Search Frictions and Idiosyncratic Price Dispersionin the US Housing Market∗

Nadia Kotova†and Anthony Lee Zhang‡

February 2020

Abstract

This paper studies idiosyncratic price dispersion in the US housing market. Wedevelop a new strategy to measure dispersion at the level of individual house sales.We show that idiosyncratic price dispersion is countercyclical and seasonal, and isassociated with various measures of market tightness. We construct a search-and-bargaining model of the housing market, and show that the relationship between sellertime-on-market and price dispersion can be used to separately identify dispersiondriven by seller and buyer preferences. We calibrate the model to data to quantify themagnitude of market frictions and their effects on the welfare of market participants.

Keywords: Housing, search, liquidity, price dispersion

∗We appreciate comments from Mohammad Akbarpour, Sam Antill, Alina Arefeva, Dmitry Arkhangel-sky, Adrien Auclert, Lanier Benkard, Tim Bresnahan, Scarlet Chen, Daniel Chen, John Cochrane, Cody Cook,Rebecca Diamond, Evgeni Drynkin, Darrell Duffie, Liran Einav, Matthew Gentzkow, Guido Imbens, ChadJones, Eddie Lazear, Paul Milgrom, Jonas Mueller-Gastell, Sean Myers, Michael Ostrovsky, Amine Ouazad,Monika Piazzesi, Peter Reiss, Al Roth, Martin Schneider, Jesse Shapiro, Erling Skancke, Paulo Somaini,Amir Sufi, Rose Tan, Zach Taylor, Chris Tonetti, Jia Wan, Joseph Vavra, Robert Wilson, Ali Yurukoglu, andFudong Zhang, as well as seminar participants at Stanford, UChicago, the Wisconsin Business School, theTrans-Atlantic Doctoral Conference, the Urban Economics Association 2019 meeting, the Young EconomistsSymposium, the XX April International Academic Conference, Tsinghua PBCSF, and the 2019 West CoastSearch & Matching Workshop.†Stanford Graduate School of Business, 655 Knight Way, Stanford, CA 94305; nkotova@stanford.edu.‡University of Chicago Booth School of Business, 5807 S Woodlawn Ave, Chicago, IL 60637; an-

thony.zhang@chicagobooth.edu.

1 Introduction

This paper studies idiosyncratic price dispersion in the US residential housing market.Residential real estate is a very illiquid asset class: houses take many months to sell,and realtor commissions for each sale are around 6% of total house prices. Illiquidityis also reflected in price dispersion: similar houses may sell for very different pricesat similar points in time. Early work by Case and Shiller (1988) shows that the pricevolatility of individual houses is much higher than the volatility of city-wide averageprices. However, we do not yet have a strong understanding of how idiosyncratic houseprice dispersion behaves in the cross-section and over time, and whether it is associatedwith other measures of housing market liquidity.

In this paper, we develop a new strategy to measure price dispersion at the level ofindividual house sales. We regress log sale prices of houses on zipcode-month fixedeffects, house fixed effects, and a smooth function of house characteristics and time. Thisapproach combines repeat-sales and hedonic specifications for house prices: it absorbstime-invariant house quality into the house fixed effects, and also allows observable housecharacteristics to affect the evolution of house prices over time. We measure dispersionusing the squared residuals from this regression, which we can aggregate flexibly acrosslocations and over time. Quantitatively, the standard deviation of these residuals rangesfrom around 11.2% to 22.6% of house prices across zipcodes. Since homebuyers incur thisidiosyncratic shock twice, once upon purchase and once upon sale, our estimates implythat home buyers are exposed to additional idiosyncratic risk of around 15.8% to 32.0% ofhouse prices, relative to the volatility of zipcode house price indices.

We demonstrate a number of new stylized facts about idiosyncratic price dispersion.First, it is countercyclical: it decreases during the housing boom, increases during thebust, and decreases again during the subsequent recovery. Second, it is seasonal: itsystematically decreases during the summer hot season, when volume and prices are highand time-on-market is low. Third, it is negatively correlated with prices and sales volume,and is positively correlated with time-on-market, vacancy rates, and other measures ofinverse market tightness, over time and in the cross-section.

We build a search-and-bargaining model of the housing market to rationalize thesefindings. In our model, price dispersion arises from heterogeneity in seller and buyerpreferences: sellers have different holding utilities for keeping their houses on the marketper unit time, and buyers receive an independent match quality shock every time they

1

match with a house. Buyers’ match quality shocks are drawn from an exponential distri-bution, and sellers’ holding utilities are drawn from an arbitrary continuous distribution.Both dimensions of heterogeneity affect prices in equilibrium. Buyers with higher matchquality draws pay higher prices, and sellers with lower holding utilities sell for lowerprices. In an appendix, we show that the model can also accommodate an additionaldimension of heterogeneity: persistent differences in buyer urgency to buy.

Our primary theoretical result is that, in our model, price variance can be analyticallydecomposed into buyer- and seller-induced components. Moreover, the two componentsbehave differently as market tightness varies. In tight markets, the seller component ofprice variance decreases, while the buyer component remains constant. This is becausemarket tightness influences the extent to which dispersion in sellers’ flow holding utilitiestranslates into dispersion in sellers’ stock values. If markets are slack, the relative cost ofbeing a seller with low flow utility for keeping her house on the market is higher becauseall sellers expect to stay on the market for longer. Thus, low-type sellers are willing totake larger price cuts to sell more quickly, which results in higher price dispersion inequilibrium. In contrast, the buyer-induced component of variance is tied to the varianceof match quality conditional on sale, and is not affected by market tightness.

The link between market tightness and the seller component of price dispersiondepends on an observable sufficient statistic: sellers’ expected time-on-market. Formally,we show that the derivative of the seller’s value function with respect to sellers’ holdingutility is a function of the discount factor, sellers’ bargaining power, and seller’s time-on-market. This implies that, holding fixed the discount factor, bargaining power, andthe distributions of buyer and seller types, any changes in market conditions which havethe same effect on seller time-on-market have the same effect on price dispersion. Thedata support this prediction: the empirical relationship between time-on-market andprice dispersion has similar magnitudes in time-series, seasonal, and panel regressionspecifications. Together, these results imply that the relationship between time-on-marketand price dispersion is informative about the size of the seller component of pricedispersion, and the underlying distribution of seller holding utilities.

We calibrate the model to match various moments of the US housing market. We findthat, in the average US county in 2013, sellers with 75th percentile holding utilities keeptheir houses on market for 1.4 months, or 60%, longer than sellers with 25th percentileholding utilities. As a result of waiting longer, high utility sellers achieve average saleprices that are $11,341 (6%) higher than low utility sellers. This “liquidity discount”

2

depends on average market tightness: in slack markets with average time-on-market of4.8 months, the liquidity discount increases to $15,731, and in tight markets with averagetime-on-market of 2 months, it decreases to $7,549.

Under our calibrated model, we find that 4.5% of total price dispersion in the data isattributable to seller heterogeneity, and 15.3% to buyer heterogeneity. We attribute theremaining 80.2% to unobserved house-level heterogeneity. Thus, our calibration suggeststhat buyer- and seller-induced dispersion in frictional markets can account for a nontrivial,but modest, share of the total idiosyncratic price variance that we observe in the data.

Since real estate represents a large component of household wealth in the US, under-standing idiosyncratic price dispersion is important from the perspective of householdportfolio choice. Real estate is not necessarily a poor investment choice. Jordà, Schularickand Taylor (2019) argue that the risk-adjusted total rate of return on house price indices ishigher than that of stock market indices in many countries. We replicate this result andfind that a diversified portfolio of US residential real estate outperforms the S&P 500 overour sample period, 2000-2017. However, the average household in the US only owns asingle house, their primary place of residence. Under our estimates, the return variance ofan individual house purchase is around 2.1-3.3 times higher than the variance of US-wideaverage returns. Accounting for transaction costs such as realtor fees, purchasing a singlehouse only slightly outperforms the S&P 500 over this time period.

1.1 Related literature

This paper is most directly related to a number of papers that study idiosyncratic houseprice volatility. Case and Shiller (1988) is one of the early papers to show that prices ofindividual houses are much more volatile than city-wide average prices. Giacoletti (2017)studies the residential real estate market in California and shows that idiosyncratic houseprice risk does not scale with holding period, which suggests that much of idiosyncratichouse price volatility is due to liquidity risk. In addition, using an instrument for zipcode-level shocks to mortgage credit, Giacoletti shows that decreasing local credit availabilityincreases idiosyncratic risk. Sagi (2015) studies commercial real estate and documentsa similar result to Giacoletti (2017) that the idiosyncratic component of commercialreal estate price risk does not scale with holding period. Peng and Thibodeau (2017)calculates price dispersion at the zipcode level using a hedonic regression specificationand documents relationships between idiosyncratic price dispersion and characteristics of

3

zipcodes such as average income. Two other papers which measure idiosyncratic pricedispersion are Anenberg and Bayer (2015) and Landvoigt, Piazzesi and Schneider (2015).We discuss in detail the relationship between our method for measuring idiosyncratichouse price dispersion, and these prior papers in subsection 6.1.

In terms of the modeling approach, our work fits into the literature on applying search-and-bargaining models1 to housing markets2 and to financial markets more generally.3

Relative to this literature, an innovation of our model is that we allow for continuouslydistributed persistent seller values as well as match-specific quality shocks. In an appendix,we show that the model can also accommodate continuously distributed persistent buyervalues, without significantly affecting the results. Thus, our model is rich enough that itcan be calibrated to data realistically, but remains analytically tractable and is relativelytransparent in its mapping from primitives to outcomes. In particular, our model admitsan analytical decomposition of price variance into buyer- and seller-induced components,and we show that these components can be empirically separated since only the sellercomponent of price dispersion depends on market tightness.

A few papers which have closely related models to ours are Albrecht et al. (2007), Al-brecht, Gautier and Vroman (2016), Anenberg and Bayer (2015), and Sagi (2015). Albrechtet al. (2007) study a random search model of the housing market with two seller types.Albrecht, Gautier and Vroman (2016) construct a directed housing search model with twoseller types, which can predict sellers’ asking prices and the sale-to-ask spread. Anenbergand Bayer (2015) allows continuous match quality shocks and a discrete number of sellertypes. Sagi (2015) allows a discrete number of seller types.

The fact that volume, prices, and time-on-market are correlated in housing markets isthe subject of a number of papers; see, for example, Stein (1995), Krainer (2001), Genesoveand Mayer (2001), Leung, Lau and Leong (2002), Clayton, Miller and Peng (2010), Diazand Jerez (2013), and DeFusco, Nathanson and Zwick (2017). Our contribution to thisliterature is to show how idiosyncratic price dispersion co-moves with these variables at

1For earlier applications to labor markets, see, for example, Mortensen and Pissarides (1994), or thesurvey article Rogerson, Shimer and Wright (2005)

2See, for example, Wheaton (1990), Albrecht et al. (2007), Piazzesi and Schneider (2009), Genesoveand Han (2012), Anenberg and Bayer (2015), Ngai and Tenreyro (2014), Head, Lloyd-Ellis and Sun (2014),Piazzesi, Schneider and Stroebel (2015), Sagi (2015), Albrecht, Gautier and Vroman (2016), Guren (2018),Gabrovski and Ortego-Marti (2019a), Gabrovski and Ortego-Marti (2019b), Arefeva (2019), Guren andMcQuade (2019), and Ouazad and Rancière (2019).

3See, for example, Duffie, Gârleanu and Pedersen (2005), Duffie, Gârleanu and Pedersen (2007), Vayanosand Wang (2007), Vayanos and Weill (2008), Weill (2008), Afonso (2011), Duffie, Qiao and Sun (2017),Gavazza (2016), Trejos and Wright (2016), Hugonnier, Lester and Weill (2018).

4

the aggregate level. Two other papers that analyze the relationship between list prices andtime-on-market at the level of individual house sales are Drenik, Herreno and Ottonello(2019), which studies Spanish commercial real estate, and Guren (2018). Drenik, Herrenoand Ottonello (2019) attempts to rationalize the data using adverse selection, withoutsearch frictions, and Guren (2018) emphasizes strategic complementarity in sellers’ price-setting decisions. Our model abstracts away from both of these forces, focusing on markettightness and its effects on price dispersion.

1.2 Outline

The rest of the paper proceeds as follows. Section 2 describes the data and how weconstruct our measure of price dispersion. Section 3 contains our empirical results.Section 4 describes our model and theoretical results. Section 5 describes our calibration.Section 6 discusses implications of our findings, and Section 7 concludes.

2 Data and measurement

2.1 Data sources

We use microdata on single-family house sales and house characteristics from the CoreL-ogic tax and deed database, spanning the time period 2000-2017. Other datasets we use areRealtor.com data for listings and time-on-market at the zipcode level; Zillow Research datafor total listings and time-on-market at the county level, as well as the Zillow Home ValueIndex (ZHVI); and the ACS 1-year and 5-year samples for demographic characteristicsof zipcodes and counties. Further details of the steps we take to clean and merge thedatasets are described in appendix A.

Since we estimate price dispersion using a repeat-sales specification, we filter tozipcodes with a fairly large number of sales; details of how we select zipcodes aredescribed in appendix A.1, and descriptive statistics for zipcodes in our primary estimationsample dataset are shown in table 1. Our primary dataset comprises 36 million housesales within 3,870 zipcodes, over the period 2000-2017. As appendix table 8 shows, ourestimation sample contains 12.0% of all zipcodes, but covers 45.4% of the US population.Our sample is concentrated in relatively large, dense, and high-income zipcodes, but isrepresentative in terms of other demographic characteristics.

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Table 1: Descriptive statistics, zipcode sample

Name Mean SD P10 P90Sales 9314 4829 4969 15653Average monthly sales 46 23 25 76Mean price (x1000 USD) 425.3 742.8 140.3 691.9Mean TOM (Months), 2012-2016 2.29 0.67 1.52 3.16Total zipcodes 3870Total sales (mil) 36.05

Notes. Monthly sales and mean price data are from the CoreLogic Deed dataset, and meantime-on-market data is from Realtor.com.

2.2 Measuring idiosyncratic price dispersion

Let i index properties, as determined by assessor parcel numbers (APNs). Let xi representa vector of time-invariant characteristics of house i; we use the house’s geographic location,square footage, age, and numbers of bedrooms and bathrooms. Let t denote months, zdenote zipcodes, and let pit denote the log price of house i sold in month t in zipcodez. We index individual houses only by i, and we omit the subscript z for notationalconvenience. We assume the following specification for log house prices:

pit = γi + ηzt + fz (xi, t) + εit (1)

In words, (1) says that prices are determined by a time-invariant house fixed effect, γi, azip-month fixed effect, ηzt, a smooth function fz (xi, t) of observable house characteristicsxi and time t, and a mean-0 error term εit.

Specification (1) combines repeat-sales and hedonic specifications. The γi term absorbsall variation in the level of sale prices for a given house, capturing both observed andunobserved features of houses that have a time-invariant effect on the log price of agiven house. The ηzt term absorbs parallel shifts in log prices within a zipcode over time.Thus, absent the fz (xi, t) term, specification (1) behaves like a repeat-sales specification,implying that the conditional expectations of log house prices within a zipcode shouldfollow parallel trends. The fz (xi, t) term relaxes parallel trends by allowing houses withdifferent observable characteristics xi to appreciate at different rates. This is importantbecause markets may be segmented within zipcodes: houses in different geographicallocations may appreciate differentially, or houses that are older or larger may appreciate

6

differentially within zipcodes.4

Effectively, specification (1) absorbs all time-invariant house quality, whether or not itis captured by observables xi, into the house fixed effects γi, then allows characteristics xito smoothly affect the evolution of house prices over time; it is a repeat-sales model forhouse price levels and a hedonic model for price differences over time. The error term,εit, is our measure of idiosyncratic house price variation.

We use (1) as our baseline specification because it is the most flexible model forpredicting house prices that is possible given our data limitations, so it will tend toproduce the smallest estimates of idiosyncratic dispersion. However, our stylized factsare robust to different ways of estimating price dispersion. In appendix B.2, we usea pure repeat-sales specification for log house prices, omitting the fz (xi, t) term, andin appendix B.3, we use a pure hedonic specification, omitting the γi term; our time-series and regression results are qualitatively and quantitatively similar to the baselinespecification under both robustness checks.

2.2.1 Measurement concerns

There are two aspects of house prices which specification (1) cannot capture. First, wecannot account for house-level unobservable characteristics with time-varying effects onprices. Suppose houses have some characteristic yi, which is observed and valued bymarket participants but is not in our data. Any time-invariant effects of yi on log houseprices will be absorbed into our house fixed effects γi, but any time-varying effect of yiwill not be captured by specification (1), so we will incorrectly attribute these effects to theidiosyncratic error term εit. Second, we observe characteristics at a single point in time,so we cannot account for time-varying house characteristics. Thus, we cannot account forthe effects of house renovations or improvements on house prices. Again, these effectsmay have predictable effects on house prices from the perspective of market participants,but we will incorrectly attribute these effects to εit.

4Landvoigt, Piazzesi and Schneider (2015) show that returns for houses at different price points in SanDiego differ substantially, implying that the fz (xi, t) term is potentially important. We do not allow thelevel of log prices to affect returns directly; this is econometrically difficult since log prices are themselvesaffected by idiosyncratic price dispersion εit. However, we can capture the differential appreciation trendsnoted by Landvoigt, Piazzesi and Schneider (2015) as long as most factors that affect average house pricesare included in our vector xi of observable characteristics. xi contains location, square footage, bedrooms,bathrooms, and year built, so we believe xi should capture most first-order features which affect the relativeprices of houses within a zipcode.

7

Both of these issues will tend to bias our estimates of idiosyncratic price dispersionupwards. However, since most of our analysis in section 3 consists of regressing εit againstvarious covariates, measurement error in εit only affects our regression results to theextent that it is correlated with our regression covariates. Moreover, the magnitude of bothissues seems to be fairly small. In appendix B.2, we estimate specification (1) excludingthe fz (xi, t) term. This decreases the estimated standard deviation of εit by approximately0.6% of house prices on average, implying that observable characteristics have relativelysmall time-varying effects on house prices. Assuming xi contains most characteristicsof houses relevant for valuation, unobservable characteristics should also have relativelysmall time-varying effects on prices. Giacoletti (2017) observes data on remodelingexpenditures for houses in California, and performs a similar exercise: comparing figureE.vi to figure 2 in Giacoletti (2017), excluding remodeled properties decreases Giacoletti’sestimated standard deviation of house returns by at most around 2% of house prices,which is also a relatively small fraction of total idiosyncratic house price variation.

2.3 Implementation

When we estimate specification (1), it is computationally infeasible to estimate a fullyinteracted polynomial in all house characteristics for fz (xi, t), so we use an additivefunctional form for:

fz (xi, t) = glatlongz (t, lati, longi) + gsqftz (t, sqfti) + gyrbuiltz (t,yrbuilti)+

gbedroomsz (t,bedroomsi) + gbathroomsz (t,bathroomsi) (2)

The functions glatlongz , gsqftz , and gyrbuiltz are fully-interacted third-order polynomials

in their constituent components, and the functions gbedroomsz and gbathroomsz interactdummies for a given house having 1, 2, 3 or more bedrooms and 1, 2, 3 or more bathroomsrespectively with third-order polynomials in time.

Additivity in specification (2) rules out many interaction effects between characteristics.Older or larger houses can appreciate faster or slower than newer or smaller houses.However, houses which are both large and old are constrained to appreciate at a ratewhich is the sum of the “old house” and “large house” effects on prices. The onlyinteraction term we include is the glatlongz (t, lati, longi) function, which interacts latitudeand longitude. This is important because it is implausible that latitude and longitude

8

have additive effects on prices; effectively, this specification allows house prices to varysmoothly as a function of a house’s geographic location over time.

