employment density decreases from 0.013 to 0.0038, although it remains significant. This

likely reflects the high correlation between distance to the CBD and employment density.

Next, we stratify zip codes by two indicators of supply elasticity and compare the betas

estimated from pooling observations within stratifications. We classify zip codes into city centers

and suburbs as well as into high- and low-density locations. City centers are defined to be within

1 mile of a CBD while suburbs are defined to be 4-8 miles from a CBD. We are able to define

city centers and suburbs for 134 CBSAs. In part, this number is limited because we only work

19

with CBSAs for which Holian and Kahn (2015) specify the coordinates associated with the

center of the CBD. We also define high-density zip codes as those with employment density of at

least 3,500 workers per square mile (about the 95th percentile density for our sample) and low-

density zip codes as those with less than 1,400 workers per square mile (about the mean density

of our sample). Due to the specified cutoff for employment density, not all CBSAs have a high-

density location. In this instance, we are able to identify both high- and low-density locations in

243 CBSAs.

Table 12 reports the mean beta for each grouping noted above (central city versus suburb

and high versus low density). For both comparisons, average beta is roughly 7% higher in the

more supply constrained areas (central city and high density). Also of note, city center betas are

larger than their suburban counterparts in 61.7% of CBSAs, while high density betas are larger

than their low density counterparts in 63.9% of CBSAs.

Figure 6 provides a more complete picture by plotting the distribution of cross-CBSA

betas for each of the groupings above. A striking pattern is that supply elastic locations – suburbs

and low density – have very tight beta distributions centered close to one. In contrast, the

distribution of betas for supply constrained areas is right shifted and exhibits much greater

dispersion. As a general characterization, while relative risk exposure is elevated on average in

supply constrained areas within CBSAs, the extent to which this occurs differs considerably

across urban areas.

Finally, we examine the relationship between price volatility and asset risk. As in the

previous section price volatility is measured as log variance in home prices. Regressing zip code

level betas on price volatility we find that a one percent increase in volatility is associated with a

0.054 increase in beta (Table 7, panel B). While the magnitude of this coefficient is smaller than

20

the CBSA level regression, it is still highly significant. Running this regression CBSA-by-

CBSA, 75% of significant coefficients are positive.

5. Home price appreciation and CAPM betas

We now return to the central question of the paper – to what extent is spatial variation in

home price appreciation over our sample period (1996-2019) driven by differences in risk across

locations? As above, we consider this question both across and within CBSAs.

5.1 Across CBSAs

This section examines the degree to which CBSA-level measures of beta help to explain

cross-CBSA differences in the rate at which home prices appreciate. We begin by measuring

year-over-year home price appreciation for each CBSA using monthly frequency data. For each

CBSA, year-over-year appreciation is then averaged over the entire sample horizon, 1996-2019,

and scaled by 100. Our analysis is then based on a cross section with one observation from each

location. Across CBSAs, year-over-year home price appreciation averages 3.1 percent.

Table 13 presents a series of regressions for which the dependent variable is the CBSA

average annual rate of appreciation as just described. Column 1 controls for the beta associated

with that CBSA. Column 2 adds in controls for year 2000 CBSA population and median income.

Column 3 adds additional controls for whether the CBSA is growing or shrinking (Growing,

Shrinking). Column 4 adds in a dummy measure for whether Gyourko et al (2013) code the

CBSA as a superstar city. Column 5 adds a final control, the quality of life index for the urban

area as measured by Chen and Rosenthal (2008).

21

The dominant and most notable result in Table 13 is that the coefficient on beta is always

positive, quite significant, and very robust to inclusion of other controls in the model. In the most

robust specification in column 5, the coefficient on beta is roughly 0.7 which suggests that a 0.1

increase in beta is associated with a 0.07 percentage point increase in a CBSAs average annual

rate of home price appreciation over the sample horizon.

5.2 Within CBSAs

Tables 14 and 15 mirror the analysis above but in this instance we focus on within CBSA

patterns. In both tables, our dependent variable is the same as above but measured at the zipcode

level. Table 14 controls for distance to the center of the CBD in addition to the zipcode level

beta. Table 15 controls for log employment density at the zipcode level instead of distance. In

both tables, Panel A reports estimates having pooled CBSAs together and including CBSA fixed

effects. Panels B-G report analogous estimates for groupings of CBSAs by size of urban

population and again controlling always for CBSA fixed effects10.

