2013 V41 4: pp. 887–924 DOI: 10.1111/1540-6229.12020

REAL ESTATE ECONOMICS

Is Selection Bias Inherent in Housing Transactions? An Equilibrium Approach Anna Chernobai* and Ekaterina Chernobai** We develop an equilibrium model for residential housing transactions in an economy with houses that differ in their quality and households that differ in their planned holding horizon. We show that, in equilibrium, a clientele effect persists, with long-horizon buyers overwhelmingly choosing higher quality properties and short-horizon buyers settling for lower quality properties. This clientele effect creates a sample selection bias: the properties that are on the market are predominantly of lower quality. Since these are the preferred choice of short-horizon buyers, they demonstrate a faster turnover. Both the clientele effect and the selection bias are more pronounced with an increase in the variance of house quality and in the variance of the planned holding horizon. Our theoretical model supports empirical evidence on the existence of such bias in home price indices and explains it by the differences in ex ante holding horizons.

The majority of the literature on residential housing pricing and liquidity focuses on sell-side variables such as physical, location and neighborhood attributes of the property, as well as seller characteristics. However, pricing and liquidity are likely to also be influenced by buyer-side characteristics, regarding which the literature has been more silent. As a result of buyer-side differences, clienteles may form in that a certain class of houses would be preferred over other classes by a particular type of buyers. Clienteles may differ in their holding periods. In this case, the types of properties on and off the market would be systematically misrepresenting the entire housing stock. This phenomenon has been observed empirically and is commonly referred to as “sample selection bias” or “transaction bias” present in house price indices. Repeat sales house price indices attempt to correct for this bias by controlling for the time between sales; see, for example, Bourassa, Hoesli and Sun (2006), Case, Pollakowski and Wachter (1991), Case and Shiller (1987), Costello and Watkins (2002), Dreiman and Pennington-Cross (2004), Englund, Quigley and Redfearn *M.J. Whitman School of Management, Department of Finance, Syracuse University, Syracuse, NY 13244 or [email protected]. **California State Polytechnic University Pomona, Pomona, CA 91768 or [email protected].  C

2013 American Real Estate and Urban Economics Association

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(1999), Goetzmann (1992), Jansen, de Vries, Coolen, Lamain and Boelhouwer (2008) and Shiller (1991). This bias reported in past empirical studies is based on realized holding periods observed ex post at the time of a repeat sale of properties. The realized holding period can be viewed simply as the expected holding horizon plus a noise component that was unexpected at the time of home purchase. This paper develops a theoretical equilibrium model of housing transactions that investigates the determination of the selection bias ex ante. In our model, households differ in their expected holding horizons at the time of home search, and they visit houses that differ in quality. Therefore, if the clienteles form at equilibrium in which buyers with certain planned horizons predominantly buy houses of a certain quality, this would provide theoretical evidence of the bias ex ante. This would mean that the selection bias is an inherent feature of the residential housing market, with properties for sale systematically underrepresenting the entire housing stock. This would have implications for empirical research and may help predict selection bias in future house price indices based on currently observed buyer clienteles. One can think of many factors that influence the expected holding horizon; they include expectations regarding future income uncertainty and job stability, relocation plans, family size and age. For instance, high income uncertainty would shorten the expected holding horizon and vice versa. As such, a household that foresees job termination or relocation in a certain number of years would have an expected holding period equal to the same number of years. Similarly, a large household is less mobile and is more likely to remain in a house for a longer period of time than a smaller household. Other examples of short-tenure households would be singles and professionals and examples of long-tenure households would be retirees. Due to these household-specific factors, each household’s expected holding horizon is fixed (exogenous) at the time of house search. It influences a buyer’s decision regarding the type of property that would fit her or him best. Of course, various exogenous shocks (such as unexpected income shock or a sudden relocation) may cause a household to sell the property earlier or later than expected. In other words, the actual occupancy period observed ex post may depart from the expected one known ex ante. The events that are expected at the time of house search are already incorporated in the expected holding horizon, but unexpected events do not influence expected holding horizon and therefore do not impact the buyer’s decision-making process at the time of the house search. Not only are households heterogeneous in their expected holding horizons, but they also have idiosyncratic valuations of houses. A house feature that is highly valuable to one household may be worthless to another household. A buyer