Given this functional form for fz (xi, t), specification (1) is a standard fixed effectsregression, and we estimate specification (1) using OLS separately for each zipcode in oursample. Once we have estimated specification (1), we estimate squared residuals ε̂2

it foreach house sale as:

ε̂2it =

Nz

Nz −Kz(pit − p̂it)

2 (3)

where Nz is the number of house sales in zipcode z, and Kz is the number of parametersestimated from specification (1). The term Nz

Nz−Kzis a degrees-of-freedom correction, which

causes variance estimates to be unbiased at the zipcode level; this is important to includebecause most houses are sold relatively few times, so the number of parameters Kz isnontrivially large relative to the number of house sales Nz in our dataset.5

Expression (3) thus gives us a measure of idiosyncratic price dispersion, ε̂2it, at the level

of each individual house sale. We can then average the estimated errors ε̂2it at various

levels of aggregation. We estimate the standard deviation of εit at the zipcode level, overthe period 2012-2016, as:

σ̂z =

√ ∑i ε̂

2it

Nz,2012−2016(4)

where (4) sums over all house sales that happened in zipcode z over the time period2012-2016, and Nz,2012−2016 is the number of observations in zipcode z over this period.We filter to 2012-2016 because this corresponds to the zipcode sample we use in ourcross-sectional regressions in subsection 3.4 below; we do not observe time-on-marketfor zipcodes prior to 2012. Figure 1 shows a density plot of σ̂z. The mean of σ̂z is 16.8%of house prices, and the standard deviation is 4.6%. The distribution of σ̂z is slightlyright-skewed: the 10th percentile is 11.2% and the 90th percentile is 22.6%. A homeownerwho purchases and then sells a house incurs the idiosyncratic shock εit twice, once uponpurchase and once upon sale. Multiplying these estimates by a factor

√2, these estimates

imply that a homeowner who buys and sells an individual house is exposed to additionalidiosyncratic risk of around 15.8% to 32.0% of house prices, relative to the volatility of the

5More formally, assuming homoskedasticity within zipcodes, σ2it = σ2

z, the degrees-of-freedom correctionin expression (3) causes the expectation of ε̂2

it to be equal to the true variance, σ2z. We thus apply the

homoskedastic variance adjustment term here, as we are not aware of any computationally tractable way toimplement a degrees-of-freedom correction in the general heteroskedastic case. However, in appendix B.4,we further adjust the estimated residuals ε̂it to account for the number of times a house is sold and theaverage time-between-sales, and show that our results are robust to this adjustment.

9

Figure 1: Distribution of σ̂z

Notes: Distribution of σ̂z across zipcodes, 2012-2016. Each data point is a zipcode.

zipcode house price index. In subsection 6.3 of the discussion, we show how accountingfor idiosyncratic risk affects the attractiveness of housing as an asset class.

In the following section, we will construct analogs of (4) at different levels of aggrega-tion. Since aggregates such as σ̂z can be interpreted as the standard deviation of εit, theidiosyncratic component of log house prices, we will often refer to these aggregates as the“logSD” of house prices in the following sections.

3 Empirical results

In this section, we document new stylized facts about idiosyncratic house price dispersion:we show that price dispersion is countercyclical, seasonal, and correlated with time-on-market and other measures of market tightness in panel and cross-sectional regressions.Details of the construction of the datasets used in this section are discussed in appendicesA.5, A.6 and A.7.

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3.1 Idiosyncratic price dispersion over the business cycle

In figure 2, we show the behavior of indexed total sales, logSD, prices, and time-on-market at the yearly level. Figure 2 shows that idiosyncratic house price dispersion iscountercyclical. Price dispersion is decreasing from 2000-2004, as the housing market isbooming and volume and prices are increasing. From 2004 onwards, idiosyncratic pricedispersion reverses direction and starts increasing, peaking in 2010, in the midst of thehousing bust, as sales and prices are both low due to the housing bust. Price dispersionthen starts decreasing once again as sales volume and average prices recover. Whilewe only observe time-on-market from 2010 onwards, time-on-market is also smoothlydecreasing throughout the 2010-2016 recovery.

Quantitatively, total housing sales in our sample increased 41% from 2000 to 2005,dropped to a trough of 69% of its 2000 value in 2011, and recovered to 4.5% above its 2000value in 2016. Average prices increase 72% from 2000 to 2007, drop to 25% above their2000 value in 2011, and increase to 64% above their 2000 value in 2016. While the peaksand troughs of the logSD series do not align perfectly with the time series for sales orprice series, logSD drops to 4.1% (0.7% of house prices) below its 2000 value near the peakof the boom in 2004, increases to 8.7% (1.4%) above its 2000 value in 2009, and decreasesto 2.2% (0.3%) above its 2000 value in 2016. From 2010 to 2016, time-on-market decreasesby 26%.

We believe that it is a novel contribution of our paper to empirically document thecountercyclicality of idiosyncratic house price dispersion.6 The cyclical changes in pricedispersion are nontrivially large – the logSD line moves by 2% of house prices, fromits trough in 2004 to its peak in 2009, which is approximately 43% of a cross-sectionalstandard deviation in logSD across zipcodes.

3.2 Seasonality

Figure 3 shows the seasonal behavior of prices, total sales, time-on-market, and logSD,aggregated to the level of calendar months over the period 2010-2016. Total sales, time-on-market, and logSD are from our data. For prices, we use the FHFA US-wide monthlyhouse price index; we use the FHFA index rather than the ZHVI because the ZHVI is

6Landvoigt, Piazzesi and Schneider (2015) use a different methodology to demonstrate that idiosyncraticprice dispersion increased in San Diego from 2005-2007, but does not note the decrease in price dispersionfrom 2009 onwards.

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Figure 2: Time-series variation in prices, sales, logSD, and time-on-market

Notes. Indexed total sales, average prices, and logSD, 2000-2016, and time-on-market(TOM), 2010-2016. All variables are indexed so that they are equal to 1 in the first yearthat we observe them.

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Figure 3: Monthly variation in sales, prices, logSD, and time-on-market

Notes. Indexed total sales, average prices, logSD, and time-on-market (TOM) by calendarmonth, over the time period 2010-2016. All variables are indexed by dividing by theirJanuary level.

seasonally adjusted and cannot capture price variation over calendar months. We filter outlow-frequency trends in all four time-series through a procedure we describe in appendixA.5.

Figure 3 shows that all four variables are seasonal. Quantitatively, there are on average82% more house sales in June than in January. Prices are around 3.4% higher in June,time-on-market is around 35% lower, and logSD is around 7.2% (in terms of house prices,1.2%) lower. The seasonality of sales, prices, and time-on-market is known in the literature(Ngai and Tenreyro, 2014), but we believe we are the first to show that idiosyncratichouse price dispersion is also seasonal. The magnitude of the seasonal variation is alsonontrivially large, at around 35% of the seasonal effect on average house prices.

3.3 County-year panel regressions

Table 2 shows county-year panel regressions of logSD on various covariates. We runpanel regressions at the year level instead of the month level to avoid picking up seasonal

13

variation in our regressions. Similarly to subsection 3.1 above, we use Zillow’s single-family residence ZHVIs to measure prices at the county-year level. Data on county-yeartime-on-market also comes from Zillow and is available for the 2010-2016 time period.

Columns 1 and 2 show that logSD is negatively correlated with prices and sales volume,and column 3 shows that price dispersion is positively correlated with time-on-market.Thus, the panel regressions support the findings in figures 2 and 3. Note that the panelvariation is formally distinct from the variation shown in figures 2 and 3: figure 2 focuseson aggregate trends in outcome variables, on average across counties within a givenyear, and figure 3 takes average within months; in contrast, the estimates in table 2absorb correlated variation into year fixed-effects. Thus, for example, the interpretation ofcolumns 1-2 is that, when a county in a given year experiences an unusually large increasein prices or volume, relative to other counties in the same year, it also experiences anunusual decline in price dispersion.

Columns 4-5 regress price dispersion on two measures of market tightness: column 4uses the fraction of the total housing stock which is vacant, and column 5 uses the yearlypopulation growth rate. Price dispersion is positively correlated with vacancy rates andis negatively correlated with population growth rates. Together, these two specificationssuggest that price dispersion is lower in tighter markets with more buying pressure.Column 6 includes all variables in a single specification. The coefficients on vacancy ratesand log sales remain significant with unchanged signs, whereas other coefficients losesignificance.

3.4 Zipcode cross-sectional regressions

In this section, we study variation in idiosyncratic price dispersion in the cross-sectionof zipcodes.7As described in appendix A.7, we aggregate price dispersion and a varietyof independent variables to the zipcode level over the period 2012-2016. We regressprice dispersion on various independent variables, controlling for third-order polyno-mials in a number of control variables: the average age, square footage, bedroom andbathroom counts of sold houses, average log income, and the fractions of the zipcode’s

7The availability of time-on-market data restricts our choice of the main geographical unit of study.Realtor.com provides time-on-market data on a zipcode level and covers the time period from May 2012 toApril 2018. Time-on-market data from Zillow covers a longer period of time, 2010-2018, but is only availableat the county level. In the main text, we report regression results for the panel of counties (2010-2016) andthe cross-section of zipcodes (2012-2016). We report county-level cross-sectional and zipcode-level panelresults in appendix B.1 as well.

14

Tabl

e2:

Cou

nty-

year

pane

lreg

ress

ions

LogS

Dx

100

(1)

(2)

(3)

(4)

(5)

(6)

Log

ZH

VI

−1.

051∗∗∗

−0.

720

(0.3

13)

(0.7

52)

Log

sale

s−

1.10

8∗∗∗

−1.

687∗∗∗

(0.2

26)

(0.4

52)

Tim

eon

mar

ket

(mon

ths)

0.50

1∗∗∗

0.13

9(0

.149

)(0

.142

)

Vac

ancy

rate

16.3

28∗∗∗

10.4

02∗∗∗

(1.8

01)

(2.5

89)

Pop

grow

thra

te−

8.64

0∗∗∗

−5.

650

(1.9

53)

(3.5

18)

Cou

nty

fixed

effe

cts

XX

XX

XX

Year

fixed

effe

cts

XX

XX

XX

Sam

ple

peri

od20

00-2

016

2000

-201

620

10-2

016

2006

-201

620

06-2

016

2010

-201

6N

6,13

76,

443

1,87

65,

555

5,12

01,

862

Adj

uste

dR

20.

890

0.89

30.

932

0.91

60.

913

0.93

4

Not

es.E

ach

dat

apo

int

isa

coun

ty-y

ear.

Reg

ress

ions

are

wei

ghte

dby

the

aver

age

num

ber

ofsa

les

ina

give

nco

unty

over

ally

ears

we

obse

rve.

Stan

dard

erro

rsar

ecl

uste

red

atth

eco

unty

leve

l.

15

population which are aged 18-35, 35-64, black, high school and college graduates, married,unemployed, and homeowners. We do not include total sales in the zipcode as a control;zipcodes are not isolated geographical markets, so rate and fraction variables are morereasonable to use in regressions than stock variables. The results are shown in table 3.

Column 1 of table 3 shows that price dispersion is negatively correlated with averagezipcode-level prices. Columns 2-4 show that, analogous to the time-series and panelspecifications in the previous subsections, price dispersion is correlated with time-on-market and two other measures of market tightness: vacancy rate and population growthrate, measured as the percentage change in each zipcode’s population between the2008-2012 and 2012-2016 ACS 5-year samples. Column 5 includes all variables together;the signs of all coefficients are unchanged, though the magnitudes change somewhat.Columns 6 and 7 add state and CBSA fixed effects; the signs of all coefficients areunchanged, and most coefficients remain significant.

3.5 Robustness checks

We run a variety of robustness checks for our main empirical results. In our main specifi-cations, we run cross-sectional regressions at the zipcode level and panel regressions at thecounty-year level; appendix B.1 shows that results from zipcode-year panel regressionsand county cross-sectional regressions are quantitatively similar to those in the main text.

In appendix B.2, we estimate price dispersion without including the polynomial termfz (xi, t) in house characteristics and time. This represents the results from a pure repeat-sales regression, with house and zipcode-month fixed effects. This has quantitatively smalleffects on the estimated residuals, and also small effects on all plots and regression results.Conversely, in appendix B.3, we estimate price dispersion including only the polynomialterm fz (xi, t) and a zipcode-time trend, omitting house fixed effects, corresponding to apure hedonic specification; this dramatically increases our estimates of price dispersion,but all time-series plots and all regression specifications retain their signs and approximatemagnitudes.

The method we use to estimate idiosyncratic price variance is potentially vulnerableto two flaws. First, specification (1) implies that idiosyncratic dispersion is invariant toholding period; this is incorrect if there is a random walk component of price dispersion,which scales with the time between sales. Second, specification (1) will mechanicallyinfer lower idiosyncratic price dispersion for houses which are sold more times in our

16

Tabl

e3:

Zip

code

cros

s-se

ctio

nalr

egre

ssio

ns

LogS

Dx

100

(1)

(2)

(3)

(4)

(5)

(6)

(7)

Log

ZH

VI

−3.

705∗∗∗

−3.

425∗∗∗

−1.

025∗∗∗

−1.

458∗∗∗

(0.2

18)

(0.2

01)

(0.2

73)

(0.2

32)

Tim

eon

mar

ket

(mon

ths)

2.31

9∗∗∗

1.60

2∗∗∗

2.92

3∗∗∗

2.21

9∗∗∗

(0.0

96)

(0.1

05)

(0.1

41)

(0.1

08)

Vac

ancy

rate

17.2

47∗∗∗

10.9

67∗∗∗

2.47

5∗∗∗

5.51

3∗∗∗

(0.8

61)

(0.9

38)

(0.8

66)

(0.8

69)

Pop

grow

thra

te−

2.14

1∗∗

−0.

684

−1.

397∗∗

−1.

291∗

(0.9

30)

(0.8

54)

(0.6

52)

(0.7

32)

Con

trol

sX

XX

XX

XX

Fixe

def

fect

sSt

ate

CBS

AN

3,39

43,

514

3,51

43,

514

3,39

43,

394

3,39

4A

djus

ted

R2

0.47

80.

573

0.55

30.

502

0.56

80.

772

0.69

4

Not

es.E

ach

data

poin

tis

azi

pcod

e.R

egre

ssio

nsar

ew

eigh

ted

byth

enu

mbe

rof

sale

sw

ithin

the

zipc

ode

over

the

time

peri

od20

12-2

016.

17

dataset, as the house fixed effects γi are estimated more precisely, and we only do thedegrees-of-freedom correction at the zipcode level. In appendix B.4, we account for theseeffects by removing all components of price variance, at the zipcode level, which can beexplained by a flexible function of average time-between-sales and the number of times ahouse is sold. We show that most of our empirical results continue to hold, though themagnitudes of estimated coefficients tend to decrease.

In appendix B.5, we show that most of our county-level results are robust to usingeither Realtor.com or Zillow as a source for time-on-market, although the county panelcoefficient on time-on-market loses significance. In appendix B.6, we remake figures 2and 3 weighting by the size of a county’s housing stock, rather than average sales, andin appendix B.7, we re-run our zipcode and county panel regressions with unbalancedpanels; in both cases, results are qualitatively and quantitatively very similar.

4 Model

In this section, we construct a search-and-bargaining model to rationalize the empiricalresults described in the preceding section. In the model, price dispersion is generated bytwo forces: heterogeneity in buyers’ idiosyncratic match utility for houses, and persistentheterogeneity in sellers’ flow utilities for keeping their house on the market. We showhow search frictions amplify the effects of seller value heterogeneity on price variance,rationalizing the empirical relationships between market tightness measures and pricedispersion.

Proofs and theoretical extensions are contained in appendix C.

4.1 Primitives

We model a single, geographically isolated housing market. There is a unit mass of houseswhich are ex-ante identical. Time is continuous, and all agents discount the future at rater. There are three kinds of agents in the model: sellers, buyers, and matched homeowners.The lifecycle of agents in the model is as follows: buyers purchase houses and becomematched homeowners, matched homeowners receive separation shocks to become sellers,and sellers leave the market upon successfully selling their houses.

18

4.1.1 Sellers

We use MS to denote the mass of sellers in the market who are waiting to sell their housesto buyers. As we describe in subsection 4.1.3, sellers are matched homeowners who havereceived separation shocks from their houses. Once a seller successfully sells her house,she permanently leaves the market, attaining a continuation value which we normalize to0.

Sellers are heterogeneous, characterized by some time-invariant holding utility, v > 0,which she receives as a flow per unit time that she keeps her house on the market. Aswe will describe in subsection 4.1.3, holding utilities v are drawn independently from adistribution F (v), at the time that a matched owner chooses to unmatch with her houseand become a seller. Holding utilities v play an important role in our model: sellerswith higher values of v are more patient, so they are more willing to wait longer to sellfor better prices. This will generate price dispersion in our model. How much pricedispersion this generates will depend on market tightness.

Differences in holding utilities v represent differences in sellers’ urgency to sell, whichmay come from several sources. Sellers who are moving within the same city may bewilling to hold on to their houses for longer than sellers who are moving out of the city.8

Certain sellers may be credit-constrained, so they cannot afford to wait for high prices,while others may be renting their houses out while selling, so they are more willing towait for higher sale prices.

In equilibrium, a seller with holding utility v attains some expected value VS (v), whichwe will characterize in subsection 4.2.2 below. A seller who matches with a buyer andsells her house for some price P receives utility P and continuation value 0, so her netgain in expected value is:

P− VS (v)

We use Feq (v) denote the distribution of holding utilities among sellers in stationaryequilibrium. Feq (v) will, in general, differ from F (v), the distribution of holding utilitiesamong sellers entering the market, because sellers with higher holding utilities will waitlonger to sell their houses. We describe how Feq (v) is determined in subsection 4.2.1below.

8This is related to Anenberg and Bayer (2015), who study a model in which sellers may buy housesbefore attempting to sell their houses; sellers who buy before selling and who sell before buying will thusendogenously have different flow utilities from their houses.

19

4.1.2 Buyers

We use MB to denote the mass of buyers who are present in the market, who are tryingto purchase a house. Buyers waiting to match with sellers are identical and receiveflow utility normalized to 0 while waiting. Buyers enter the market at some exogenousflow rate ηB. We assume that ηB < λM, where λM is the rate at which matched ownersreceive separation shocks and become homeowners; we show in subsection 4.2.1 that thiscondition is necessary for the model to admit a stationary equilibrium. Buyers can onlyexit the market by purchasing houses from sellers.

Buyers meet sellers through a process we describe in subsection 4.1.4 below. When abuyer meets a seller, he draws, independently across matches, some match utility ε ∼ G (·)for the house. We will occasionally refer to ε also as representing match quality. If thebuyer buys the house, he becomes a matched homeowner, receiving ε from the house perunit time, until he receives a separation shock and becomes a seller.

We assume that the distribution of match utilities G (·) is exponential, with standarddeviation σε, and lower bound ε0, which may differ from 0. The assumption of exponentialvalues is required for proposition 2, which analytically decomposes price variance intocomponents attributable to seller and buyer values; appendix C.3 shows how the resultschange when we allow for general match utility distributions.

Since unmatched buyers are undifferentiated, the expected value of an unmatchedbuyer in stationary equilibrium is some scalar VB, which we will characterize in subsection4.2.2 below. If a buyer matches with a seller, draws match utility ε, and purchases thehouse at price P, he becomes a matched homeowner with value VM (ε), so he achieves anet gain in the expected value of:

VM (ε) − P− VB

Our baseline model assumes buyers are undifferentiated for simplicity, as it impliesthat VB is a scalar, so the model is easier to solve computationally. A natural extensionof the model is to allow buyers to have persistent heterogeneous types so that buyers’flow utilities for remaining unmatched can differ. The model is robust to includingbuyer heterogeneity: appendix C.6 derives equilibrium conditions under persistent buyerheterogeneity, and shows that all of the main theoretical results of the model concerningprice dispersion – propositions 2, 3, and corollary 1 – are unchanged.