Several core patterns are evident in these tables. First, the qualitative patterns are largely

identical for the two tables regardless of whether we proxy for housing supply restrictions using

distance to the CBD or local employment density. Second, beta is, with one exception, a positive

and significant predictor of local house price appreciation regardless of the size of the urban area.

The only case in which this does not hold is for CBSAs with population 250,000 to 500,000 for

which the associated estimates are not very precise.

10 We also ran each of these with a partial linear specification, controlling for beta parametrically and allowing the

relationship between average growth and distance to the center of the CBD (log employment density) to vary

nonparametrically. In all cases the nonparametric relationship was roughly linear and thus we have only included the

linear results.

22

Focusing on the spatial patterns, among CBSAs up to 500,000 residents we see little

evidence of within-CBSA variation in home price appreciation both based on distance to the

CBD and local employment density. This is the case regardless of whether beta is included in the

models or omitted. A different pattern is evident among larger CBSAs. For CBSAs with

population over 500,000, omitting beta (column 1), home prices appreciate faster in zipcodes

closer to the CBD and in denser locations. Adding controls for beta (as in column 3), the

coefficient on distance to the CBD (in Table 13) and density (in Table 15) is notably reduced in

magnitude while the coefficient on beta remains highly significant. This pattern indicates that

controlling for risk-return tradeoffs – as summarized by beta – appears to explain an important

portion of systematic spatial variation in home price appreciation observed within cities.

6 Conclusion

This paper presents a new stylized fact and offers an explanation that links to other

previously identified patterns in the literature. We show that housing appreciates faster in central

city markets than in suburban markets and especially so in large cities. Previous literature has

documented unusually high long run rates of home price appreciation in select superstar cities as

characterized by Gyourko et al (2013). We also document persistent differences in home price

appreciation rates across cities.

Our analysis relies on Zillow data which we draw upon both at CBSA and zip code levels

of geography. Using these data, we show that within- and cross-CBSA spatial variation in the

volatility of home prices in a given location increases with indicators of supply restrictions,

consistent with the view that housing supply constraints increase investor exposure to risk.

Building on that pattern, we also demonstrate that home price appreciation rates also increase

with supply constraints and that elevated rates of home price appreciation are partly explained by

23

the local market’s CAPM beta that measures systematic risk relative to the broader geographic

market to which the area belongs.

Our findings support the idea that risk-return tradeoffs help to explain both cross-CBSA

and within-CBSA variation in home price appreciation rates, and that these results are robust to

controls for other possible mechanisms. Differences in home price appreciation rates, even

within a given city, therefore, do not necessarily present opportunities for arbitrage.

24

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Gyourko, J., C. Mayer, and T. Sinai, (2013). “Superstar cities”. American Economic Journal:

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26

Tables and Figures

Figure 1

Panel A: Distance to the CBD on percent year-over-year ZHVI growth

Observations are limited to zip codes within 30 miles of a CBD. Estimates are based on a local polynomial

regression of degree 0 using the epanechnikov kernel and bandwidth of 1.11 as determined by the Rule of Thumb

bandwidth selection process.

Panel B: Log employment density on percent year-over-year ZHVI growth

Observations are limited to zip code with log employment density above 0 and are truncated at the top 1% in terms

of employment density. Estimates are based on a kernel regression of degree 0 using the epanechnikov kernel and

bandwidth of 0.27 as determined by Rule of Thumb bandwidth selection process.

27

Figure 2: Normalized single-family home value index growth for CBSAs at the top, middle, and bottom of the

growth distribution

ZHVI is normalized such that April 1996 equals 100.

28

Figure 4: Distribution of CAPM Betas at the CBSA and Zip code Levels

Estimates are based on a kernel density using the epanechnikov kernel and bandwidth determined by the Silverman

(1986) optimal bandwidth. For visual clarity the zip code level distribution is truncated at -6 and 6, this excludes 13

zip codes, less than one percent of the sample.