20

4.1.3 Matched homeowners

Matched homeowners are buyers who have purchased houses, and have not yet receivedseparation shocks to become sellers. Since there is a unit mass of houses, and each house isowned either by a matched homeowner or a seller, the total mass of matched homeownersis 1 −MS. Matched homeowners with match utility ε receive flow utility ε from theirhouse per unit time. Matched homeowners receive separation shocks at some exogenousPoisson rate λM; upon receiving a separation shock, a matched homeowner draws somev from the exogeneous distribution F (·), and becomes a seller with flow utility v. Aseparation shock can be thought of as, for example, a job offer in a different city, whichforces the homeowner to decide to sell her house and move out of the market.

We use Geq (ε) denote the distribution of match utilities among matched homeownersin stationary equilibrium. This will, in general, differ from G (ε), the distribution ofmatch utilities that buyers draw when they meet sellers, because higher draws of ε aremore likely to result in a successful trade. We will describe how Geq (ε) is determined insubsection 4.2.1 below.

4.1.4 Matching

Matches between buyers and sellers are generated at a flow rate m (MB,MS), whichdepends on the masses of unmatched buyers MB and unmatched sellers MS present inthe market. As is standard in the classical search-and-matching literature, we assume thatm (MB,MS) is Cobb-Douglas with constant returns to scale:

m (MB,MS) = αMφBM

1−φS

From the perspective of any given buyer or seller, matching occurs at Poisson rates λBand λS respectively, given by:

λB =m (MB,MS)

MB, λS =

m (MB,MS)

MS

Since buyer match utilities are drawn randomly from G (ε) each time a match occurs,sellers meet buyers with match utilities ε ∼ G (·); buyers meet sellers with holding utilitiesdrawn from Feq (v), the stationary distribution of seller holding utilities.

21

4.1.5 Price determination

Suppose that a buyer is matched with a seller with holding utility v, and the buyer drawsmatch utility ε. If the buyer and seller trade, the sum of their value functions is thebuyer’s continuation value VM (ε), plus the seller’s continuation value, which we havenormalized to 0. If they do not trade, the buyer receives the expected value from beingan unmatched buyer, VB, and the seller receives VS (v). Thus, the bilateral match surplusfrom trade when the buyer’s match utility is ε and the seller’s holding utility is v is:

VM (ε) − VB − VS (v) (5)

In stationary equilibrium, trade occurs if and only if the bilateral match surplus isnonnegative. This implies that a seller with holding utility v will trade with any buyerthat draws a match utility higher than some trade cutoff ε∗ (v). Rearranging expression (5),ε∗ (v) must satisfy:

VM (ε∗ (v)) = VB + VS (v)

We will assume that ε0 is sufficiently low that, in equilibrium,

ε∗ (v) > ε0 ∀v

that is, no seller type wishes to trade with all buyer types. This is an assumption onequilibrium objects, so it is hard to verify; we do not impose it in the calibration, but weimpose it in this section because it is needed to prove our theoretical results.

When trade occurs, we assume that prices are set through Nash bargaining. When aseller with holding utility v meets a buyer with match utility ε, the price that the sellerreceives is equal to her outside option, VS (v), plus a share θ of the bilateral match surplus;that is,

P (v, ε) = VS (v) + θ (VM (ε) − VB − VS (v)) (6)

22

4.2 Stationary equilibrium

4.2.1 Flow equality

There are four flows in our model: buyers enter the market, buyers turn into matchedowners, matched owners turn into sellers, and sellers sell and leave the market. Weanalyze the model in a stationary equilibrium, so all flows of all agent types must beequal.

First, consider flow equality for sellers. In equilibrium, the rate at which matchedhomeowners receive separation shocks and become sellers of type v is:

(1 −MS) λMf (v) (7)

In words, this is the product of the total mass of matched homeowners, 1−MS; the rate atwhich shocks homeowners receive separation shocks, λM; and the density f (v) of enteringsellers with value v.9 The equilibrium rate at which sellers of type v sell their houses andleave the market is:

MSfeq (v) λS (1 −G (ε∗ (v))) (8)

In words, this is the product of the mass of sellers, MS; the density of values among sellersin equilibrium, feq (v); the rate at which sellers are matched to buyers in equilibrium, λS;and the probability that the match utility draw ε exceeds the trade cutoff ε∗ (v) for a sellerof type v, which is 1 −G (ε∗ (v)). In stationary equilibrium, expressions (7) and (8) mustbe equal.

Flow equality for individual seller types implies that the total rate at which matchedhomeowners become sellers is equal to the total rate at which sellers sell and exit; that is,integrating (7) and (8) over v, we have:

(1 −MS) λM =

ˆvλSMS (1 −G (ε∗ (v))) feq (v)dv (9)

Moreover, since each successful sale turns a buyer into a matched homeowner, the RHS of(9) is also equal to the rate at which buyers turn into matched homeowners.

Second, inflows and outflows for matched homeowners with match utility ε mustbe equal. Matched homeowners’ separation rate λM does not depend on their match

9Since the distribution F (v) of holding utilities v does not depend on matched homeowners’ match utilityε, we do not need to explicitly integrate over the distribution Geq (ε) in expression (7).

23

utility ε, so the distribution of match utilities among matched homeowners is equal to thedistribution of match utilities among successful home buyers, which is:

Geq (ε) =

´v λSMS

[´ εε̃=ε0

1 (ε̃ > ε∗ (v))dG (ε̃)]dFeq (v)´

v λSMS (1 −G (ε∗ (v)))dFeq (v)

In words, the numerator is the flow rate at which a seller of value v successfully tradeswith a buyer with match utility below ε, integrated over the equilibrium distributionfeq (v) of holding utilities v among sellers. The denominator is the RHS of (9), the totalflow rate at which buyers become matched homeowners.

Finally, the rate at which buyers enter the market must be equal to all other flow rates;using (9), this condition can be written as:

(1 −MS) λM = ηB (10)

Since MS cannot be negative, expression (10) shows why we require ηB < λM in order forthe model to admit a stationary equilibrium.

4.2.2 Value functions

Given expression (6) for prices, we can write the seller, buyer, and matched ownercontinuous-time Bellman equations. Given the buyer match rate λB, trade cutoffs ε∗ (v),the equilibrium distribution of seller values Feq (v), and the seller value function VS (v),the equilibrium value of buyers, VB, must satisfy:

rVB = λB

ˆ ˆε>ε∗(v)

[(1 − θ) (VM (ε) − VB − VS (v))]dG (ε)dFeq (v) (11)

In words, expression (11) can be interpreted as follows. At rate λB, the buyer is matchedto a seller with type randomly drawn from Feq (·), and the buyer draws match quality εfrom G (·). If the buyer’s match quality draw, ε, is higher than the seller’s match qualitycutoff, ε∗ (v), trade occurs, and the buyer receives a share (1 − θ) of the bilateral matchsurplus.

Similarly, given the seller match rate λS, trade cutoffs ε∗ (v), and the buyer value VB,

24

the seller value function VS (v) satisfies:

rVS (v) = v+ λS

ˆε>ε∗(v)

θ (VM (ε) − VB − VS (v))dG (ε) (12)

In words, expression (12) states that a seller of type v receives flow value v from theirhouse while they are waiting for buyers. At rate λS, the seller meets a buyer with matchvalue ε randomly drawn from G (·). If ε > ε∗ (v), trade occurs, and the seller receives ashare θ of the bilateral match surplus.

The expected value VM of matched owners is determined by the Bellman equation:

rVM (ε) = ε+ λM

(ˆVS (v)dF (v) − VM (ε)

)(13)

In words, expression (13) states that matched owners get flow value ε while matched totheir house and receive separation shocks at rate λM, at which point they become sellersand attain the expectation of the seller value function VS (v) over the seller holding utilitydistribution F (v).

4.2.3 Equilibrium conditions

Collecting equilibrium conditions from earlier subsections, in our model, stationaryequilibrium is characterized by the following set of equations.

Proposition 1. Given primitives

r,α,φ, θ, λM,ηB, F (v) , ε0,σ2ε

a stationary equilibrium of the model is described by buyer and seller masses MB,MS, stationarydistributions Feq (v) and Geq (ε), matching rates λS, λB, value functions VS (v) ,VM (ε) ,VB, anda trade cutoff function ε∗ (v), which satisfy the following conditions:

Buyer, seller, and matched owner Bellman equations:

rVB = λB

ˆ ˆε>ε∗(v)

[(1 − θ) (VM (ε) − VB − VS (v))]dG (ε)dFeq (v) (14)

rVS (v) = v+ λS

ˆε>ε∗(v)

θ (VM (ε) − VB − VS (v))dG (ε) (15)

25

rVM (ε) = ε+ λM

(ˆVS (v)dF (v) − VM (ε)

)(16)

Trade cutoffs:VM (ε∗ (v)) = VS (v) + VB (17)

Matching rates:MSλS =MBλB = αMφ

BM1−φS (18)

Flow equality:(1 −MS) λMf (v) = λSMSfeq (v) (1 −G (ε∗ (v))) (19)

Geq (ε) =

´v λSMS

[´ εε̃=ε0

1 (ε̃ > ε∗ (v))dG (ε̃)]dFeq (v)´

v λSMS (1 −G (ε∗ (v)))dFeq (v)(20)

(1 −MS) λM = ηB (21)

4.3 Price dispersion

Since houses in the model are identical, we measure price dispersion in the model as thevariance of prices, P (v, ε), with respect to the joint distribution of v and ε among buyersand sellers whose meetings result in trade. Proposition 2, which we prove in appendixC.2, shows that equilibrium price dispersion admits a simple analytical characterization.

Proposition 2. In stationary equilibrium, the variance of P (v, ε) among trading sellers andbuyers is:

Varv∼F(·) (VS (v))︸ ︷︷ ︸Seller holding value

+

(θ

r+ λM

)2

σ2ε︸ ︷︷ ︸

Buyer match value

(22)

Expression (22) shows that equilibrium price dispersion arises from two sources: dif-ferences in sellers’ value functions VS (v), caused by differences in sellers’ holding utilities;and differences in buyers’ match utilities ε. The buyer value term only depends onθ, r, λM,σ2

ε, which are primitives of the model. The seller value term VS (v) is an equilib-rium object which we cannot analytically characterize, but there is a simple analyticalexpression for its derivative V ′S (v). Define the expected time-on-market in equilibrium fora seller of type v, TOM (v), as:

TOM (v) ≡ 1λS (1 −G (ε∗ (v)))

(23)

26

In words, time-on-market is the inverse of λS (1 −G (ε∗ (v))), which is the product of theequilibrium rate at which sellers meet buyers, λS, and the fraction of meetings for a sellerof type v that result in trade, (1 −G (ε∗ (v))).

Proposition 3. The seller value function VS (v) satisfies:

V ′S (v) =TOM (v)

rTOM (v) + θ(24)

Qualitatively, proposition 3 shows that V ′S (v) is higher, and hence, the dispersion inseller values is higher, when equilibrium time-on-market TOM (v) is higher. Thus, thepass-through of dispersion in holding utility v into sellers’ value functions, and thusprice dispersion, is tightly linked to the equilibrium time-on-market function, TOM (v).Intuitively, differences in v represent differences in sellers’ flow utility per unit time forkeeping their houses on the market. How much differences in v affect sellers’ stock valuefunctions VS (v) depends on how long sellers expect to spend on the market. If MB is high,so all sellers sell quickly, all sellers are better off, and the difference between expectedvalues of high- and low-v sellers is low. If MB is low, so sellers expect to wait on themarket for a long time, all sellers are worse off, but this disproportionately harms sellerswith low holding utility v.

Propositions 2 and 3 also show why it is necessary to have heterogeneous seller holdingutilities in order to link price dispersion to market tightness. Without dispersion in sellervalues, if buyer values are exponential, price dispersion is a constant determined bythe variance in match quality, σε, together with the parameters θ, r, λM. Appendix C.3shows that this is not an artifact of our assumption that values are exponential; in general,without heterogeneous seller holding utility, price variance may increase or decrease asmarket thickness changes.

To formalize the link between time-on-market and price dispersion, appendix C.5shows that, if time-on-market TOM (v) increases, pointwise for every v, then equilibriumprice dispersion will also increase.

Corollary 1. Fix r, θ, F (v) ,σ2ε. Consider two sets of model parameters,

Θ1 =(α1,φ1, λ1

M,η1B, ε1

0

), Θ2 =

(α2,φ2, λ2

M,η2B, ε2

0

)such that time-on-market is uniformly higher in stationary equilibrium under Θ1; that is, letting

27

TOMΘ1 (v) denote the equilibrium time-on-market function under Θ1,

TOMΘ1 (v) > TOMΘ2 (v) ∀v

Then equilibrium price dispersion will also be higher under Θ1 than Θ2.

Together, proposition 3 and corollary 1 show that there is a tight link between time-on-market and price dispersion: fixing the interest rate r, bargaining power θ, the distributionof seller holding utilities F (v) and the variance of match quality σ2

ε, the equilibriumtime-on-market function TOM (v) is a sufficient statistic for equilibrium price dispersion.

Proposition 3 and corollary 1 also show how to empirically distinguish between theseller and buyer components of price dispersion. Since the relationship between time-on-market and price dispersion is driven only by seller value heterogeneity, the TOM-PDrelationship in the data is informative about the magnitude of the seller value componentof price dispersion. In the following section, we calibrate the model to match the TOM-PDrelationship in the data, to quantitatively measure the buyer and seller components ofprice dispersion.

4.4 Comparative statics

To illustrate how our model maps primitives to outcomes, we solve the model compu-tationally and show comparative statics with respect to various input parameters. Weparametrize F (v) as uniform on the interval

[v̄−∆v, v̄+∆v]

so that v̄ is the mean of entering seller values, and ∆v is a parameter governing the spreadof seller values. Choices for other parameters are described in appendix C.8, and appendixD.1 describes how we computationally solve the model. In figure 4, we plot comparativestatics of various model outputs as we vary the input parameters

λM,ηB, v̄,∆v, ε0,σε

Specifically, figure 4 shows average prices, time-on-market, and price variance; figure 5further decomposes price variance into its buyer- and seller-value induced components,using the decomposition of proposition 2.

28

Increasing ε0 and increasing ηB have similar effects on outcomes: prices increase, andtime-on-market and price variance decrease. This is because ηB causes the equilibriummass of buyers MB to increase, and increasing ε0 makes buyer match utility draws morefavorable; both effects increase tightness, increasing prices, decreasing time-on-marketand thus decreasing price dispersion. Decreasing v̄ increases prices, but also increasestime-on-market; as a result, price variance increases.

The effects of ε0,ηB, v̄ on price variance act entirely through the seller value channel;in the top row of figure 5, the buyer value component of price dispersion is flat. This isbecause the buyer value component of price dispersion only depends on σε and λM. As aresult, the relative slopes of the TOM and Var (P) lines are similar in all three plots in thetop row of figure 4: holding fixed λM, ∆v, and σε, different ways to move time-on-markethave similar effects on price dispersion.

Increasing ∆v slightly increases prices and time-on-market, but dramatically increasesprice dispersion, and the entirety of the increase is through the seller value channel. Thisis because, as it follows from proposition 3, total dispersion in seller values VS (v) is theintegral of the time-on-market function over the range of seller flow holding utilities;increasing the dispersion of seller flow utilities thus increases the dispersion in VS (v), andthus equilibrium price variance.

Increasing λM decreases prices and increases time-on-market, as it increases theequilibrium mass of sellers and decreases tightness. However, it actually decreases pricedispersion. Figure 5 shows that, while increasing λM increases seller-induced pricedispersion, it dramatically decreases the buyer value component of price dispersion: ifbuyers expect to receive separation shocks at a higher rate, their match utilities ε affecttheir values for houses less, decreasing equilibrium price dispersion.

Finally, increasing σε increases prices, time-on-market, and price variance. The buyercomponent of price variance increases directly, and the seller component increases becauseequilibrium time-on-market is higher.

4.5 Buyer value heterogeneity and price dispersion

In section 3, we showed that time-on-market and price dispersion are correlated overyears, calendar months, and in panel and cross-sectional regressions. Table 4 showsthe correlation between TOM and PD across these specifications. To extract impliedcoefficients from the yearly and seasonal plots of figures 2 and 3, we run univariate

29

Figure 4: Comparative statics

Notes. Average price, time-on-market, and price variance, as model parameters vary. Allvariables are indexed so they are equal to 1 at the lowest value of the x variable.

30

Figure 5: Comparative statics, buyer and seller variance components

Notes. Buyer and seller components of price variance as model parameters vary. Buyerand seller variance are both indexed so they are equal to 1 at the lowest value of the xvariable.

31

regressions of price dispersion on time-on-market. The “county panel” row of table 4reports the coefficient in column 3 of table 2,10 and the “zipcode cross-sectional” rowreports the range of estimates of the time-on-market coefficient from tables 3 and 5

Table 4 shows that the implied regression coefficients between price dispersion andtime-on-market from the yearly, calendar month, and panel specifications are similar, butthe zipcode cross-sectional regressions produce a much larger coefficient. The modelsuggests a tentative explanation for this finding. Dispersion in buyer match quality, σε,is likely to be fairly stable over time within a location but may vary substantially acrosslocations. In locations where the housing stock is fairly homogeneous, σε will be low, ashouses are relatively good substitutes; if the housing stock is more heterogeneous, matchquality matters more and σε should be higher.

Under this interpretation, the coefficient in a regression of price dispersion on time-on-market reflects variation in market tightness over time, that is, variables in the top row offigures 4 and 5, which hold fixed the buyer value component of price dispersion. We thusexpect the implied correlation between time-on-market and price dispersion to be lowerin relative terms, and fairly consistent across specifications. The cross-sectional regressioncoefficient, in contrast, contains both variation in tightness and variation in buyer valuedispersion, σε. The bottom-right panels of figures 4 and 5 show that variation in σε willtend to move price dispersion more, per unit that time-on-market moves, because it movesboth the seller and buyer value components of price dispersion.

We can empirically test this hypothesis as follows. If σε varies across zipcodes, figure4 predicts that time-on-market should be higher in markets with higher match qualitydispersion. Moreover, if we attempt to control for σε, comparing zipcodes with moresimilar dispersion in match quality, the coefficient in a regression of price dispersion ontime-on-market should decrease.

Table 5 tests these predictions. As proxies for σε, we use the standard deviation inyear built and square footage across houses sold in 2012-2016 for each zipcode. Theseare measures of house heterogeneity; we believe they are reasonable proxies for σεbecause buyers’ values will be less disperse if houses in a given market are more similar.In columns 1-3, we regress time-on-market on SD year built and SD square footage,separately and combined; in all specifications, the coefficient is negative, so zipcodeswith more heterogeneous housing stocks have higher time-on-market, confirming our

10We exclude the time-on-market coefficient in column 7 of table 4, as it is not significantly distinguishablefrom 0.

32

Table 4: Correlation between TOM and PD across different specifications

Type CoefYearly 0.775Seasonal 0.667County panel 0.501Zipcode cross-sectional 1.419 - 2.923

Notes. Estimates of the correlation between price dispersion and time-on-market fromyearly, seasonal, panel and cross-sectional analyses.

first prediction. In columns 4 and 5, we regress logSD on time-on-market, controllingfor SD year built and SD square footage; as predicted, the coefficient on time-on-marketdecreases.

This suggests that buyer value heterogeneity may partially explain why the cross-sectional relationship between logSD and TOM is steeper than the panel and time-seriesrelationships. The coefficient in column 5 is still around twice as large as the paneland time-series relationships; however, our proxies for σε are far from perfect, so it isconceivable that the TOM coefficient could be reduced further with better proxies for σε.

5 Calibration

In this section, we fit the model to the data to measure the relative sizes of buyer- and seller-induced price variance and to quantify the tradeoff sellers face between time-on-marketand prices. The overall logic of the calibration is as follows. Proposition 2 implies that,holding fixed r, λM,σε, changes in other primitives affect price dispersion only throughthe seller component of price variance. Moreover, figure 4 shows that different driversof tightness imply very similar slopes for the TOM-PD relationship; this is theoreticallysupported by proposition 3. Thus, the relationship between time-on-market and pricedispersion is informative about how large the dispersion in seller values is. The calibrationthus essentially chooses the dispersion in seller values to match the TOM-PD relationshipobserved in the data; other parameters are chosen to match empirical moments in waysthat are relatively standard in the housing literature.

33

Tabl

e5:

Zip

code

cros

s-se

ctio

nalr

egre

ssio

nsw

ith

hous

ehe

tero

gene

ity

mea

sure

s

Tim

eon

mar

ket

(mon

ths)

LogS

Dx

100

(1)

(2)

(3)

(4)

(5)

Tim

eon

mar

ket

(mon

ths)

1.60

2∗∗∗

1.41

9∗∗∗

(0.1

05)

(0.1

10)

Nor

mSD

yrbu

ilt0.

169∗∗∗

0.14

2∗∗∗

0.54

5∗∗∗

(0.0

11)

(0.0

11)

(0.0

74)

Nor

mSD

sqft

0.22

7∗∗∗

0.17

4∗∗∗

−0.

093

(0.0

18)

(0.0

18)

(0.1

15)

Log

ZH

VI

−0.

204∗∗∗

−0.

207∗∗∗

−0.

198∗∗∗

−3.

425∗∗∗

−3.

426∗∗∗

(0.0

32)

(0.0

32)

(0.0

31)

(0.2

01)

(0.1

99)

Vac

ancy

rate

4.39

8∗∗∗

4.04

7∗∗∗

4.20

0∗∗∗

10.9

67∗∗∗

12.2

23∗∗∗

(0.1

31)

(0.1

33)

(0.1

30)

(0.9

38)

(0.9

47)

Pop

grow

th−

0.55

2∗∗∗

−0.

309∗∗

−0.

470∗∗∗

−0.

684

−1.

337

(0.1

36)

(0.1

37)

(0.1

34)

(0.8

54)

(0.8

53)

Con

trol

sX

XX

XX

N3,

394

3,39

43,

394

3,39

43,

394

Adj

uste

dR

20.

514

0.50

60.

528

0.56

80.

575

Not

es.E

ach

data

poin

tis

azi

pcod

e.R

egre

ssio

nsar

ew

eigh

ted

byth

eav

erag

enu

mbe

rof

sale

sin

agi

ven

zipc

ode

over

the

tim

epe

riod

2012

-201

6.“N

orm

SDyr

built

”an

d“N

orm

SDsq

ft”

are,

resp

ecti

vely

,the

stan

dar

dd

evia

tion

ofye

arbu

iltan

dsq

uare

foot

age

amon

gho

uses

sold

duri

ng20

12-2

016

with

inth

egi

ven

zipc

ode,

norm

aliz

edac

ross

allz

ipco

des

inou

rsa

mpl

e.C

olum

n4

isid

enti

calt

oco

lum

n5

ofta

ble

3.

34

5.1 Methodology

To bring the model to data, we impose the following parametric assumptions. Time ismeasured in years. As in the model simulations of subsection 4.4, we parametrize F (v) asa uniform distribution on the interval

[v̄−∆v, v̄+∆v]

Thus, v̄ is the mean of seller values, and ∆v controls the dispersion in seller values.

Then, we calibrate the following parameters externally:

r,α,φ, θ

and estimate the remaining parameters by matching the moments in the data:

λM,ηB, v̄,∆v, ε0,σ2ε

In particular, to estimate ∆v, we first construct a grid of time-on-market and average pricepairs using the data on the cross-section of counties. Then, we vary:

ηB, v̄

at the county level, to exactly match time-on-market and average prices for each county,while holding the other parameters in the model fixed:

r,α,φ, θ, λM,∆v, ε0,σ2ε

For each county, we generate model-implied log price dispersion. We calibrate ∆v suchthat the relationship between time-on-market and model-implied log price dispersionacross counties matches the one observed in the data exactly.

5.1.1 Externally calibrated parameters

We set the yearly discount rate r = 0.052, so that the annual discount factor is 0.95.We assume symmetric bargaining power, so θ = 0.5; this is also used by, for example,Anenberg and Bayer (2015) and Arefeva (2019). We choose the matching function elasticity

35

φ to 0.84, based on Genesove and Han (2012), who estimate this elasticity using theNational Association of Realtors survey data, which captures both buyer and seller time-on-market; this estimate is also used by Anenberg and Bayer (2015). Since we do notobserve buyer time-on-market, we cannot separately identify the matching efficiencyparameter α and the mass of buyers MB, so we normalize the match efficiency parameterto α = 1.

5.2 Moment matching

The remaining parameters in our model are λM,ηB, v̄,∆v, ε0,σ2ε. We match these param-

eters to data moments using a moment matching procedure, with an outer loop formatching ∆v, and an inner loop for matching all other parameters conditional on ∆v. Inthe inner loop, for any guess for ∆v, we choose other parameters to exactly match fivemoments, and then generate a model-implied relationship between time-on-market andlog price dispersion using a procedure described in subsection 5.2.2. In the outer loop, wechoose ∆v to match the model-implied TOM-PD relationship to the data. Computationaldetails of how we solve the model for a given choice of parameters are described inappendix D.1.

5.2.1 Inner loop

Given any value of ∆v, we choose λM, ε0,σ2ε, v̄,ηB to match the average values of five

moments. Three of the target moments come from a sample of counties in 2013. Werestrict to counties that have mean prices below the 95th percentile.11 The three momentsthat we calculate from our data are the sales-weighted sample averages of average houseprice (Zillow’s ZHVI) and average time-on-market (Zillow), and the turnover rate – totalhouse sales as a fraction of the total housing stock (Corelogic deed and tax).

We target two moments from other papers in the literature. The first is the averagenumber of houses that buyers visit before buying, which Genesove and Han (2012) findto be 9.96, in the US. The second is the dispersion in buyer values for houses, fromAnundsen, Larsen and Sommervoll (2019). Using data on Norwegian residential real

11The distribution of county mean prices is quite long-tailed. In 2013, the median county has meanprice of $170, 942, and the 99th percentile mean price is $669, 584. To calibrate ∆v, we hold many modelparameters fixed across counties, and only use ηB, v̄ to match the average price and time-on-market acrosscounties. It is hard, therefore, to match the right tail of the mean price distribution, so we only includecounties with mean prices below the 95th percentile.

36

estate auctions,12 Table 1 of Anundsen, Larsen and Sommervoll (2019) reports a variety ofsummary statistics about bid-ask, bid-appraisal, and bid-sell spreads. To conservativelyestimate the buyer component of price variance, we target the lowest spread in Table 1,the difference between the opening bid price and the ask price, which is 6.45% of houseprices on average. We match this in the model by requiring the square root of buyer matchutility-induced price variance to equal 6.45% of house prices; that is, from (25) in the maintext, we will choose σε such that:

1E (P (v, ε))

√(θ

r+ λM

)2

σ2ε = 0.0645

While the mapping between moments and parameters is complex, roughly speaking,the input parameters determine the output moments as follows. The lower bound, ε0,and dispersion, σ2

ε, of buyer utilities jointly determine the level of buyer-induced pricedispersion and the overall level of house prices. The mean of seller holding utilities, v̄,and the lower bound, ε0, affect average gains-from-trade and thus move prices and theaverage number of house visits by buyers before purchasing. The entry rate ηB of buyersdetermines market tightness, which determines average time-on-market. The separationrate λM is tightly linked to the turnover rate.

5.2.2 Outer loop

In the data, we have several ways to measure the relationship between price dispersionand time-on-market. As table 4 shows, this coefficient is relatively similar in the panel,seasonal and time-series regression specifications. For our calibration, we re-estimatethe panel regression using the restricted sample of counties used for estimating the datamoments for the inner loop over the period 2010-2016. The coefficient is 0.5277 log pointsper month time-on-market, which is very similar to the estimates from section 3.

To most closely match the data, we simulate a grid of values of time-on-market andaverage prices in our model which matches the joint cross-sectional distribution of thesevariables in 2013. Specifically, holding fixed the values of other parameters estimated intable 6, we choose ηB and v̄ for each county to exactly match time-on-market and averageprice. The model then generates price dispersion in dollar terms for each simulated county.

12We use these data because we are unaware of publically available bid data on housing auctions in theUS.

37

We use a Taylor approximation, which we describe in appendix D.2, to calculate predictedlog price dispersion. We then regress predicted log price dispersion from our model oncounty time-on-market and prices, to generate a model-predicted regression coefficient.We choose ∆v to match this model-predicted TOM-logSD relationship to the county panelcoefficient.

We note that the goal of this exercise is not to match all target moments in all of thecounties in our subsample; we only use the counties to create a realistic grid of valuesfor prices and time-on-market, to generate a regression coefficient in the model, whichis roughly comparable to the coefficient in the data. The regression in the model iscross-sectional, while the data regression is a panel regression; this is because our modeldoes not have county and time-level unobservables, so it is inappropriate for simulating apanel regression.

5.2.3 Parameter estimates

Table 6 shows the values of empirical moments that we target and the parameter valuesthat we estimate. We find that match quality ε, per month of homeownership, is drawnfrom an exponential distribution with a lower bound of $426, and a standard deviation of$205. Since trade happens if and only if the drawn ε is high enough, the distribution ofrealized match qualities, conditional on house purchase, is different from the underlyingdistribution of match quality draws. Specifically, conditional on house purchase, theaverage ε that buyers attain is around $1,093 per month. We find that seller utilities, permonth, are drawn from a uniform distribution with mean -$3,390 and standard deviation$1,114.

Using our parameter estimates, we can calculate buyers’ average value from homeown-ership as the integrated flow value of match utility ε, until buyers receive a separationshock: ˆ

E[ε|ε > ε∗(v)]

r+ λmdF(v)

Likewise, we can calculate sellers’ average total value from keeping their houses on themarket as: ˆ

v · TOM(v)dF(v)

We find that buyers attain an average value of $132,174 from homeownership before theyreceive separation shocks and that sellers experience an average loss of around $9,835

38

during the time that their houses are on the market.

5.3 Quantifying liquidity discounts

Our model allows us to quantify the tradeoff between time-on-market and sale price thatsellers with different holding utilities face.13 Sellers with higher holding utilities waitlonger to sell their houses but achieve higher prices. The left panel of figure 6 quantifiesthis tradeoff. Agents with 75th percentile value of holding utility v spend 3.83 monthsselling their houses on average and attain an expected sale price of $196,663. At the sametime, agents with 25th percentile value of holding utility v spend 2.42 months selling thehouse, and attain 6.12% lower expected prices, averaging at $185,322. This price gap canbe thought of as a “liquidity discount”, which sellers with high holding costs bear to selltheir houses more quickly.

In tighter markets, where all sellers sell their houses relatively quickly, these liquiditydiscounts also decrease. The right panel of figure 6 illustrates how the p75-p25 gap inexpected prices varies with average market tightness. To make this plot, we vary thebuyer entry rate, ηB, holding fixed other parameters, and plot how the p75-p25 gap inexpected prices varies with average time-on-market. The p75-p25 gap, in dollar terms,decreases from $15,731 when average the time-on-market is 4.8 months to $7,549 whenthe average time-on-market is 2.04 months.

5.4 Decomposing price variance

Using the estimates from our model, together with the variance decomposition fromproposition 2, we can quantify how much of price dispersion is attributable to buyer andseller values respectively. Formally, taking (22), and using a linear approximation to thelogarithm, we can write the variance of prices within the model as approximately:

Var (logP (v, ε)) ≈ Var(P (v, ε))

(E (P (v, ε)))2

13This is somewhat an abuse of terminology; figure 6 does not truly represent a “tradeoff”, because, in thecontext of Nash bargaining models, agents’ types are assumed to be common knowledge, so a given sellercannot replicate the price and time-on-market pair achieved by another seller with different holding utility.

39

Table 6: Moment and parameter values

Moment Value Parameter ValuePrice 191.99 λM 0.047

Buyer-induced SD 6.45% ε0 0.426TOM (Months) 3.197 σε 0.2047

Turnover rate 0.046 ηB 0.046Num. visits 9.960 v̄ -3.387

PD-TOM Corr 0.528 ∆v 1.929

Notes. Target moments and estimated parameter values. ε0,σε, v̄,∆v are reported inthousands of US dollars per month. Turnover rate and λM are yearly turnover andseparation rates respectively, and ηB is a fraction of the unit mass of houses per year.

Figure 6: Liquidity discounts

Notes. The left panel shows the average price and time-on-market achieved by sellerswith different holding utilities v. The red stars represent, from left to right, the 25th, 50th,and 75th percentiles of seller holding utilities. The right panel shows, as we vary markettightness using ηB, the difference in expected prices attained by sellers with 75th and 25thpercentile values of v vs average time-on-market.

40

Var (logP (v, ε)) ≈Varv∼F(·) (VS (v))

(E (P (v, ε)))2︸ ︷︷ ︸Seller holding value

+1

(E (P (v, ε)))2

(θ

r+ λM

)2

σ2ε︸ ︷︷ ︸

Buyer match value

(25)

Our model is designed to capture all price variance which is driven by buyers’ and sellers’preferences. We do not use logSD in the data as a target moment, and we find that logSDin the calibrated model is lower than the average logSD in the data, which is 16.7% ofhouse prices. One explanation for the difference is unobserved heterogeneity: componentsof house quality which are observed by market participants, and thus affect prices, butare not in our data. If we attribute the entire difference between model and data logSDto unobserved heterogeneity, we can further decompose total logSD in the data intocomponents attributable to buyer values, seller values, and unobserved heterogeneity:

(0.165)2 ≈Varv∼F(·) (VS (v))

(E (P (v, ε)))2︸ ︷︷ ︸Seller holding value

+1

(E (P (v, ε)))2

(θ

r+ λM

)2

σ2ε︸ ︷︷ ︸

Buyer match value

+σ2unobserved het (26)

For ease of interpretation, we can report the square roots of each component of (26),which are in units of percentages of prices. We find that the seller component has SDof 3.52% of prices, the buyer component is 6.45% of house prices, and the unobservedheterogeneity component is 14.77% of house prices. In terms of variance fractions, 4.5% oftotal logSD is attributable to seller values, 15.3% to buyer values, and 80.2% to unobservedheterogeneity. The calibrated model thus implies that seller value-induced price dispersionaccounts for a nontrivial, but fairly small, fraction of total idiosyncratic price dispersionwhich we observe in the data; buyer value dispersion accounts for a somewhat largerfraction, and we infer that the majority of observed price dispersion is probably due tounobserved house-level heterogeneity.

As a robustness check, we explore an alternative calibration method in appendix D.3:we drop the buyer value heterogeneity moment from Anundsen, Larsen and Sommervoll(2019), and we instead require model-implied price dispersion to exactly match the averageLogSD in our data. The parameter estimates are somewhat less realistic: we find a muchhigher value of buyer match quality heterogeneity σε, which then implies that v̄ and ε0

must be very negative to rationalize the prices and time-on-market which we observe inthe data. However, our estimates of ∆v, liquidity discounts, and the magnitude of sellervalue-induced price dispersion are similar across both calibration methodologies.

41

6 Discussion

6.1 Prior literature on price dispersion

A number of other papers have attempted to measure idiosyncratic house price dispersion.Giacoletti (2017) uses the same Corelogic data that we use to measure idiosyncratic pricedispersion in the metropolitan areas of San Francisco, San Deigo, and Los Angeles. Unlikeour specification (1), Sagi (2015) and Giacoletti use returns, rather than individual housesales, as the primary unit of analysis.14 Using returns, rather than sales, as the unit ofanalysis is more appropriate to the extent that the difference between an individual house’sprice and the zipcode index follows a random walk. However, Sagi (2015) and Giacolettishow that the random walk assumption is rejected in the data; idiosyncratic variance doesscale with holding periods, but much more slowly than under a random walk model. Arelated paper is Carrillo, Doerner and Larson (2019), which finds that excess returns ofindividual houses over market averages are mean-reverting in subsequent transactions.

The results of these papers thus support the use of our specification (1) to measureprice dispersion. Specification (1) goes further, assuming that idiosyncratic variance has norelationship with holding period; this is violated in the data, but we relax this in appendixB.4. The benefit of our measurement strategy is that, since we can measure errors at thelevel of individual house sales, rather than pairs of purchases and sales, our estimates ofidiosyncratic price dispersion can be flexibly aggregated cross-sectionally and over time.This is necessary for us to produce our stylized facts, which we believe are new to theliterature: that idiosyncratic price dispersion is countercyclical, seasonal, and correlatedwith time-on-market and other measures of market tightness. These results build on andcomplement Giacoletti (2017), who shows that contractions to mortgage credit availabilityat the zipcode level are associated with increased idiosyncratic variance, and Landvoigt,Piazzesi and Schneider (2015), who show that idiosyncratic variance increased in SanDiego following the 2008 housing bust.

Peng and Thibodeau (2017) uses a purely hedonic specification to measure pricedispersion, analyzing the relationship between idiosyncratic price dispersion and various

14There are a number of other differences between Giacoletti’s methodology and ours. First, Giacolettimeasures returns with respect to Zillow’s home value index, rather than adding zipcode-month fixed effectsas we do in this paper. Second, Giacoletti does not allow returns to flexibly vary over time as a function ofhouse characteristics – characteristics are allowed to affect returns, but not in a time-dependent manner.Third, Giacoletti incorporates data on remodeling expenses in measuring price dispersion, which we do notdo in this paper.

42

other variables in the cross-section of zipcodes. To address the possibility that the hedonicmodel determining prices changes over time, Peng and Thibodeau (2017) runs separatehedonic regressions for different time periods. We address this issue through the hedonicfz (xi, t) term in specification (1), which effectively allows the hedonic coefficients ondifferent characteristics to change continuously over time. In appendix B.3, we measureprice dispersion using a purely hedonic specification for log prices, similar to Peng andThibodeau (2017); this does not substantially change our results.

Two other papers which measure idiosyncratic price dispersion are Anenberg andBayer (2015) and Landvoigt, Piazzesi and Schneider (2015). Anenberg and Bayer (2015), asan input moment for estimating their structural model, estimate the idiosyncratic volatilityof house prices using a repeat-sales specification with zipcode-month and house fixedeffects, without allowing characteristics to affect prices over time. Landvoigt, Piazzesiand Schneider (2015) estimates idiosyncratic price dispersion assuming that the onlycharacteristic that affects mean returns is a house’s previous sale price. Our specification(1) does not nest that of Landvoigt, Piazzesi and Schneider (2015), since we do not includeprevious sale prices in specification (1); however, as we discuss in footnote 4, to the extentthat the factors which affect prices are summarized by our house characteristics xt, ourspecification will also be able to capture these trends.