29

Figure 5: Local Constant Regression of Zip Code Betas on

Within-CBSA Indicators of Supply Elasticity

Panel A: Relationship between Beta and Distance to the CBD

Observations are limited to zip codes within 30 miles of a CBD. Estimates are based on a local polynomial

regression of degree 0 using the epanechnikov kernel and the Rule of Thumb bandwidth (2.7).

Panel B: Relationship between Beta and Log Employment Density

Observations exclude the top 1% in terms of employment density and zip codes with log employment density less

than 0. Estimates are based on a local polynomial regression of degree 0 using the epanechnikov kernel and the

Rule of Thumb bandwidth (0.7).

30

Figure 6: Distribution of within-CBSA stratified betas

Panel A: Distributions of city centers and suburbs betas

Estimates are based on a kernel density using the epanechnikov kernel and bandwidth of determined by the

Silverman (1986) optimal bandwidth. For visual clarity, the St. Louis city center beta has been omitted. It’s value is

-7.69.

Panel B: Distribution of high-density and low-density betas

Estimates are based on a kernel density using the epanechnikov kernel and bandwidth of determined by the

Silverman (1986) optimal bandwidth: 0.0692 for the high-density betas and 0.0056 for the low-density betas.

31

Table 1: Single-family home index growth for

bottom and top 20 CBSAs: 1996-2019

Bottom 20 Top 20

Average one-

year growth

Population in

2000

Average

one-year

growth Population in 2000

Youngstown, OH 1.27 602,964 Washington, DC 4.30 4,849,948

Dayton, OH 1.43 805,816 New York, NY 4.36 18,323,002

Cleveland, OH 1.53 2,148,143 North Port, FL 4.43 589,959

Memphis, TN 1.75 1,205,204 Orlando, FL 4.45 1,644,561

Akron, OH 1.84 694,960 Phoenix, AZ 4.57 3,251,876

Jackson, MS 1.93 546,955 Tampa, FL 4.66 2,395,997

Birmingham, AL 1.94 981,525 Portland, OR 4.71 1,927,881

Scranton, PA 1.95 560,625 Boston, MA 4.77 4,391,344

Toledo, OH 1.96 659,188 Honolulu, HI 4.79 876,156

Chicago, IL 1.98 9,098,316 Fresno, CA 4.84 799,407

Greensboro, NC 1.99 643,430 Denver, CO 4.94 2,157,756

El Paso, TX 2.00 682,966 Miami, FL 5.17 5,007,564

Indianapolis, IN 2.05 1,658,462 Seattle, WA 5.35 3,043,878

Columbia, SC 2.08 647,158 Sacramento, CA 5.48 1,796,857

Rochester, NY 2.20 1,062,452 San Diego, CA 5.88 2,813,833

Albuquerque, NM 2.23 729,649 Riverside, CA 6.17 3,254,821

Wichita, KS 2.23 571,166 Los Angeles, CA 6.33 12,365,627

Cincinnati, OH 2.29 2,016,981 Stockton, CA 6.57 563,598

Winston, NC 2.29 569,207 San Francisco, CA 7.22 4,123,740

Baton Rouge, LA 2.38 729,361 San Jose, CA 7.23 1,735,819

CBSAs in this this table are limited to those with population greater than 500,000 in 2000. The name of each CBSA is

limited to its primary city.

32

Table 2: CBSA level ZHVI growth and variance regressed on indicators of housing supply elasticity

Panel A: Average of one-year SF ZHVI growth

(1) (2)

WLURI 0.535*** -

(0.104) -

Siaz supply elasticities - -0.646***

- (0.138)

Constant 3.198*** 4.698***

Observations

298 82

𝑅2 0.079 0.204

Panel B: variance of SF ZHVI

(1) (2)

WLURI 2.92e09*** -

(5.05E08) -

Siaz supply elasticities - -3.89e09***

- (1.19E09)

Constant 2.49e09*** 1.17e10***

Observations 298 82

𝑅2

0.098 0.106

* p<0.1, ** p<0.05, *** p<0.01

Standard errors are in parenthesis.