Quantitatively, Giacoletti (2017), using data from 1989 to 2013, finds that the standarddeviation of idiosyncratic component of returns is approximately 9.6%-11.8% in San Diego,13.9%-16.5% in Los Angeles, and 13.7-17.6% in San Francisco. Landvoigt, Piazzesi andSchneider (2015) finds a similar SD of 8.8%-13.8% for San Diego over the time horizon1999-2007. In our sample, over the time period 2000-2017, we estimate return standarddeviations of 15.7% for San Diego, 16.8% for Los Angeles, and 19.1% for San Francisco.15

Our estimates are thus roughly in line with the estimates from Giacoletti (2017) andLandvoigt, Piazzesi and Schneider (2015), preserving the ordering of idiosyncratic pricedispersion between the three regions, although our estimates are somewhat higher thantheirs. Moreover, similar to our findings in subsection 3.1 below, Landvoigt, Piazzesiand Schneider (2015) finds that price dispersion increased during the 2008 housing bust,though their sample does not include the subsequent recovery. Thus, our paper, Giacoletti

15To calculate these quantities, we take sales-weighted averages of σ̂2z for all zipcodes within the San

Diego-Carlsbad-San Marcos, San Francisco-Oakland-Fremont, and Los Angeles-Long Beach-AnaheimCBSAs. We then multiply σ̂z by a factor of

√2, to convert standard deviations of prices at each sale to

standard deviations of returns, which can then be compared directly to the estimates in Giacoletti (2017)and Landvoigt, Piazzesi and Schneider (2015).

43

(2017), and Landvoigt, Piazzesi and Schneider (2015) arrive at similar estimates usingdifferent methodologies, datasets, time horizons, and geographic definitions, suggestingthat the stylized facts we document are fairly robust to different measurement strategies.

6.2 Quantifying microstructure frictions in housing markets

This paper relates to literature studying the microstructure of the housing market, sur-veyed in Han and Strange (2015). Relative to this literature, one direction to extend ourresults is to analyze the effects of different market mechanisms on idiosyncratic pricedispersion. We use Nash bargaining as a simple reduced-form model of price-setting;it is attractive as a model because it flexibly allows prices to depend on buyers’ andsellers’ outside options, but it is quite stylized. In practice, price-setting mechanisms differsomewhat in different housing markets; most houses are sold via bilateral bargainingbased on a posted list price,16 but in some markets explicit auctions are used.17 A naturalextension of our results is to analyze whether different trading mechanisms are associatedwith higher or lower levels of idiosyncratic price dispersion.

Another direction for further inquiry is the interaction of idiosyncratic price dispersionwith the composition of housing market participants. A number of papers show thatdifferent classes of participants in housing markets achieve different prices and averagereturns.18 Housing markets may be more efficient, and thus idiosyncratic price dispersionmay be lower if participants are more sophisticated. In particular, a number of papersstudy the effects of house flippers (Bayer et al. (2011), Giacoletti and Westrupp (2017));given that we find that idiosyncratic price dispersion decreases when volume increases, itis plausible that flippers may decrease idiosyncratic price dispersion, even if they increasethe volatility of aggregate house prices.

16See Han and Strange (2016) for a discussion of the role of house list prices. Guren (2018) analyzes therelationship between house list prices and sale prices, arguing that sellers face strategic complementaritiesin adjusting list prices.

17Han and Strange (2014) analyzes time-series and cross-sectional trends in the prevalence of auctions.Arefeva (2019) constructs a dynamic search model which suggests that auctions amplify house pricevolatility.

18Chinco and Mayer (2015) shows that out-of-town second-home buyers achieve lower capital gains thanlocal buyers, Myers (2004), Ihlanfeldt and Mayock (2009), and Bayer, Ferreira and Ross (2016) study racialprice gaps in the housing market, and Goldsmith-Pinkham and Shue (2019) that men attain higher returnsin housing markets than women.

44

6.3 Idiosyncratic risk and household welfare

How important is local and idiosyncratic risk in accounting for total risk associated withhousing investments? Table 7 attempts to decompose house price variation into national,local, and idiosyncratic components; appendix A.8 describes how we construct this table.The first row of table 7 shows the standard deviation of housing returns, for a 10-yearholding period, at different levels of aggregation. The “national” column corresponds tothe return standard deviation of a hypothetical consumer holding a diversified portfolioof houses within the US; the “CBSA” and “zipcode” columns show the return SD for aconsumer holding a portfolio within a CBSA or zipcode, and the “Ind. house” columnconsiders a consumer who buys and sells a single house 10 years apart. The “Varianceratio” row shows the ratios of return variance at each level of diversification with nationalreturn variance.

Considering local and idiosyncratic components of house price variance dramaticallyincreases house price risk. The zipcode-level 10-year return SD of is 33.9%, which is77% higher than national return variance; accounting for idiosyncratic variance increasesvariance further, totaling between 116% to 235% of national variance. Put another way,for a representative household who purchases and sells a house 10 years apart, onlyaround 1/3-1/2 of her risk is national house price risk; the remainder is due to local andidiosyncratic factors, which have similar magnitudes.

The third row shows the annualized Sharpe ratios associated with housing investmentsat each level of diversification. Echoing Jordà, Schularick and Taylor (2019), we find thata national portfolio of residential housing, diversified across the US, has an annualizedSharpe ratio of 0.87 from 2000-2017, which is substantially higher than the S&P 500 Sharperatio over 1950-2017. A portfolio that is concentrated only within a single CBSA or zipcodehas a much lower Sharpe ratio, of approximately 0.66-0.70. Purchasing a single househas a Sharpe of 0.48-0.59, depending on how high idiosyncratic price dispersion is in thezipcode in question. The “Ind. + fees” column shows that, if we subtract an additional10% total return in realtor fees, the Sharpe ratio for individual houses decreases further to0.41-0.51. This is still somewhat higher than the Sharpe ratio of the S&P 500 over the sametime period, but accounting for idiosyncratic variance and transaction costs substantiallydecreases the attractiveness of housing as an asset class.

45

Table 7: Performance of residential housing investments at different levels of aggregation

National CBSA Zip Ind. house Ind + fees SP500SD 10yr 25.5% 32.0% 33.9% 37.5%-46.7%Var Ratio 1.00 1.57 1.77 2.16-3.35Ann. Sharpe 0.87 0.70 0.66 0.48-0.59 0.41-0.51 0.34

Notes. 10-year holding period return SD, variance ratios, and annualized Sharpe ratios forhousing portfolios at different levels of diversification, and the S&P 500. See appendixA.8 for details.

7 Conclusion

In this paper, we have created a new methodology, combining repeat-sales and hedonicmodels of house prices, to measure idiosyncratic house price dispersion at the level ofindividual house sales. Our estimates of price dispersion can be flexibly aggregated acrossregions and over time. We demonstrate a number of new stylized facts about idiosyncratichouse price dispersion: it is countercyclical, seasonal, and is robustly correlated withvarious measures of market tightness. We construct a simple search-and-bargainingmodel to rationalize these findings and fit the model to the data to quantify liquiditydiscounts that sellers face in housing markets. These results help us understand themicrostructure of housing markets and the determinants of idiosyncratic house pricedispersion, which have important implications for understanding the total risk associatedwith US households’ investments in residential housing.

46

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51

Appendix

A Data description and cleaning

A.1 CoreLogic tax and deed data

Our data on house sales comes from the CoreLogic deed dataset, which is derived fromcounty government records of house transactions. Corelogic records the price and date ofeach sale, and housing units are uniquely identified, within a FIPS county code, by anAssessor Parcel Number (APN), which is assigned to each plot of land by tax assessors.Our data on house characteristics comes from the CoreLogic tax assessment data for thefiscal year 2016-2017, which contains, for each APN, its latitude, longitude, year built,square footage, and numbers of bedrooms and bathrooms, as of 2016-2017. We merge thetax data to the CoreLogic deed data by APN and FIPS county code.

We clean the datasets using a number of steps. First, we use only arms-length newconstruction or resales of single-family residences, which are not foreclosures, which havenon-missing sale price, date, APN, and zipcode in the CoreLogic deed data, and whichhave non-missing year built and square footage in the CoreLogic tax data. As mentionedin the main text, we use only data from 2000 onwards, as we find that CoreLogic’s dataquality is quite low prior to around 2000. Even after throwing out pre-2000 data, we findthat some counties have very low total sales for early years, suggesting that some data ismissing. To address this, we filter out all county-years for which the total number of saleswe observe is below 20% of average annual sales over all county-years in our sample.

We use the dataset that results from these cleaning steps to measure monthly salesby zipcode. This subsample is, however, unsuitable for estimating price dispersion, andwe apply a few additional cleaning steps for the subsample we use to estimate pricedispersion regressions in subsection 2.2.

First, our measurement of price dispersion uses a repeat-sales specification, so we canonly use houses that were sold multiple times. Moreover, we wish to filter for “houseflips”, as well as instances where reported sale price seems anomalous. To filter for flips,if a house is ever sold twice within a year, we drop all observations of the house; these arelikely flips, which are known to be a peculiar segment of the real estate market (Bayeret al. (2011), Giacoletti and Westrupp (2017)). To filter for potentially anomalous prices,

52

Table 8: Characteristics of zipcodes in the primary dataset

Sample mean All zipcodes meanPopulation 35597 9724Pop / Sq mile 2604 1307Housing units 14192 4095Avg income 86372 69190% Age 18-35 22.6% 20.3%% Married 50.5% 52.1%% Black 11.2% 7.69%Total zipcodes 3971 33120Total pop (1000’s) 146267 322072

Notes. Characteristics of zipcodes in primary estimation sample, compared to all zipcodesin ACS 2012-2016 5-year sample.

similarly to Landvoigt, Piazzesi and Schneider (2015) and Giacoletti (2017), if we everobserve a property whose annualized appreciation or depreciation is above 50% for anygiven pair of sales, we drop all observations of the property.

Second, specification (1) involves a fairly large number of parameters: house andzipcode-month fixed effects, as well as many parameters in the fz (xi, t) polynomial term.We thus require a fairly large number of house sales in order to precisely estimate (1), andthus we filter to zipcodes with at least 1000 house sales remaining, and with at least 10sales per month on average, after applying the filtering steps described above.

Appendix table 8 shows characteristics of the zipcodes in our estimation sample,compared to the universe of zipcodes in the 2012-2016 ACS 5-year dataset. Our datasetconstitutes approximately 11.6% of all zipcodes. Zipcodes in our sample are larger anddenser than average, so our sample constitutes around 48% of the total US population. Interms of demographics, the average income is somewhat higher than average for zipcodesin our sample, but our zipcodes are quite representative in terms of age, race, and thefraction of the population that is married.

A.2 ACS

We use demographic information about zipcodes and counties from the ACS. For county-years, we use ACS 1-year samples spanning the years 2006-2017; for zipcodes, we use

53

the ACS 5-year sample spanning the years 2012-2016. We accessed the data using SocialExplorer, a commercial provider of pre-aggregated ACS data. The demographic andhousing stock characteristics we use are total population, population growth rate, totalnumber of housing units, log average income, unemployment rate (calculated as one minusthe fraction of population which is employed, divided by the fraction of the populationin the labor force), the vacancy rate (calculated as the fraction of all surveyed houseswhich are vacant), the fraction of population aged 18-35 and 35-64, and the fractions ofthe population which are black, married, high school graduates, and college graduates.

A.3 Zillow Research

At the county level, we use Zillow Research data on time-on-market, daily listings, and theZillow Home Value Index (ZHVI) for single-family homes at the zipcode-month level. TheZHVI is available from 1997-2018, time-on-market is available from 2010-2018, and dailylistings data is available from 2013-2018. We use a version of the ZHVI data downloadedin April of 2019; the ZHVI methodology was revised in November 2019, and as of January2020, ZHVI data prior to 2008 is not available for download on Zillow’s website.

A.4 Realtor.com

For our zipcode-level results, we use data on time-on-market and total housing listingsfrom Realtor.com. Realtor.com calculates median time-on-market and total listings atthe zipcode-month level, using microdata from multiple listing services. The dataset wedownloaded covers the time period May 2012 to April 2018. Zillow.com and Realtor.comtime-on-market differ somewhat, because Zillow’s measure is based on how long housesspend on the Zillow platform, whereas Realtor.com’s data is based on multiple listingservices. Figure 7 plots average time-on-market on Realtor.com and Zillow, respectively,for counties between the years 2012-2016. The time-on-numbers are strongly correlated,although Zillow’s time-on-market numbers tend to be higher than Realtor.com’s, andthere is some independent variation. Appendix B.5 shows results from running ourcounty-level regression specifications using time-on-market numbers from Realtor.cominstead of Zillow.

54

Figure 7: Zillow vs Realtor.com time-on-market

Notes. Zillow vs Realtor.com time-on-market. Each data point is a county. Time-on-marketis calculated as an average over county-months from 2012-2016.

55

A.5 Yearly and seasonal data construction

To construct the dataset used in figure 2, we first filter to counties which we observeevery year from 2000-2016; this leaves us with 361 counties, comprising approximately38.8 million home sales. To construct the LogSD line in figure 2, we average ε̂2

it over allobservations within a given county-year, then take the square root of the resultant average.The time-on-market line represents the sales-weighted average of time-on-market acrosscounty-months in a given year, and the price line represents the sales-weighted averageof the Zillow Home Value Index for single-family residences across county-months in agiven year. We only have time-on-market data from Zillow for seven complete years, from2010-2016, but we are able to measure total sales, logSD, and prices for the longer timehorizon 2000-2016.

To construct the seasonal dataset, we filter to the years 2010-2016, as we only observetime-on-market for these years. This leaves us with 262 counties, comprising approx-imately 13.8 million home sales over this time period. We first collapse the data toyear-month level. To collapse, we take the sum over sales in all counties, the mean overall ε̂2

it terms that we estimate, and the sales-weighted average of time-on-market. For theprice line, we take the monthly FHFA house price index for the entire US, also for thetime horizon 2010-2016. The Zillow ZHVI is seasonally adjusted, hence is not appropriatefor studying seasonal variation in prices.

Since all four variables – prices, price dispersion, sales, and time-on-market – havelow-frequency trends over time, we filter out these trends by regressing the collapsedyear-month dataset outcomes on a third-order polynomial in year-month, subtractingaway the predicted values, and adding back the mean. We then average the filtered seriesover years to the calendar month level, index each series to its January level, and plot theresultant series in figure 3.

A.6 County-year and county-year-month data construction

To construct the county-year dataset, we first aggregate the estimated errors ε̂it fromspecification (1) to the county-year level. We merge to aggregated Zillow Research datato the county-year level, taking averages of time-on-market, ZHVI, and daily listingsfor all county-months within a county year, and we merge to ACS county-year datafrom 2006-2016. We construct the fraction of houses which are listed as Zillow’s daily

56

listings divided by total housing units for a given county-year from the ACS. We use anunbalanced panel for the panel regressions of subsection 3.3. For figure 3.1, we balance thepanel, keeping only county-years that we see for the entire period 2000-2017. Balancingthe panel decreases the number of counties represented from 733 to 361, but only dropsaround 18% of house sales from the dataset.

To construct the monthly dataset, used in figure 3, we aggregate total house sales fromCorelogic and estimated errors ε̂it from specification (1) to the county month level, forall months within 2010-2018. We then merge to Zillow Research data on county-monthtime-on-market and ZHVIs. We balance the panel, keeping only county-months that weobserve for all 84 months from January 2010 to December 2018. We then aggregate allvariables to the calendar month level, weighting average time-on-market, prices, and thesquare of ε̂it by total sales, and we plot the resultant averages in figure 3.

The zipcode-year level dataset used in appendix table 10 is constructed analogously,by aggregating price dispersion to the zipcode-year level, then merging to time-on-marketfrom Realtor.com and the ZHVI from Zillow. Zillow’s ZHVI is available at the zipcode-month level, but Zillow’s time-on-market measure is not.

A.7 Zipcode cross-sectional data construction

To construct the zipcode-level dataset for the cross-sectional regressions of subsection 3.4,we take the average of the estimated residuals ε̂2

it for each zipcode in our sample for thetime period 2012-2016. We then take listings and time-on-market data from Realtor.com,filter to the same period 2012-2016, and aggregate to the zipcode level; we calculatezipcode-level listings averaging over all zipcode-months we observe, and we calculatezipcode-level time-on-market by taking the listings-weighted average of median time-on-market for each zipcode-month. We measure total housing units and other demographiccovariates for zipcodes using the 2012-2016 ACS 5-year sample, as described in appendixA.2 above.

We construct the county cross-sectional dataset used in appendix table 9 analogously,using Corelogic data for sales and price dispersion, Zillow and Realtor.com data fortime-on-market, total listings, and the ZHVI, and the ACS 2012-2016 5-year sample fordemographic covariates. We filter the Corelogic, Zillow, and Realtor.com datasets to thetime period 2012-2016 to match the time period of the ACS dataset.

The county cross-sectional dataset used in appendix table 9 is constructed analogously.

57

A.8 Variance decomposition

To calculate quantities in table 7, we use Zillow’s ZHVI data, aggregated to the zipcode-year level, from 2000-2017. We first filter to the largest 171 CBSAs, covering 95% ofcumulative house sales in our data. We construct log returns at the zipcode-year level asthe first difference of yearly averaged ZHVIs. We then regress zipcode-year returns onCBSA-year fixed effects:

rzt = rcbsa,t + εzt (27)

weighting by zipcode sales averaged over time; εzt, the estimated variance of the residual,is our measure of yearly zipcode-specific price volatility, the “Zip” column of table 7.

We then run the following regression:

rcbsa,t = rt + εcbsa,t

again weighting by average total CBSA sales across years. The term rt can be thought ofas the mean sales-weighted return US-wide, and εcbsa,t as the county-specific component.Finally, we calculate the volatility and mean of rt as the sample mean and SD of theestimated year fixed effects.

The first row of table 7, “SD 10 year”, means the total volatility on a 10-year holdingperiod. For the national, CBSA, and zip columns, these are calculated as

√10 times the

annual volatility of rt, rcbsa,t and rzt respectively. For the upper and lower bounds in the“total” column, we take the variance of rzt and add

√2 times the p10 and p90 estimates

of zipcode-level idiosyncratic variance, which are respectively 11.2% and 22.6% (sincevariance is incurred upon sales, then take the square root). We multiply by

√2 because

housing returns will experience the idiosyncratic ε̂it shock twice, once upon purchaseand once upon sale.

The second row of 7, “Var rat”, is the ratio of return variances at each level to nationalreturn variances; formally, each column is the square of the “SD 10yr” row divided by thesquare of the national SD 10yr.

The “Sharpe 10yr” column is the annualized Sharpe ratio. For the national, CBSA, zip,and total columns, we take 10 times the mean yearly return on the national average ZHVI,subtract the average 10-year T-bill return from 2000-2017, and add a rental yield of 0.0533per year, which is the Jordà, Schularick and Taylor (2019) estimate for the US averagerental yield, 1870 to present, net of taxes and other maintainence costs. We then divide

58

Figure 8: Yearly and monthly variation in price dispersion, robustness checks

Notes. Figures 3 and 2, where “LogSD” represents the original estimate√σ̂2t , “LogSD, TBS

adj” represents√σ̂2TBSadj,t, “LogSD, Repeat sales” represents

√σ̂2Repeatsales,t, and “LogSD,

Hedonic” represents√σ̂2Hedonic,t.

by the SD 10yr row in each column, then further divide by√

10 to convert to annualizedSharpe ratios. For comparison, the first column shows the realized Sharpe ratio of theS&P 500, inclusive of dividends, from 2000-2017, using data from CRSP.