33

Table 3: Zip Code Level ZHVI Growth Pooling Across CBSAs

Panel A: Miles to the city center

One Year Growth Rate Five Year Growth Rate

(1) (2) (3) (4) (5) (6)

Miles to CBD -0.0183*** -0.0259*** -0.019*** -0.110*** -0.168*** -0.132***

(.000546) (.000558) (.000429) (.00258) (.002548) (.00176)

Fixed Effects

CBSA - 352 352 - 352 352

Month - - 237 - - 237

Observations 2,949,703 2,949,703 2,949,703 2,358,634 2,358,634 2,358,634

𝑅2 0.0004 0.0321 0.4270 0.0008 0.0956 0.5682

Observations are restricted to zip codes within 30 miles of a CBD.

Panel B: Log employment density

One Year Growth Rate Five Year Growth Rate

(1) (2) (3) (4) (5) (6)

Log employment density 0.185*** 0.0811*** 0.0415*** 1.441*** 0.604*** 0.352***

(.00142) (.00170) (.00130) (.00685) (.00793) (.00541)

Fixed Effects

CBSA - 352 352 - 352 352

Month - - 237 - - 237

Observations 3,953,325 3,953,325 3,953,325 3,151,022 3,151,022 3,151,022

𝑅2 0.0043 0.0327 0.435 0.0139 0.0977 0.581

Observations are restricted to zip codes with employment density greater than 0.

* p<0.1, ** p<0.05, *** p<0.01

Standard errors are in parenthesis.

34

Table 4: Zip code level regressions of ZHVI on within-CBSA indicators of supply elasticity by CBSA

Panel A: Miles to the CBD

Population

groups

Total

Significant

Coefficients

Negative

Coefficients

Percent

Negative

Coefficients

75th

Percentile

Coefficient

50th

Percentile

Coefficient

25th

Percentile

Coefficient

100K – 250K 85 44 52 0.051 -0.022 -0.064

250K – 500K 55 24 44 0.058 0.024 -0.042

500K – 1M 35 20 57 -0.023 -0.019 -0.046

1M-2.5M 28 24 86 -0.018 -0.033 -0.046

Over 2.5 M 17 17 100 -0.049 -0.065 -0.084

Total 236 137 58 0.044 -0.026 -0.054

Counts and percentiles are based on significant coefficients only.

Panel B: Log employment density

Population

groups

Total

Significant

Coefficients

Positive

Coefficients

Percent

Significant

Positive

25th

Percentile

Coefficient

50th

Percentile

Coefficient

75th

Percentile

Coefficient

100K – 250K 90 44 49 -0.165 -0.051 0.178

250K – 500K 48 22 46 -0.260 -0.070 0.148

500K – 1M 35 19 54 -0.101 0.090 0.180

1M-2.5M 24 20 83 0.050 0.109 0.180

Over 2.5 M 15 15 100 0.144 0.256 0.401

Total 229 125 55 -0.147 0.072 0.178

Counts and percentiles are based on significant coefficients only.

35

Table 5: Regressions of home price index variance on supply elasticity indicators

Panel A: Distance to the CBD

(1) (2) (3)

Miles to the city center -3.02E08*** -2.99E08*** -2.99E08***

(2.72E06) (2.54E06) (2.54E06)

Fixed Effects

CBSA - 352 352

month - - 237

Observations 3,519,180 3,519,180 3,519,180

𝑅2 0.004 0.189 0.189

Observations are restricted to zip codes within 30 miles of a CBD.

Panel B: Log employment density

(1) (2) (3)

Log employment density 3.09E09*** 1.37E09*** 1.37E09***

(7.92E06) (8.77E06) (8.77E06)

Fixed Effects

CBSA - 352 352

month - - 237

Observations 4,523,805 4,523,805 4,523,805

𝑅2 0.033 0.172 0.172

* p<0.1, ** p<0.5, *** p<0.01

Observations are restricted to zip codes with employment density greater than 1.

36

Table 6: CBSAs with the highest and lowest betas

Bottom 20 Top 20

Beta Population Beta Population

Wichita, KS 0.221 571,166 Seattle, WA 1.649 3,043,878

Tulsa, OK 0.282 859,532 Providence, RI 1.687 1,582,997

Oklahoma City, OK 0.331 1,095,421 Detroit, MI 1.816 4,452,557

Baton Rouge, LA 0.332 729,361 Jacksonville, FL 1.862 1,122,750

Pittsburgh, PA* 0.375 2,431,087 Washington, DC 1.881 4,849,948

McAllen, TX 0.388 569,463 Tucson, AZ 1.991 843,746

Little Rock, AR 0.393 610,518 San Francisco, CA 1.996 4,123,740

Winston, NC 0.397 569,207 San Diego, CA 2.134 2,813,833

Greenville, SC 0.430 725,680 Los Angeles, CA 2.392 12,365,627

El Paso, TX 0.430 682,966 Tampa, FL 2.511 2,395,997

Columbia, SC 0.430 647,158 Phoenix, AZ 2.619 3,251,876

Rochester, NY 0.434 1,062,452 North Port, FL 2.648 589,959

Buffalo, NY* 0.444 1,170,111 Orlando, FL 2.696 1,644,561

Greensboro, NC 0.454 643,430 Miami, FL 2.821 5,007,564

Oxford, NC 0.455 753,197 Sacramento, CA 2.858 1,796,857

Syracuse, NY 0.474 650,154 Bakersfield, CA 3.057 661,645

Harrisburg, PA 0.475 509,074 Fresno, CA 3.091 799,407

Louisville, KY 0.487 1,090,024 Riverside, CA 3.115 3,254,821

Jackson, MS 0.492 546,955 Las Vegas, NV 3.249 1,375,765

Scranton, PA* 0.497 560,625 Stockton, CA 3.282 563,598

CBSAs shown here are limited to those with population greater than 500 thousand in 2000. Stars indicate CBSAs

with population loss between 1990 and 2010. The North Port, FL CBSA contains Sarasota.

37

Table 7: Regression of CAPM-Betas on local house price volatility

Panel A: CBSA level beta observations

(1) (2)

Log variance in location i house prices 0.324*** -

(.0199) -

Variance of location i house prices - 4.01E-11***

- (6.01E-12)

Constant -5.567*** 0.880***

(0.403) (0.0428)

Observations 362 362

𝑅2 0.424 0.111

Panel B: Zip Code level beta observations

(1) (2)

Log variance in location i house prices 0.0539*** -

(.00316) -

Variance of location i house prices - -1.48E-13

- (1.02E-13)

CBSA Fixed Effects 364 364

Observations 17,837 17,837

𝑅2 0.0262 0.0101

* p<0.1, ** p<0.05, *** p<0.01

38

Table 8: Summary Statistics for CBSA Characteristics

N. Obs. Mean Std. Dev. Min Max

Log population in 2000 362 12.63 1.02 11.52 16.72

Percent change in population: 1990 to 2010 362 28.61 26.51 -12.68 163.17

Log median income 2000 362 9.88 0.17 9.20 10.55

Wharton Land Use Index 298 -0.028 0.737 -1.936 2.982

Chen and Rosenthal (2008) Quality of Life Index 261 0.00 1.97 -4.11 8.90

Note: We divided the Chen and Rosenthal Quality-of-Life Index by 1000 so that it was the same order of

magnitude as the other coefficients.

39

Table 9: Regression of CAPM-Betas on the WLURI

(1) (2) (3) (4)

Wharton Land Use Regulation Index 0.260*** 0.218*** 0.182*** 0.108

(0.0639) (0.0657) (0.0628) (0.0828)

Log Population 2000 0.112** 0.0906** 0.148***

(0.0458) (0.0436) (0.0548)

GROWING 0.00848*** 0.00580*

(0.00183) (0.00233)

SHRINKING -0.0618** -0.0795**

(0.0298) (0.0400)

Chen and Rosenthal Quality of Life Index 0.0783**

(0.0315)