B Robustness checks

B.1 Zipcode-year and county cross-sectional regressions

Table 9 runs cross-sectional regressions at the county level, instead of the zipcode level asin table 3 in the main text. Most coefficients are significant, and the results are qualitativelysimilar: price dispersion is positively correlated with time-on-market and vacancy rate.The results remain significant when including state fixed effects. Similarly, table 10reports results from zipcode-year panel regressions. Coefficients on prices, sales, andtime-on-market are significant and have the same signs as those in table 2 in the maintext, although the magnitudes of coefficients change somewhat. Since the publicallyavailable ACS 1-year samples do not include zipcode information, we are unable tomeasure vacancy rates and population growth rates at the zipcode-year level.

59

Tabl

e9:

Cou

nty

cros

s-se

ctio

nalr

egre

ssio

ns

LogS

Dx

100

(1)

(2)

(3)

(4)

(5)

(6)

Log

ZH

VI

−3.

875∗∗∗

−4.

790∗∗∗

−3.

138∗∗∗

(0.8

83)

(0.8

79)

(1.0

92)

Tim

eon

mar

ket

(mon

ths)

1.06

5∗∗∗

0.32

51.

393∗∗∗

(0.3

97)

(0.4

07)

(0.4

90)

Vac

ancy

rate

17.8

86∗∗∗

22.0

07∗∗∗

13.2

75∗∗∗

(3.4

53)

(4.5

94)

(4.0

61)

Pop

grow

thra

te−

13.3

78−

1.73

0−

10.8

02(8

.276

)(9

.748

)(1

0.34

2)

Con

trol

sX

XX

XX

XFi

xed

effe

cts

Stat

eN

257

261

393

393

257

257

Adj

uste

dR

20.

493

0.46

50.

524

0.49

10.

553

0.72

0

Not

es.E

ach

dat

apo

int

isa

coun

ty.

Reg

ress

ions

are

wei

ghte

dby

the

num

ber

ofsa

les

wit

hin

the

coun

tyov

erth

eti

me

peri

od20

12-2

016.

60

Tabl

e10

:Zip

code

-yea

rpa

nelr

egre

ssio

ns

LogS

Dx

100

(1)

(2)

(3)

(4)

Log

ZH

VI

−1.

203∗∗∗

−1.

701∗∗∗

(0.0

98)

(0.3

54)

Log

sale

s−

0.93

6∗∗∗

−0.

191

(0.1

12)

(0.1

69)

Tim

eon

mar

ket

(mon

ths)

0.26

0∗∗∗

0.23

8∗∗∗

(0.0

71)

(0.0

73)

Zip

fixed

effe

cts

XX

XX

Year

fixed

effe

cts

XX

XX

Sam

ple

peri

od20

00-2

016

2000

-201

620

13-2

016

2013

-201

6N

51,5

7851

,578

12,2

1212

,212

Adj

uste

dR

20.

851

0.85

20.

923

0.92

4

Not

es.E

ach

data

poin

tis

azi

pcod

e-ye

ar.

Reg

ress

ions

are

wei

ghte

dby

the

aver

age

num

ber

ofsa

les

ina

give

nzi

pcod

eov

eral

lyea

rsw

eob

serv

e.St

anda

rder

rors

are

clus

tere

dat

the

zipc

ode

leve

l.

61

B.2 Pure repeat-sales specification

As a robustness check, we estimate idiosyncratic price dispersion using an alternativespecification in which we omit the polynomial fz (xi, t) term; this corresponds to a purerepeat-sales specification, in which characteristics cannot have time-varying effects onprices:

pit = γi + ηzt + εit (28)

Call the resultant estimates ε̂2Repeatsales,it. Figure 9 compares ε̂2

Repeatsales,it to errors esti-mated from the baseline specification. The left panel aggregates the idiosyncratic errors,from 2012-2016, to the zipcode level and plots the resultant estimates, with σ̂2

Repeatsales,z onthe y-axis and the baseline specification on the x-axis. As expected, most points in figure9 lie above the line y = x, indicating that residuals from specification (28) are generallyhigher than those in the baseline specification, (1), as the baseline specification is strictlymore flexible than (28). This is not always true – since the DF correction term

Nz

Nz −Kz

differs between the two specifications, it is possible for specification (28) to produce smallererror estimates than the baseline specification; thus, there are a few points in figure 9which lie below the y = x line. The difference between residual estimates is quantitativelysmall; the average ratio between residual standard deviations from (1) and (28) is 1.038,and figure 9 shows that the estimates from the two specifications are highly correlated.The right panel of figure 9 shows density plots of zipcode-level idiosyncratic standarddeviations from the baseline specification and specification (28), further supporting thatthe difference between the two specifications is quantitatively small.

To verify that the inclusion of the fz (xi, t) term does not drive our results, we re-runthe analyses of section 3 using ε̂2

Repeatsales,it in place of ε̂2it. The line in figure 8 labeled

“LogSD, No poly”, shows the seasonal and time-series behavior of ε̂2Repeatsales,it. We find

that ε̂2Repeatsales,it tracks ε̂2

it closely, seasonally and over the business cycle. Tables 11, 12,13 and 14 re-run all zipcode and county cross-sectional regressions using ε̂2

Repeatsales,it inplace of ε̂2

it; results are qualitatively and quantitatively similar to those in the main text.

62

Tabl

e11

:Zip

code

cros

s-se

ctio

nalr

egre

ssio

ns,r

epea

t-sa

les

LogS

Dx

100

(1)

(2)

(3)

(4)

(5)

(6)

(7)

Log

ZH

VI

−3.

749∗∗∗

−3.

510∗∗∗

−1.

157∗∗∗

−1.

647∗∗∗

(0.2

23)

(0.2

05)

(0.2

87)

(0.2

40)

Tim

eon

mar

ket

(mon

ths)

2.27

6∗∗∗

1.47

0∗∗∗

2.85

2∗∗∗

2.08

8∗∗∗

(0.0

98)

(0.1

08)

(0.1

48)

(0.1

12)

Vac

ancy

rate

18.2

30∗∗∗

12.6

54∗∗∗

3.62

1∗∗∗

6.69

2∗∗∗

(0.8

76)

(0.9

59)

(0.9

08)

(0.9

00)

Pop

grow

thra

te−

1.80

6∗−

0.39

5−

1.07

8−

0.96

7(0

.951

)(0

.873

)(0

.683

)(0

.758

)

Con

trol

sX

XX

XX

XX

Fixe

def

fect

sSt

ate

CBS

AN

3,39

43,

514

3,51

43,

514

3,39

43,

394

3,39

4A

djus

ted

R2

0.48

50.

574

0.56

30.

509

0.57

30.

764

0.68

9

Not

es.

Re-

runn

ing

all

spec

ifica

tion

sin

tabl

e3

usi

nga

pu

rere

pea

t-sa

les

spec

ifica

tion

.E

ach

dat

ap

oint

isa

zip

cod

e.R

egre

ssio

nsar

ew

eigh

ted

byth

enu

mbe

rof

sale

sw

ithi

nth

ezi

pcod

eov

erth

eti

me

peri

od20

12-2

016.

63

Tabl

e12

:Zip

code

-yea

rpa

nelr

egre

ssio

ns,r

epea

t-sa

les

LogS

Dx

100

(1)

(2)

(3)

(4)

Log

ZH

VI

−1.

592∗∗∗

−1.

331∗∗∗

(0.1

03)

(0.4

03)

Log

sale

s−

0.89

3∗∗∗

−0.

444∗∗

(0.1

11)

(0.1

74)

Tim

eon

mar

ket

(mon

ths)

0.53

1∗∗∗

0.48

1∗∗∗

(0.0

74)

(0.0

76)

Zip

fixed

effe

cts

XX

XX

Year

fixed

effe

cts

XX

XX

Sam

ple

peri

od20

00-2

016

2000

-201

620

13-2

016

2013

-201

6N

51,2

5551

,255

12,2

1212

,212

Adj

uste

dR

20.

834

0.83

40.

917

0.91

7

Not

es.R

e-ru

nnin

gal

lspe

cific

atio

nsin

tabl

e10

usin

ga

pure

repe

at-s

ales

spec

ifica

tion.

Each

data

poin

tis

azi

pcod

e-ye

ar.

Reg

ress

ions

are

wei

ghte

dby

the

aver

age

num

ber

ofsa

les

ina

give

nzi

pcod

eov

eral

lyea

rsw

eob

serv

e.St

anda

rder

rors

are

clus

tere

dat

the

zipc

ode

leve

l.

64

Tabl

e13

:Cou

nty

cros

s-se

ctio

nalr

egre

ssio

ns,r

epea

t-sa

les

LogS

Dx

100

(1)

(2)

(3)

(4)

(5)

(6)

Log

ZH

VI

−3.

976∗∗∗

−4.

970∗∗∗

−3.

457∗∗∗

(0.9

10)

(0.9

09)

(1.1

41)

Tim

eon

mar

ket

(mon

ths)

1.02

9∗∗

0.30

11.

220∗∗

(0.4

10)

(0.4

21)

(0.5

12)

Vac

ancy

rate

18.1

67∗∗∗

22.3

12∗∗∗

13.3

54∗∗∗

(3.5

44)

(4.7

48)

(4.2

43)

Pop

grow

thra

te−

11.2

500.

695

−9.

779

(8.5

00)

(10.

074)

(10.

807)

Con

trol

sX

XX

XX

XFi

xed

effe

cts

Stat

eN

257

261

393

393

257

257

Adj

uste

dR

20.

494

0.46

50.

525

0.49

20.

552

0.71

4

Not

es.

Re-

runn

ing

all

spec

ifica

tion

sin

tabl

e9

usi

nga

pu

rere

pea

t-sa

les

spec

ifica

tion

.E

ach

dat

ap

oint

isa

cou

nty.

Reg

ress

ions

are

wei

ghte

dby

the

num

ber

ofsa

les

wit

hin

the

coun

tyov

erth

eti

me

peri

od20

12-2

016.

65

Tabl

e14

:Cou

nty-

year

pane

lreg

ress

ions

,rep

eat-

sale

s

LogS

Dx

100

(1)

(2)

(3)

(4)

(5)

(6)

Log

ZH

VI

−1.

347∗∗∗

−2.

367∗∗∗

(0.3

18)

(0.8

93)

Log

sale

s−

0.87

2∗∗∗

−1.

258∗∗

(0.2

69)

(0.5

16)

Tim

eon

mar

ket

(mon

ths)

0.30

0∗0.

055

(0.1

74)

(0.1

65)

Vac

ancy

rate

19.7

48∗∗∗

14.2

40∗∗∗

(1.9

38)

(2.9

60)

Pop

grow

thra

te−

10.0

08∗∗∗

−6.

286

(2.9

43)

(4.8

97)

Cou

nty

fixed

effe

cts

XX

XX

XX

Year

fixed

effe

cts

XX

XX

XX

Sam

ple

peri

od20

00-2

016

2000

-201

620

10-2

016

2006

-201

620

06-2

016

2010

-201

6N

6,13

76,

443

1,87

65,

555

5,12

01,

862

Adj

uste

dR

20.

865

0.86

70.

920

0.89

30.

888

0.92

4

Not

es.R

e-ru

nnin

gal

lspe

cific

atio

nsin

tabl

e2

usin

ga

pure

repe

at-s

ales

spec

ifica

tion

.Eac

hda

tapo

int

isa

coun

ty-y

ear.

Reg

ress

ions

are

wei

ghte

dby

the

aver

age

num

ber

ofsa

les

ina

give

nco

unty

over

ally

ears

we

obse

rve.

Stan

dard

erro

rsar

ecl

uste

red

atth

eco

unty

leve

l.

66

Figure 9: Repeat-sales versus baseline estimates of price dispersion

Notes. On the left plot, the y-axis shows estimates of σ̂z using the baseline specification,(1), and the x-axis shows estimates from specification (28), excluding controls for charac-teristics. Each data point represents a zipcode. The right axis plots density plots of σ̂zusing the baseline specification and specification (28).

B.3 Pure hedonic specification

To complement the results of appendix B.2, we estimate idiosyncratic price dispersionusing a specification in which we include the polynomial fz (xi, t) term and zipcode-monthfixed effects, but omit house fixed effects:

pit = ηzt + fz (xi, t) + εit (29)

This is a hedonic regression specification, in which the hedonic price function is allowed tovary smoothly over time. Call the resultant estimates ε̂2

Hedonic,it. Analogously to figure 9,figure 10 compares ε̂2

Hedonic,it, aggregated to the zipcode level from 2012-2016, to estimatesfrom the baseline specification. Excluding house fixed effects has a much larger effect onestimated idiosyncratic price dispersion, increasing the average standard deviation of εitacross zipcodes from 16.8% to 27.7% of house prices. However, the estimates from (29)are still very positively correlated with those from the baseline specification.

Analogous to appendix B.2, we plot the time-series and seasonal behavior of ε̂2Hedonic,it

in figure 8, and show the results of all regression specifications using ε̂2Hedonic,it in tables

15, 16, 17 and 18. The seasonal behavior of ε̂2Hedonic,it is very similar to that of the baseline

specification. The time-series movements are somewhat different, perhaps because ourspecification for fz (xi, t) is not completely capturing time-series variation in hedonic

67

Figure 10: Hedonic

Notes. On the left plot, the y-axis shows estimates of σ̂z using the baseline specification,(1), and the x-axis shows estimates from specification (29), excluding controls for charac-teristics. Each data point represents a zipcode. The right axis plots density plots of σ̂zusing the baseline specification and specification (29).

coefficients. However, broadly speaking, ε̂2Hedonic,it shows a similar pattern of decreasing

during the boom and increasing during the bust. Moreover, regression coefficients largelyhave the same signs as the baseline specification and remain significant.

B.4 Adjusting for time-between-sales and number of sales

Our specification (1) implies that idiosyncratic price variance does not depend on theholding period. This is rejected, for example, if house prices follow a random walk.However, Giacoletti (2017), using the same dataset as ours, and Sagi (2015) using dataon commercial real estate, reject the hypothesis that house prices follow a random walk;both papers show that the dispersion of returns increases with the holding period, butmore slowly than the random walk model predicts. Another problem with specification(1) is that ε̂2

it will tend to be larger for houses which are sold more times, because thehouse fixed effect γi is estimated more precisely; we correct for this at the zipcode level,through the degrees-of-freedom correction term in (3), so that estimated residuals areunbiased once aggregated to the zipcode level, but residual estimates will still be biasedfor individual house sales.

Let tbsi be the average time between sales for house i, and let salesi be the totalnumber of times we see house i being sold. Figure 11 plots kernel regression fit of ε̂2

it

68

Tabl

e15

:Zip

code

cros

s-se

ctio

nalr

egre

ssio

ns,h

edon

ic

LogS

Dx

100

(1)

(2)

(3)

(4)

(5)

(6)

(7)

Log

ZH

VI

−5.

085∗∗∗

−4.

633∗∗∗

−3.

259∗∗∗

−4.

428∗∗∗

(0.2

28)

(0.1

98)

(0.3

31)

(0.2

59)

Tim

eon

mar

ket

(mon

ths)

3.14

1∗∗∗

2.42

1∗∗∗

3.10

0∗∗∗

2.43

4∗∗∗

(0.0

97)

(0.1

04)

(0.1

71)

(0.1

20)

Vac

ancy

rate

19.5

30∗∗∗

10.1

26∗∗∗

4.58

2∗∗∗

6.26

7∗∗∗

(0.9

17)

(0.9

27)

(1.0

49)

(0.9

69)

Pop

grow

thra

te−

3.35

4∗∗∗

−0.

760

−0.

710

−1.

104

(0.9

96)

(0.8

44)

(0.7

89)

(0.8

16)

Con

trol

sX

XX

XX

XX

Fixe

def

fect

sSt

ate

CBS

AN

3,39

43,

514

3,51

43,

514

3,39

43,

394

3,39

4A

djus

ted

R2

0.48

90.

565

0.50

00.

436

0.62

20.

700

0.65

8

Not

es.

Re-

runn

ing

all

spec

ifica

tion

sin

tabl

e3

usi

nga

pu

rehe

don

icsp

ecifi

cati

on.

Eac

hd

ata

poi

ntis

azi

pco

de.

Reg

ress

ions

are

wei

ghte

dby

the

num

ber

ofsa

les

wit

hin

the

zipc

ode

over

the

tim

epe

riod

2012

-201

6.

69

Tabl

e16

:Zip

code

-yea

rpa

nelr

egre

ssio

ns,h

edon

ic

LogS

Dx

100

(1)

(2)

(3)

(4)

Log

ZH

VI

−4.

867∗∗∗

−1.

628∗∗∗

(0.1

45)

(0.4

57)

Log

sale

s−

0.24

7∗∗

−1.

181∗∗∗

(0.1

05)

(0.1

96)

Tim

eon

mar

ket

(mon

ths)

0.83

2∗∗∗

0.69

8∗∗∗

(0.0

89)

(0.0

90)

Zip

fixed

effe

cts

XX

XX

Year

fixed

effe

cts

XX

XX

Sam

ple

peri

od20

00-2

016

2000

-201

620

13-2

016

2013

-201

6N

51,5

7851

,578

12,2

1212

,212

Adj

uste

dR

20.

824

0.80

80.

913

0.91

4

Not

es.R

e-ru

nnin

gal

lspe

cific

atio

nsin

tabl

e10

usin

ga

pure

hed

onic

spec

ifica

tion

.E

ach

dat

apo

int

isa

zipc

ode-

year

.R

egre

ssio

nsar

ew

eigh

ted

byth

eav

erag

enu

mbe

rof

sale

sin

agi

ven

zipc

ode

over

ally

ears

we

obse

rve.

Stan

dard

erro

rsar

ecl

uste

red

atth

ezi

pcod

ele

vel.

70

Tabl

e17

:Cou

nty

cros

s-se

ctio

nalr

egre

ssio

ns,h

edon

ic

LogS

Dx

100

(1)

(2)

(3)

(4)

(5)

(6)

Log

ZH

VI

−3.

715∗∗∗

−4.

233∗∗∗

−4.

801∗∗∗

(0.7

20)

(0.6

47)

(0.8

71)

Tim

eon

mar

ket

(mon

ths)

2.18

8∗∗∗

1.59

9∗∗∗

1.69

4∗∗∗

(0.3

01)

(0.3

00)

(0.3

91)

Vac

ancy

rate

15.3

01∗∗∗

16.2

77∗∗∗

14.6

72∗∗∗

(2.8

37)

(3.3

79)

(3.2

38)

Pop

grow

thra

te−

21.3

17∗∗∗

−6.

001

−21

.266∗∗

(6.7

52)

(7.1

69)

(8.2

48)

Con

trol

sX

XX

XX

XFi

xed

effe

cts

Stat

eN

257

261

393

393

257

257

Adj

uste

dR

20.

651

0.68

40.

637

0.61

70.

750

0.81

6

Not

es.

Re-

runn

ing

all

spec

ifica

tion

sin

tabl

e9

usi

nga

pu

rehe

don

icsp

ecifi

cati

on.

Eac

hd

ata

poi

ntis

aco

unt

y.R

egre

ssio

nsar

ew

eigh

ted

byth

enu

mbe

rof

sale

sw

ithi

nth

eco

unty

over

the

tim

epe

riod

2012

-201

6.

71

Tabl

e18

:Cou

nty-

year

pane

lreg

ress

ions

,hed

onic

LogS

Dx

100

(1)

(2)

(3)

(4)

(5)

(6)

Log

ZH

VI

−4.

610∗∗∗

−2.

757∗∗∗

(0.4

45)

(0.9

84)

Log

sale

s−

0.11

5−

2.20

5∗∗∗

(0.2

67)

(0.5

39)

Tim

eon

mar

ket

(mon

ths)

0.42

3∗∗

0.03

9(0

.180

)(0

.175

)

Vac

ancy

rate

22.1

80∗∗∗

8.51

6∗∗∗

(2.2

95)

(3.0

22)

Pop

grow

thra

te−

8.55

4∗∗

−4.