Constant 1.043 -0.384 -0.338 -0.997

Observations 298 298 298 222

𝑅2 0.0497 0.0654 0.157 0.152

* p<0.1, ** p<0.5, *** p<0.01

40

Table 10: CBSA Beta Estimates Grouping Areas by Housing Supply Restrictions

WLURI 1st

Quartile CBSAs

WLURI 2nd

Quartile CBSAs

WLURI 3rd

Quartile CBSAs

WLURI 4th

Quartile CBSAs

Beta 0.749*** 0.909*** 0.981*** 1.45***

(0.009) (0.008) (0.007) (0.010)

Observations 12,200 18,457 23,509 20,418

𝑅2 0.348 0.395 0.446 0.494

* p<0.1, ** p<0.05, *** p<0.01

The unevenness in the size of the quartile groups in due to our restriction of the sample to CBSAs

with population greater than 100 thousand.

41

Table 11: Zip code level Betas regressed on within-CBSA measures of supply elasticity

(1) (2) (3) (4) (5) (6)

Miles to the CBD -0.0052*** -0.0058*** - - -0.0040*** -0.0046***

(0.0006) (0.0006) - - (0.0007) (0.0008)

Log employment density - - 0.0127*** 0.0189*** 0.0135*** 0.0128***

- - (0.0014) (0.00185) (0.0022) (0.0033)

CBSA fixed effects - 286 - 286 - 286

Observations 6,114 6,114 9,853 9,853 5,883 5,883

𝑅2 0.0121 0.0607 0.0083 0.0661 0.0216 0.0719

* p<0.1, ** p<0.05, *** p<0.01

Observations have been Winsorized at the top and bottom 1% in terms of log employment density. We have also

limited the observations to be within 30 miles of a CBD.

42

Table 12: Average Betas for Supply Constrained and Supply Elastic Areas Within CBSAs

Grouping Within-CBSA Location

into City Center and Suburban

Grouping Within-CBSA Location

into High and Low Density

City Center Suburb High Density Low Density

Mean across CBSAs 1.115 1.054 1.075 0.989

Std Deviation 0.408 0.189 0.532 0.027

Observations (No. CBSAs) 128 128 241 241

Note: Two betas are estimated for each CBSA, one for supply constrained and one for supply elastic

areas as defined in the two pairs of columns. Betas for each classification are then averaged across

CBSAs in the sample. St. Louis city center Beta was an extreme outlier and omitted for that reason.

43

Table 13: CBSA-Level Percent Year-Over-Year Home Price Appreciationa

CBSA Attributes (1) (2) (3) (4) (5)

Beta 0.979*** 0.938*** 0.871*** 0.835*** 0.723***

(.0732) (.0757) (0.0810) (0.0969) (0.0897)

Log CBSA population 2000 0.0137 0.00706 0.0637 0.239**

(.0678) (0.0676) (0.0875) (0.0855)

Median CBSA income 2000 0.0000362* 0.0000361* 0.0000309 0.000024

(.0000203) (0.0000203) (0.0000287) (0.0000266)

GROWING 0.00393 0.00484 -0.000335

(0.00254) (0.00322) (0.00305)

SHRINKING -0.0672 -0.0257 -0.044

(0.0419) (0.0489) (0.0502)

Superstar status 0.721** 0.153

(0.321) (0.323)

Quality of Life Index 0.298***

(0.0393)

Observations 362 362 362 243 226

𝑅2 0.330 0.335 0.343 0.361 0.502 a The dependent variable has a mean of 3.1 which indicates that on average, CBSA home prices increase 3.1

percent per year over our sample horizon. Measured in this fashion, if the coefficient on beta above was equal to

1.0, a 0.1 increase in beta would increase house average CBSA home price appreciation 0.1 percentage points.

* p<0.1, ** p<0.05, *** p<0.01

Standard errors are in parentheses.

44

Table 14: Distance from the CBD and Zip Code Level House Price Growth on a Year-Over-Year Basis

Panel A: All CBSAs

(1) (2) (3)

Miles to CBD -0.0258*** - -0.0199***

(.00168) - (.00161)

Beta - 0.953*** 0.920***

- (.0260) (.0260)

CBSA FE 364 364 364

Observations 12,348 12,348 12,348

R2 0.487 0.529 0.535

Panel B: Population between 100K & 250K Panel C: Population between 250K & 500K

(1) (2) (3) (1) (2) (3)

Miles to CBD -0.00222 - -0.00628* 0.00432 - 0.0043

(.00375) - (.00351) (.00420) - (.00420)

Beta - 1.246*** 1.253*** - -0.0471 -0.0467

- (.0666) (.0667) - (.0557) (.0557)

CBSA FE 194 194 194 79 79 79

Observations 2,628 2,628 2,628 2,243 2,243 2,243

𝑅2 0.480 0.544 0.545 0.433 0.433 0.433

Panel D: Population between 500K & 1M Panel E: Population between 1M & 2.5M

(1) (2) (3) (1) (2) (3)

Miles to CBD -0.00775** - -0.00176 -0.0368*** - -0.0134***

(.00378) - (.00361) (.00351) - (.00334)

Beta - 0.651*** 0.647*** - 1.364*** 1.295***

- (.0554) (.0560) - (.0539) (.0565)

CBSA FE 45 45 45 29 29 29

Observations 2,004 2,004 2,004 2,427 2,427 2,427

𝑅2 0.374 0.414 0.414 0.493 0.582 0.585

Panel F: Population between 2.5M & 5M Panel G: Population over 5M

(1) (2) (3) (1) (2) (3)

Miles to CBD -0.0700*** - -0.0611*** -0.0569*** - -0.0470***

(.00458) - (.00382) (.00515) - (.00457)

Beta - 2.0556*** 1.961*** - 1.534*** 1.447***

- (.0754) (.0706) - (.0776) (.0751)

CBSA FE 13 13 13 5 5 5

Observations 1,734 1,734 1,734 1,312 1,312 1,312

𝑅2 0.479 0.587 0.641 0.289 0.401 0.446

* p<0.1, ** p<0.05, *** p<0.01.

Standard errors are in parentheses. Zip codes are limited to be within 30 miles of a CBD.

45

Table 15: Log Employment Density and Zip Code Level House Price Growth on a Year-Over-Year Basis

Panel A: All CBSAs

(1) (2) (3)

Log emp density 0.0855*** - 0.0574***

(.00559) - (.00538)

Beta - 1.001*** 0.977***

- (.0237) (.0248)

CBSA FE 364 364 364

Observations 15665 16217 15665

R2 0.505 0.527 0.551

Panel B: Population between 100K & 250K Panel C: Population between 250K & 500K

(1) (2) (3) (1) (2) (3)

Log emp density -0.0267** - -0.0175 0.00226 0.0012

(.0132) - (.0129) (.0141) (.0141)

Beta - 0.788*** 0.685*** 0.108* 0.116**

- (.0567) (.0576) (.0573) (.0588)

CBSA FE 194 194 194 79 79 79

Observations 3288 3508 3288 2338 2444 2338

R2 0.498 0.477 0.520 0.470 0.430 0.471

Panel D: Population between 500K & 1M Panel E: Population between 1M & 2.5M

(1) (2) (3) (1) (2) (3)

Log emp density 0.035*** - 0.00838 0.0988*** - 0.0424***

(.0129) - (.0128) (.0113) - (.0107)

Beta - 0.678*** 0.673*** - 1.27*** 1.14***

- (.0510) (.0607) - (.0491) (.0513)

CBSA FE 45 45 45 29 29 29

Observations 2356 2443 2356 2972 3069 2972

R2 0.364 0.392 0.397 0.460 0.534 0.537

Panel F: Population between 2.5M & 5M Panel G: Population over 5M

(1) (2) (3) (1) (2) (3)

Log emp density 0.201*** - 0.114*** 0.250*** - 0.187***

(.0146) - (.0125) (.0171) - (.0158)

Beta - 2.28*** 2.11*** - 1.66*** 1.47***

- (.0642) (.0655) - (.0667) (.0663)

CBSA FE 13 13 13 5 5 5

Observations 2427 2459 2427 2284 2294 2284

R2 0.427 0.585 0.599 0.519 0.581 0.605

* p<0.1, ** p<0.05, *** p<0.01

Standard errors are in parentheses. Zip codes are Winzorized at 1% employment density.