963

(3.4

31)

(5.0

79)

Cou

nty

fixed

effe

cts

XX

XX

XX

Year

fixed

effe

cts

XX

XX

XX

Sam

ple

peri

od20

00-2

016

2000

-201

620

10-2

016

2006

-201

620

06-2

016

2010

-201

6N

6,13

76,

443

1,87

65,

555

5,12

01,

862

Adj

uste

dR

20.

867

0.84

60.

895

0.83

80.

843

0.90

2

Not

es.

Re-

runn

ing

all

spec

ifica

tion

sin

tabl

e2

usi

nga

pu

rehe

don

icsp

ecifi

cati

on.

Eac

hd

ata

poi

ntis

aco

unt

y-ye

ar.

Reg

ress

ions

are

wei

ghte

dby

the

aver

age

num

ber

ofsa

les

ina

give

nco

unty

over

ally

ears

we

obse

rve.

Stan

dard

erro

rsar

ecl

uste

red

atth

eco

unty

leve

l.

72

against tbsi, separately for salesi equal to 2, 3 and 4, for houses with tbsi between the1st and 99th percentiles for each value of salesi. Both distortions are visible in figure11: the estimated logSD, ε̂it, is on average higher for larger values of N, and for longertime-between-sales, although as shown in Giacoletti (2017) and Sagi (2015), the interceptof ε̂it at tbsi = 0 is well above 0.

It is possible that the effects of salesi and tbsi on ε̂it partially drive the results weobserve in the main text. In order to rule out this possibility, we attempt to purge ε̂2

it ofany variation which can be explained by tbsi and salesi. We do this as follows. First,we filter to houses sold at most four times over the whole sample period, with estimatedvalues of ε̂2

it below 0.25. We then run the following regression, separately for each zipcodein our sample:

ε̂2it = gz (salesi, tbsi) + ζit

Where, gz (salesi, tbsi) interacts a vector of salesi dummies with a fifth-order polynomialin tbsi. The residual ζ̂it from this regression can be interpreted as the component of thehouse’s price variance which is not explainable by the number of times a house is soldand the average time-between-sales. We then add back the mean of ε̂2

it within zipcode z,that is:

ε̂2TBSadj,it = ζ̂it + Ez

[ε̂2it

]ε̂2TBSadj,it, where the subscript stands for “time-between-sales adjusted”, can be interpreted

as our estimate ε̂2it, nonparametrically purged of all variation which is explainable by a

flexible but smooth function of salesi and tbsi. We then re-run all analyses using ε̂2TBSadj,it

in place of ε̂it.

The line in figure 8 labeled “LogSD, TBS adj” shows the seasonal and time-seriesbehavior of ε̂2

TBSadj,it; we find that ε̂2TBSadj,it tracks ε̂2

it closely seasonally and over thebusiness cycle. Tables 19, 20, 21, and 22 re-run all zipcode and county cross-sectionalregressions using ε̂2

TBSadj,it in place of ε̂2it. All coefficients have the same signs and remain

significant, though most coefficients decline somewhat in magnitude.

B.5 Zillow vs Realtor.com time-on-market

For robustness, we repeat our primary county cross-sectional and panel regressions usingRealtor.com time-on-market instead of Zillow. Time-on-market is significantly correlatedwith price dispersion in the county cross-sectional specification of column 1. However,

73

Figure 11: Effect of salesi and tbsi on ε̂2it

Notes. Predictions from kernel regressions of ε̂2it on tbsi, separately for each value of

salesi.

74

Tabl

e19

:Zip

code

cros

s-se

ctio

nalr

egre

ssio

ns,a

djus

ted

for

num

ber

ofsa

les

and

tim

e-be

twee

n-sa

les

LogS

Dx

100

(1)

(2)

(3)

(4)

(5)

(6)

(7)

Log

ZH

VI

−3.

127∗∗∗

−2.

967∗∗∗

−0.

668∗∗∗

−1.

168∗∗∗

(0.1

69)

(0.1

57)

(0.2

07)

(0.1

73)

Tim

eon

mar

ket

(mon

ths)

1.63

8∗∗∗

1.03

0∗∗∗

2.20

6∗∗∗

1.60

4∗∗∗

(0.0

76)

(0.0

83)

(0.1

07)

(0.0

80)

Vac

ancy

rate

13.2

40∗∗∗

9.38

5∗∗∗

2.41

9∗∗∗

5.06

1∗∗∗

(0.6

71)

(0.7

36)

(0.6

55)

(0.6

48)

Pop

grow

thra

te−

1.13

2−

0.08

1−

0.69

9−

0.58

1(0

.725

)(0

.670

)(0

.493

)(0

.546

)

Con

trol

sX

XX

XX

XX

Fixe

def

fect

sSt

ate

CBS

AN

3,39

43,

514

3,51

43,

514

3,39

43,

394

3,39

4A

djus

ted

R2

0.50

30.

559

0.55

00.

500

0.57

90.

794

0.73

0

Not

es.R

e-ru

nnin

gal

lspe

cific

atio

nsin

tabl

e3

usin

gε̂TBSadj,it

inst

ead

ofε̂it

.Eac

hda

tapo

int

isa

zipc

ode.

Reg

ress

ions

are

wei

ghte

dby

the

num

ber

ofsa

les

wit

hin

the

zipc

ode

over

the

tim

epe

riod

2012

-201

6.

75

Tabl

e20

:Zip

code

-yea

rpa

nelr

egre

ssio

ns,a

djus

ted

for

num

ber

ofsa

les

and

tim

e-be

twee

n-sa

les

LogS

Dx

100

(1)

(2)

(3)

(4)

Log

ZH

VI

−0.

751∗∗∗

−1.

707∗∗∗

(0.0

63)

(0.2

90)

Log

sale

s−

0.67

8∗∗∗

−0.

069

(0.0

69)

(0.1

35)

Tim

eon

mar

ket

(mon

ths)

0.13

5∗∗

0.12

7∗∗

(0.0

59)

(0.0

61)

Zip

fixed

effe

cts

XX

XX

Year

fixed

effe

cts

XX

XX

Sam

ple

peri

od20

00-2

016

2000

-201

620

13-2

016

2013

-201

6N

51,5

7851

,578

12,2

1212

,212

Adj

uste

dR

20.

884

0.88

60.

925

0.92

6

Not

es.

Re-

runn

ing

all

spec

ifica

tion

sin

tabl

e10

usi

ngε̂TBSadj,it

inst

ead

ofε̂it

.E

ach

dat

ap

oint

isa

zip

cod

e-ye

ar.

Reg

ress

ions

are

wei

ghte

dby

the

aver

age

num

ber

ofsa

les

ina

give

nzi

pcod

eov

eral

lyea

rsw

eob

serv

e.St

anda

rder

rors

are

clus

tere

dat

the

zipc

ode

leve

l.

76

Tabl

e21

:Cou

nty

cros

s-se

ctio

nalr

egre

ssio

ns,a

djus

ted

for

num

ber

ofsa

les

and

tim

e-be

twee

n-sa

les

LogS

Dx

100

(1)

(2)

(3)

(4)

(5)

(6)

Log

ZH

VI

−3.

041∗∗∗

−3.

801∗∗∗

−2.

356∗∗∗

(0.6

79)

(0.6

80)

(0.7

89)

Tim

eon

mar

ket

(mon

ths)

0.68

7∗∗

0.13

70.

995∗∗∗

(0.3

08)

(0.3

15)

(0.3

54)

Vac

ancy

rate

12.6

10∗∗∗

16.6

87∗∗∗

9.89

9∗∗∗

(2.6

75)

(3.5

54)

(2.9

33)

Pop

grow

thra

te−

8.03

51.

043

−2.

835

(6.3

83)

(7.5

40)

(7.4

71)

Con

trol

sX

XX

XX

XFi

xed

effe

cts

Stat

eN

257

261

393

393

257

257

Adj

uste

dR

20.

494

0.45

90.

503

0.47

40.

549

0.75

4

Not

es.R

e-ru

nnin

gal

lspe

cific

atio

nsin

tabl

e9

usin

gε̂TBSadj,it

inst

ead

ofε̂it

.E

ach

dat

apo

int

isa

coun

ty.

Reg

ress

ions

are

wei

ghte

dby

the

num

ber

ofsa

les

wit

hin

the

coun

tyov

erth

eti

me

peri

od20

12-2

016.

77

Tabl

e22

:Cou

nty-

year

pane

lreg

ress

ions

,adj

uste

dfo

rnu

mbe

rof

sale

san

dti

me-

betw

een-

sale

s

LogS

Dx

100

(1)

(2)

(3)

(4)

(5)

(6)

Log

ZH

VI

−0.

591∗∗∗

−0.

805∗

(0.1

74)

(0.4

71)

Log

sale

s−

0.74

4∗∗∗

−1.

299∗∗∗

(0.1

46)

(0.3

12)

Tim

eon

mar

ket

(mon

ths)

0.23

9∗∗

−0.

021

(0.1

02)

(0.0

97)

Vac

ancy

rate

9.54

9∗∗∗

5.68

5∗∗∗

(1.0

98)

(1.6

58)

Pop

grow

thra

te−

6.63

6∗∗∗

−3.

672

(1.1

76)

(2.3

83)

Cou

nty

fixed

effe

cts

XX

XX

XX

Year

fixed

effe

cts

XX

XX

XX

Sam

ple

peri

od20

00-2

016

2000

-201

620

10-2

016

2006

-201

620

06-2

016

2010

-201

6N

6,10

36,

409

1,87

65,

544

5,11

01,

862

Adj

uste

dR

20.

924

0.92

50.

942

0.93

30.

931

0.94

4

Not

es.

Re-

runn

ing

all

spec

ifica

tion

sin

tabl

e2

usi

ngε̂TBSadj,it

inst

ead

ofε̂it

.E

ach

dat

ap

oint

isa

cou

nty-

year

.R

egre

ssio

nsar

ew

eigh

ted

byth

eav

erag

enu

mbe

rof

sale

sin

agi

ven

coun

tyov

eral

lyea

rsw

eob

serv

e.St

anda

rder

rors

are

clus

tere

dat

the

coun

tyle

vel.

78

Figure 12: Yearly and monthly variation in price dispersion, robustness checks

Notes. Figures 3 and 2, aggregating across counties weighting by the number of housingunits rather than the number of sales.

the coefficient on time-on-market is not significant in the county panel specifications incolumns 3 and 4.

Figure 8 shows yearly and monthly trends in time-on-market, measured using both Zil-low and Realtor.com. The qualitative movements of time-on-market are similar, althoughthe magnitudes and shapes of the curves differ somewhat. The difference is particularlynoticeable at the seasonal level. To our knowledge, Zillow does not publically disclose howit calculates time-on-market, or what its ultimate data source is. We believe that Zillowbuilds its dataset of house listings based on the MLS, which is the basis of Realtor.com’stime-on-market measure, but updates its listings database with some lag; this would alsoexplain why Zillow time-on-market is somewhat higher than Realtor.com’s.

B.6 Stock-weighted time series

In figures 2 and 3 of the main text, we aggregate across counties weighting by sales. Asan alternative, we can aggregate weighting by the total number of housing units in thecounty, according to CoreLogic’s tax database. We omit sales from these plots, as it isdifficult to interpret a stock-weighted average of total sales. Figure 12 shows the results ofthis alternative weighting scheme; results are qualitatively and quantitatively very closeto those in the main text.

79

Tabl

e23

:Rea

ltor

.com

vsZ

illow

tim

e-on

-mar

ket

for

coun

ties

LogS

Dx

100

(1)

(2)

(3)

(4)

Rea

ltor

.com

tim

eon

mar

ket

0.94

4∗∗∗

0.27

7−

0.00

60.

101

(0.3

02)

(0.4

25)

(0.1

81)

(0.1

63)

Log

ZH

VI

−4.

814∗∗∗

−2.

439∗∗∗

(0.8

79)

(0.7

23)

Vac

ancy

rate

21.7

97∗∗∗

5.73

4∗∗

(4.9

51)

(2.3

64)

Pop

grow

thra

te−

1.43

3−

15.3

54∗∗

(10.

010)

(6.0

45)

Log

sale

s0.

546

(0.4

14)

Cou

nty

fixed

effe

cts

XX

Year

fixed

effe

cts

XX

Sam

ple

peri

od20

12-2

016

2012

-201

620

13-2

016

2013

-201

6N

388

257

2,40

42,

120

R2

0.55

30.

628

0.95

10.

956

Not

es.E

ach

data

poin

tis

aco

unty

orco

unty

-yea

ror

coun

ty.

80

B.7 Unbalanced panel regressions

In Tables 24 and 25 below, we re-run the zipcode-year and county-year panel regressionswithout balancing the panels of zipcode and county years respectively. Results arequantitatively very similar to the balanced panel specifications in tables 2 and 10.

C Proofs and theoretical extensions

C.1 Expressions for VM (ε) , ε∗ (v) ,P (ε, v)

To begin with, we analytically characterize VM (ε). From expression (13), we have:

rVM (ε) = ε+ λM

(ˆVS (v)dF (v) − VM (ε)

)Solving for VM (ε), we have:

VM (ε) =ε

r+ λM+

λMr+ λM

ˆVS (v)dF (v) (30)

Using expression (30), we can also characterize the trade cutoff function ε∗ (v). Tradeoccurs if:

VM (ε) > VB + VS (v)

=⇒ λMr+ λM

ˆVS (v)dF (v) +

ε

r+ λM> VB + VS (v) (31)

Since we have assumed in subsection 4.1.5 that ε∗ (v) is greater than ε0, the lower boundof G(·), we can treat expression (31) as an equality. Solving for ε∗ (v), we have:

ε∗ (v) = (r+ λM) [VB + VS (v)] − λM

ˆVS (v)dF (v) (32)

Using (30) and (32) we can also characterize equilibrium prices. From (6), we have:

P (ε, v) = VS (v) + θ (VM (ε) − VB − VS (v))

81

Tabl

e24

:Cou

nty-

year

unba

lanc

edpa

nelr

egre

ssio

ns

LogS

Dx

100

(1)

(2)

(3)

(4)

(5)

(6)

Log

ZH

VI

−1.

056∗∗∗

−0.

834

(0.3

69)

(0.7

68)

Log

sale

s−

1.02

5∗∗∗

−1.

750∗∗∗

(0.1

92)

(0.5

41)

Tim

eon

mar

ket

(mon

ths)

0.52

5∗∗∗

0.17

0(0

.161

)(0

.175

)

Vac

ancy

rate

16.2

55∗∗∗

10.8

04∗∗∗

(1.7

75)

(2.4

07)

Pop

grow

thra

te−

8.83

3∗∗∗

−4.

758

(2.0

45)

(3.7

79)

Cou

nty

fixed

effe

cts

XX

XX

XX

Year

fixed

effe

cts

XX

XX

XX

Sam

ple

peri

od20

00-2

016

2000

-201

620

10-2

016

2007

-201

620

07-2

016

2010

-201

6N

10,3

6611

,120

2,54

96,

008

5,46

82,

492

Adj

uste

dR

20.

858

0.86

00.

912

0.89

50.

892

0.91

9

Not

es.R

e-ru

nnin

gal

lspe

cific

atio

nsin

tabl

e2

wit

hout

bala

ncin

gth

eco

unty

-yea

rpa

nel.

Reg

ress

ions

are

wei

ghte

dby

the

aver

age

num

ber

ofsa

les

ina

give

nco

unty

over

ally

ears

we

obse

rve.

Stan

dar

der

rors

are

clus

tere

dat

the

coun

tyle

vel.

82

Tabl

e25

:Zip

code

-yea

run

bala

nced

pane

lreg

ress

ions

LogS

Dx

100

(1)

(2)

(3)

(4)

Log

ZH

VI

−1.

201∗∗∗

−1.

703∗∗∗

(0.0

98)

(0.3

52)

Log

sale

s−

0.86

2∗∗∗

−0.

178

(0.1

20)

(0.1

69)

Tim

eon

mar

ket

(mon

ths)

0.26

3∗∗∗

0.24

3∗∗∗

(0.0

71)

(0.0

73)

Zip

fixed

effe

cts

XX

XX

Year

fixed

effe

cts

XX

XX

Sam

ple

peri

od20

00-2

016

2000

-201

620

13-2

016

2013

-201

6N

52,0

6152

,061

12,2

5712

,257

Adj

uste

dR

20.

848

0.85

00.

924

0.92

4

Not

es.

Re-

runn

ing

all

spec

ifica

tion

sin

tabl

e10

wit

hou

tba

lanc

ing

the

zip

cod

e-ye

arp

anel

.E

ach

dat

ap

oint

isa

zipc

ode-

year

.R

egre

ssio

nsar

ew

eigh

ted

byth

eav

erag

enu

mbe

rof

sale

sin

agi

ven

zipc

ode

over

ally

ears

we

obse

rve.

Stan

dard

erro

rsar

ecl

uste

red

atth

ezi

pcod

ele

vel.

83

Substituting for VM (ε) using (30), we have:

P (ε, v) = VS (v) + θ(

ε

r+ λM+

λMr+ λM

ˆVS (v)dF (v) − VB − VS (v)

)(33)

Now, we can write (32) as:

λMr+ λM

ˆVS (v)dF (v) − VB − VS (v) = −

ε∗ (v)

r+ λM(34)

Hence, substituting (34) into (33), we get:

P (ε, v) = VS (v) + θ(ε− ε∗ (v)

r+ λM

)(35)

C.2 Proof of proposition 2

From (6), prices are:

P (ε, v) = θ (VM (ε) − VB − VS (v)) + VS (v) (36)

We wish to take the variance of expression (36) with respect to the joint distribution ofholding utilities v and match utilities ε within the set of pairs of buyers and sellers thatmatch and trade in any given moment; call this joint distribution Ftr (v, ε).

First, let Ftr (v) be the marginal distribution of seller holding utilities v, among thestationary mass of seller types that trade in any given time period. By flow equality inexpression (19) of proposition 1, the marginal distribution of v among sellers who tradeand exit the market at any moment must be the same as the distribution of v amongsellers that enter the platform; thus, we simply have:

Ftr (v) = F (v) (37)

Thus, to characterize Ftr (v, ε), we need only characterize

Ftr (ε | v)

for all v; that is, the distributions of buyer match utilities, conditional on trade occurringand conditional on a given seller holding utility v. Each time a seller of holding utility v

84

meets a buyer, a random match quality ε ∼ G (·) is drawn; trade occurs if ε > ε∗ (v). Thus,

Ftr (ε | v) = G (ε | ε > ε∗ (v)) (38)

that is, the conditional distribution of match qualities ε, conditional on a seller havingholding utility v and trade occurring, is simply the distribution of ε conditional on itbeing above the trade cutoff ε∗ (v).

Having characterized Ftr (v, ε), we can now take the variance of expression (36) forprices. Applying the law of iterated expectations, price variance can be written as:

Var (P (ε, v)) = Ev∼Ftr(v)[Varε∼Ftr(ε|v) (P (ε, v) | v)

]+Varv∼Ftr(v)

(Eε∼Ftr(ε|v) [P (ε, v) | v]

)(39)

Substituting (37) and (38), we can write this as:

Var (P (ε, v)) = Ev∼F(v)[Varε∼G(ε|ε>ε∗(v)) (P (ε, v) | v)

]+Varv∼F(v)

(Eε∼G(ε|ε>ε∗(v)) [P (ε, v) | v]

)(40)

First, we characterize the left term on the RHS of (39). Conditional on v, the only randomterm in P (ε, v) conditional on v is the buyer’s match utility ε; thus, substituting expression(35) for P (ε, v) and ignoring constant terms, we have:

Varε∼G(ε|ε>ε∗(v)) (P (ε, v) | v) =(

θ

r+ λM

)2

Varε∼G(ε|ε>ε∗(v)) (ε)

In words,Varε∼G(ε|ε>ε∗(v)) (ε)

is the variance of an exponential random variable ε, conditional on ε being above somecutoff ε∗ (v), which is greater than its lower bound ε0. This conditional distribution hasvariance equal to the unconditional variance of ε, σ2

ε, for any cutoff ε∗ (v); thus, we have:

Varε∼G(ε|ε>ε∗(v)) (P (ε, v) | v) =(

θ

r+ λM

)2

σ2ε (41)

85

Since expression (41) is independent of v, we also have:

Ev∼F(v)

[Varε∼G(ε|ε>ε∗(v)) (P (ε, v) | v)

]=

(θ

r+ λM

)2

σ2ε (42)

Now we move to the right term in expression (39). Substituting expression (35) forprices, we have:

Varv∼F(v)

(Eε∼G(ε|ε>ε∗(v)) [P (ε, v) | v]

)=

Varv∼F(v)

(Eε∼G(ε|ε>ε∗(v))

[VS (v) + θ

(ε− ε∗ (v)

r+ λM

) ∣∣∣ v]) (43)

Rearranging, and moving VS (v) out of the conditional expectation, this is equal to:

Varv∼F(v)

(VS (v) +

θ

r+ λMEε∼G(ε|ε>ε∗(v)) [ε− ε

∗ (v) | v]

)(44)

Since we have assumed G (·) is exponential, and ε∗ (v) > ε0, the term:

Eε∼Ftr(ε|v) [ε− ε∗ (v) | v]

is equal to σε, the standard deviation of ε. It is thus constant with respect to ε∗ (v) andthus v, and can be ignored when calculating the variance in (44). Hence,

Varv∼F(v)

(Eε∼G(ε|ε>ε∗(v)) [P (ε, v) | v]

)= Varv∼F(v) (VS (v)) (45)

Substituting (42) and (45) into (40), we have (22).

C.3 Non-exponential match quality

Proposition 2 does not hold if we relax the assumption that match quality ε is exponentiallydistributed. Under a general match quality distribution G (ε), expression (40) becomes:

Var (P (ε, v)) =

Ev∼F(v)

[Varε∼G(ε|ε>ε∗(v)) (P (ε, v) | v)

]+ Varv∼F(v)

(Eε∼G(ε|ε>ε∗(v)) [P (ε, v) | v]

)

86

For the left term, since ε is the only random term in prices conditional on the sellerholding value v, substituting expression (33) for prices, we have:

Ev∼F(v)

[Varε∼G(ε|ε>ε∗(v)) (P (ε, v) | v)

]=

(θ

r+ λM

)2

Ev∼F(v)

[Varε∼G(ε|ε>ε∗(v)) (ε)

]That is, the left term is a constant times the expectation over v ∼ F (v) of the variance ofmatch utility ε, conditional on match utility being above the trade cutoff, ε > ε∗ (v). Thiscan still be interpreted as the average variance in buyer match utility, conditional on buyermatch utility being above the trade cutoff ε∗ (v); however, for general distributions, thisexpression does not simplify further.

For the right term, all algebra steps until (44) continue to hold, so we have:

Varv∼Ftr(v)

(Eε∼Ftr(ε|v) [P (ε, v) | v]

)=

Varv∼F(v)

(VS (v) +

θ

r+ λMEε∼G(ε|ε>ε∗(v)) [ε− ε

∗ (v)]

)(46)

This can be interpreted as the variance of seller values VS (v), plus a term

θ

r+ λMEε∼G(ε|ε>ε∗(v)) [ε− ε

∗ (v)]

which can be thought of as the average of the “markup” term ε− ε∗ (v), conditionalon ε being above the trade cutoff ε∗ (v). For general distributions G (ε), the markupdepends on ε∗ (v) and thus v, thus the markup term cannot be moved outside the varianceexpression, and (46) does not simplify further. Hence, an analog of proposition 2 whichholds for arbitrary G (ε) is:

Var (P (ε, v)) =(

θ

r+ λM

)2

Ev∼F(v)

[Varε∼G(ε|ε>ε∗(v)) (ε)

]︸ ︷︷ ︸

Buyer match value

+

Varv∼F(v)

(VS (v) +

θ

r+ λMEε∼G(ε|ε>ε∗(v)) [ε− ε

∗ (v)]

)︸ ︷︷ ︸

Seller holding value plus markup

(47)

Expression (47) also shows why the assumptions of persistent heterogeneous seller valuesis needed for the results of the paper to hold. If sellers’ holding values were homogeneous,

87

price variance would only consist of the buyer match quality term in (47), which is theexpectation over v of:

Varε∼G(ε|ε>ε∗(v)) (ε) (48)

This is the variance of ε, conditional on ε > ε∗ (v). For exponential distributions, (48) isconstant in ε∗ (v); for general distributions (48) can either increase or decrease as ε∗ (v)increases.

C.4 Proof of proposition 3

From expression (15) in proposition 1, the seller value function VS (v) is:

rVS (v) = v+ λS

ˆε>ε∗(v)

θ (VM (ε) − VB − VS (v))dG (ε)

Differentiating with respect to v, using Leibniz integral rule, we have:

rV ′S (v) = 1 − λSθ(VM(ε∗(v)) − VB − VS(v)

)g(ε)

dε∗(v)

dv+

λS

ˆθ(−V ′S (v)

)1 (ε > ε∗ (v))dG (ε) (49)

By definition of ε∗(v) in (17):

VM(ε∗(v)) − VB − VS(v) = 0

Computing the integral, (49) becomes:

rV ′S (v) = 1 + λSθ(−V ′S (v)

)(1 −G (ε∗ (v)))

Solving for V ′S (v), we have:

V ′S (v) =1

r+ λSθ (1 −G (ε∗ (v)))(50)

Substituting expression (23) for TOM (v) in the denominator of (50), we have:

V ′S (v) =1

r+ θTOM(v)

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Rearranging, we have (24).

C.5 Proof of corollary 1

From expression (24), if TOMΘ1 (v) > TOMΘ2 (v) for all v, and if r, θ are the same in thetwo sets of primitives, then V ′S (v) must also be strictly larger in absolute value, pointwisein v, under parameters Θ1 than Θ2, for all v. From expression (22) in claim 2, we have:

Var (P (ε, v)) = Varv∼F(·) (VS (v))︸ ︷︷ ︸Seller holding value

+

(θ

r+ λM

)2

σ2ε︸ ︷︷ ︸

Buyermatch value

Holding fixed G (ε), the buyer value term in Var (P (ε, v)) is also the same under the twosets of primitives. Hence, we must prove that, if V ′S (v) is strictly increased pointwise in v,then Varv∼F(·) (VS (v)) must also strictly increase.

To prove this, suppose a random variable X has some distribution function G (·). Itsvariance can be written as:

Var (X) = minx̄

ˆ(x− x̄)2 dG (x) (51)

To prove expression (51), note that:

ˆ(x− x̄)2 dG (x) =

ˆ(x− E (x) + E (x) − x̄)2 dG (x)

=

ˆ(x− E (x))2 + 2 (x− E (x)) (E (x) − x̄) + (E (x) − x̄)2 dG (x)

=

ˆ(x− E (x))2 + (E (x) − x̄)2 dG (x)

Thus,

minx̄

ˆ(x− x̄)2 dG (x) = min

x̄

ˆ(x− E (x))2 dG (x) +

ˆ(E (x) − x̄)2 dG (x)

=

ˆ(x− E (x))2 dG (x) = Var (X)

Now, call the distribution of VS (v) among trading sellers FVS (V). Using expression

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(51), we can write the variance of VS (v) as:

Var (VS (v)) = minV̄

ˆ (V − V̄

)2dFVS (V) (52)

Since the distribution of v among trading sellers is F (v), and VS (v) is a function of v, bychanging variables to integrate over v, we have:

Var (VS (v)) = minv̄

ˆ(VS (v) − VS (v̄))

2 dF (v)

Hence, (52) becomes:

= minv̄

ˆ (ˆ v

v̄V ′S (v)dv

)2

dF (v) (53)

A uniform increase in V ′S (v) causes the integral in (53) to strictly increase for any v̄. Thus,if V ′S (v) is uniformly higher under Θ1 than Θ2, then Var (VS (v)) must also increase, andthus Var (P (ε, v)) must also increase.

C.6 Heterogeneous buyer urgency

We can extend the main model to accomodate persistent buyer heterogeneity. Supposethat buyers have some persistent type u ∼ H (u), drawn at the point that buyers enterthe market. Unmatched buyers receive flow utility u per unit time they are waiting topurchase their houses. Transaction prices become a function of sellers’ holding utility v,buyers’ urgency u, and buyers’ match utility ε:

P (v,u, ε) = VS (v) + θ (VM (ε) − VB (u) − VS (v)) (54)

Thus, the match quality cutoff condition becomes:

VM (ε∗ (v,u)) = VB (u) + VS (v)

Analogous to subsection 4.1.5, for our theoretical results, we will need to assume that

ε∗ (v,u) > ε0 ∀v,u

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Buyers’ and sellers’ value functions become, respectively:

rVB (u) = u+ λB

ˆv

ˆε>ε∗(v,u)

[(1 − θ) (VM (ε) − VB (u) − VS (v))]dG (ε)dFeq (v)

rVS (v) = v+ λS

ˆu

ˆε>ε∗(v,u)

[θ (VM (ε) − VB (u) − VS (v))]dG (ε)dHeq (u) (55)

The flow equality conditions for sellers and matched owners must now integrate over theequilibrium distribution Heq (u) of buyer urgencies:

(1 −MS) λMf (v) = λSMSfeq (v)

ˆu[1 −G (ε∗ (v,u))]dHeq (u)

Geq (ε) =

´u

´v λSMS

[´ εε̃=ε0

1 (ε̃ > ε∗ (v,u))dG (ε̃)]dFeq (v)dHeq (u)´

u

´v λSMS (1 −G (ε∗ (v,u)))dFeq (v)dHeq (u)

Moreover, there is an additional flow equality constraint requiring inflows and outflowsof all buyer types to be equal:

ηBh (u) = λBMBheq (u)

ˆv[1 −G (ε∗ (v,u))]dFeq (v)

Somewhat surprisingly, despite these changes to stationary equilibrium conditions, propo-sitions 2 and 3, and corollary 1, continue to hold. To prove proposition 2, note that, whenbuyers have heterogeneous values, the matched owner value function is unchanged:

VM (ε) =ε

r+ λM+

λMr+ λM

ˆvVS (v)dF (v) (56)

The derivations in appendix C.1 thus imply that:

P (v,u, ε) = VS (v) + θ(ε− ε∗ (v,u)r+ λM

)(57)

Now, similar to (39), we take the variance of prices, applying the law of iterated expecta-tions with respect to v and u, to get:

Var (P (v,u, ε)) = Ev,u∼Ftr(v,u)[Varε∼Gtr(ε|v,u) (P (v,u, ε) | v,u)

]+

Varv,u∼Ftr(v,u)

(Eε∼Gtr(ε|v,u) [P (v,u, ε) | v,u]

)(58)

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Where, analogously to expression (39), Ftr (v,u) is the joint distribution of v and u amongtrading buyers and sellers, and Ftr (ε | v,u) is the conditional distribution of ε given v,uamong trading buyers and sellers. Analogously to the argument to appendix C.2, wehave:

Ftr (ε | v,u) = G (ε | ε > ε∗ (v,u))

The joint distribution Ftr (v,u) is more complicated to characterize; however, by flowequality, the marginal distributions of Ftr (v,u) must be equal to the distributions ofentering buyer and seller types, F (v) and H (u). This implies that the following steps inappendix C.2 go through essentially unchanged. Going through the steps, for the topterm of (58), we have:

Varε∼G(ε|ε>ε∗(v,u)) (P (v,u, ε) | v,u) =(θ

r+ λM

)2

Varε∼G(ε|ε>ε∗(v,u)) (ε) =

(θ

r+ λM

)2

σ2ε

For the bottom term, substituting expression (57) for prices, we have:

Varv,u∼Ftr(v,u)

(Eε∼G(ε|ε>ε∗(v,u)) [P (v,u, ε) | v,u]

)= Varv,u∼Ftr(v,u)

(Eε∼G(ε|ε>ε∗(v,u))

[VS (v) + θ

(ε− ε∗ (v,u)r+ λM

)| v,u

])

= Varv,u∼Ftr(v,u)

(VS (v) +

θ

r+ λMEε∼G(ε|ε>ε∗(v,u)) [ε− ε

∗ (v,u) | v,u])

(59)

Again, the left term of (59),

Eε∼G(ε|ε>ε∗(v,u)) [ε− ε∗ (v,u) | v,u]

is equal to σε, which is independent of v,u, so we can ignore it in the variance calculation;(59) thus simplifies to

Varv,u∼Ftr(v,u) (VS (v))

which is independent of u, so this simplifies further to the variance with respect to themarginal distribution of v, that is,

Varv∼F(v) (VS (v))

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This proves proposition 2. Similarly, differentiating (55), we have:

rV ′S (v) = 1 + λSθ

ˆu

ˆε>ε∗(v,u)

(−V ′S (v)

)1 (ε > ε∗ (v,u))dG (ε)dHeq (u) (60)

The total match rate facing a seller of type v is the inverse of time-on-market, so we have:

TOM (v) =1

λS´u

´ε>ε∗(v,u) 1 (ε > ε∗ (v,u))dG (ε)dHeq (u)

(61)

Combining (60) and (61), we have:

V ′S (v) =1

r+ θTOM(v)

=TOM (v)

rTOM (v) + θ

proving proposition 3. Finally, the proof of corollary 1 follows unchanged from theargument of appendix C.5.

C.7 Derivation of model quantities

In this appendix, we derive expressions for average prices, time-on-market, and pricedispersion, which are plotted in figure 4. The average transaction price conditional ontrade is:

ˆ ˆP (ε, v) dG (ε | ε > ε∗ (v)) dF (v)

This is the expectation of the price function P (ε, v) over the joint distribution of ε, v amongsuccessfully trading buyers and sellers.

Average time-on-market, over the distribution of realized sales, is:

ˆTOMS (v) dF (v)

This is simply the average of time on market for a seller of holding utility v over thedistribution of holding utilities F (v); note that we showed in appendix C.2 above that thedistribution of holding utilities among trading sellers is simply Ftr (v) = F (v).

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Claim 2 characterized equilibrium price variance as:

Varv∼F(·) (VS (v)) +

(θ

r+ λM

)2

σ2ε

C.8 Comparative statics

To create figure 4, we solve the model under the following assumptions on parameters:

r = 0.0526,α = 1,φ = 0.84, θ = 0.5,

λM = 0.06,ηB = 0.03,

v̄ = 1.5,∆v = 2.3, ε0 = 7.5,σε = 3.3

To make figure 4, we increase and decrease each parameter from its baseline value whileholding all other variables fixed at their baseline values.

D Calibration details

D.1 Computation

Computationally, we solve the model by varyingMB rather than ηB; this is computationallysimpler, and for any equilibrium in terms of MB, we can back out an ηB which implementsthis equilibrium, through (21). Given a vector of parameters r,α,φ, θ, λM,MB, v̄,∆v, ε0,σ2

ε

we solve the model by iteratively solving the Bellman equations and flow equality condi-tions until convergence. Given guesses for feq (v) ,MS, we calculate λS and λB, and thennumerically solve (14), (15), (16) for VS (v) ,VM (ε) ,VB, ε∗ (v). Given guesses for the tradecutoff ε∗ (v), we can then use (19) to solve for feq (v). We iterate these equations, updatingin a penalized matter; if the result of one iteration on MS implies some new value M̃S, forthe next iteration, we update MS to:

(1 − t)MS + tM̃S

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For small enough t, the iteration converges. While we were not able to prove thatthe model admits a unique solution, in our simulations, the model reached the sameequilibrium point from many different starting values.

D.2 Log price variance approximation

In our data, idiosyncratic price variance corresponds to the residual from a regression inwhich the dependent variable is the log sale price; hence, the residual can be interpretedas the variance of log prices. In the model, the variance of prices is computationally easyto calculate, using the analytical result of proposition 2, but the variance of log prices ismore complex. For computational simplicity, in generating the variance of log prices inthe model, we use the following linear approximation, based on the Taylor expansion oflog (P) around its mean P̄:

Var (log (P)) ≈ Var(

log(P̄)+P− P̄

P̄

)= Var

(P− P̄

P̄

)=

1P̄2Var (P)

Hence, we generate the variance of prices as the variance of model-generated prices,divided by the squared mean of model-generated prices, where we calculate the varianceof model-generated prices using expression (22) of proposition 2.

D.3 Alternative calibration method: choosing σ2ε to target total price

variance

As an alternative way to calibrate parameters, rather than matching the dispersion ofbuyer values using data from Anundsen, Larsen and Sommervoll (2019), we choose σ2

ε sothat model-induced price dispersion perfectly matches observed price dispersion in thedata. All other steps of the calibration proceed as in the main text.

Table 26 shows parameter estimates under this alternative approach. We estimate thatmatch quality ε, per month of homeownership, is drawn from an exponential distributionwith a lower bound of -$208, and a standard deviation of $505. Conditional on buying thehouse, the average ε that buyers attain is around $1,450 per year. Overall, buyers attainan average value of $175,480 from homeownership before receiving the separation shock.

We find that seller utilities per month are drawn from a uniform distribution withmean of −$9,805 per month, and a standard deviation $1,477. This implies that sellers

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Table 26: Moment and parameter values, alternative calibration

Moment Value Parameter ValuePrice 191.99 λM 0.047

LogSD 16.73% ε0 -0.2075TOM (Months) 3.197 σε 0.5054

Turnover rate 0.046 ηB 0.046Num. visits 9.960 v̄ -9.086

PD-TOM Corr 0.528 ∆v 2.558

Notes. Target moments and estimated parameter values. ε0,σε, v̄,∆v are reported inthousands of US dollars per month. Turnover rate and λM are yearly turnover andseparation rates respectively, and ηB is a fraction of the unit mass of houses per year.

experience an average loss of around $28,307 during the entire time that they spend onthe market. These estimates are somewhat unrealistically large; intuitively, this is becausewe estimate much higher dispersion in buyers’ values in this version of the calibration,increasing the option value to sellers of waiting for high-value buyers. To counterweightthese increased incentives to wait and to match the observed time-on-market in the data,we infer that sellers’ average costs of keeping their houses on the market must be veryhigh. The estimated dispersion is sellers’ utilities, however, does not change that muchcompared to our estimates in the main text: $1,477 vs $1,114.

The liquidity discount numbers also increase somewhat relative to the baseline cal-ibration. The p75-p25 gap in expected prices is $15,632, and the gap in expected time-on-market is 0.8 months. The liquidity discount decreases from $20,058 when averagetime-on-market is 4.17 months to $12,319 when average time-on-market is 2.49 months.

Under this version of the calibration, idiosyncratic price variance in the model and dataconcide, so the variance components in (25) exactly add up to the squared average logSDin the in the data. We find that the square root of the seller holding value component ofprice variance is 4.76% of average prices, and the square root of the buyer component is16.0% of prices. In terms of variance fractions, 8.16% of idiosyncratic price variance isattributable to seller values, and 91.84% to buyer values.

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Figure 13: Liquidity discounts, alternative calibration of σ2ε

Notes. Plots in figure 6 under our alternative calibration methodology. The left panelshows the average price and time-on-market achieved by sellers with different holdingutilities v. The red stars represent, from left to right, the 25th, 50th, and 75th percentilesof seller holding utilities. The right panel shows, as we vary market tightness using ηB,the difference in expected prices attained by sellers with 75th and 25th percentile valuesof v vs average time-on-market.

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