Is Selection Bias Inherent in Housing Transactions? 889

is willing to accept time-related search costs with the hope of finding a house with a higher net gain. A seller is also willing to keep house prices sufficiently high as he or she may be rewarded with a visit of a buyer who assigns a higher value to his or her house and wishes to pay the offered price. Therefore, delay in transaction has value on both sides of the market, and an optimal time until transaction is different from zero, making the real estate market illiquid. The compensation for illiquidity does not have to come through bidding the price down (as is done for financial assets under market clearing). When faced with houses of different levels of illiquidity, it is possible that a buyer pays more for less liquid real estate assets, for as long as her or his net life-time utility of housing services is higher.1 In empirical evidence presented by Bourassa, Haurin, Haurin, Hoesli and Sun (2009) and Haurin (1998), higher value houses are priced more expensively and spend a longer time on the market, thus exhibiting lower liquidity. Haurin (1998) provides evidence of higher prices for homes with more valuable, unique features that take longer to sell. The model that we develop in this paper accounts for all the features of the residential real estate market described above, and it shows that the clienteles that form at equilibrium are causing the selection bias. We generalize the simple search-and-match model developed by Krainer and LeRoy (2002), in which housing liquidity and selling price are jointly determined. In their model, (a) households are identical in their expected holding period preferences, (b) there are no ex ante qualitative differences across houses and (c) households make a decision to buy now or to keep searching upon visiting a single house in that period. We explore the model under more realistic conditions by relaxing these assumptions. Our model has households of short and long expected holding horizons at the time of their house search, has two types of houses that differ in the potential level of services they can offer and assumes that in every period buyers make a buying decision upon visiting two houses. A buyer’s objective is to maximize his or her net expected utility from purchasing a house by choosing an optimal fit with housing services. A seller’s objective is to maximize the value of the house by setting an optimal price. Our model shows the presence of a clientele effect at equilibrium. An agent who makes sequential decisions about purchasing a house for a longer term would demand a higher net expected life-time utility. Such agents would outbid those with shorter expected homeownership durations for houses that offer better 1 This is in contrast to the market for financial assets, such as stocks and bonds, in which asset homogeneity and market centralization make market clearing optimal. There, illiquidity is caused by constrained supply, limited information, etc. and it is captured in the transaction cost, i.e., the bid-ask spread. Less liquid securities cannot be traded quickly when liquidity needs arise, and so investors require higher returns by bidding the prices down.

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services, cost more and are relatively less liquid. A higher net expected utility compensates for the higher price and the lower liquidity. The clientele effect that is predicted by our model has direct implications for the composition of houses that are on and off the market. Our model explains why properties that are for sale would always be dominated by properties that are on average of lower quality (and, therefore, of lower price) rather than properties that are off the market, supporting the vast empirical evidence of the sample selection bias in house price indices. We refer to this bias as an inherent selection bias, as we show it to be a natural attribute of a housing market with heterogeneous houses transacted by buyers with different planned holding horizons. To our knowledge, this is the first theoretical equilibrium model that demonstrates that the selection bias in house price indices may be predetermined ex ante at the time of house purchase. The structure of the remainder of this paper is as follows. The next section presents a review of relevant literature. We then present a theoretical model of housing transactions with two types of houses and two classes of households; all necessary derivations are provided in the appendix. Equilibrium results follow. The final section summarizes the findings and concludes. Related Literature Our study is related to a body of literature on transaction bias (also known as sample selection bias) in house price indices (HPI). Transaction bias occurs when the properties on the market that are included in the estimation of a HPI misrepresent the entire housing stock. Common methods for computing HPI are the median price (e.g., Crone and Voith 1992, Gatzlaff and Ling 1994, Mark and Goldberg 1984, Wang and Zorn 1997) and mean price of houses on the market. A more advanced index is based on a hedonic regression method that controls for heterogeneity of dwelling properties and estimates the marginal contribution of housing attributes, such as physical attributes and location, on selling prices of the properties (e.g., Musgrave 1969, Gatzlaff and Ling 1994, Gelfand, Ecker, Knight and Sirmans 2004). Another index is a repeat sale index originally developed by Bailey, Muth and Nourse 1963 (see also Goetzmann 1992, Case and Shiller 1987, 1989, Dreiman and Pennington-Cross 2004, Shiller 1991). This index eliminates a potential specification error in the hedonic regression models by looking at the price change between two dates of sale of the same property, thereby holding its quality constant, at least in theory. A variation of the traditional repeat sale index is the interval-weighted repeat sale index, originally developed by Case and Shiller (1987), which gives lower weights to properties with longer times between transactions (see also Shiller 1991 and Dreiman and Pennington-Cross 2004). A commonly used S&P/Case-Shiller HPI is both interval-weighted and value-weighted (i.e., a

Is Selection Bias Inherent in Housing Transactions? 891

higher weight is assigned to higher-valued houses). This methodology corrects for heteroscedasticity caused by the increase in house prices generally observed over time. While the hedonic and the repeat sale methods improve certain statistical accuracies of house price indices, they significantly limit the number of observations by excluding those properties that have stayed off the market beyond the sample period. This problem is particularly acute in the repeat sale method that excludes all single-sale transactions. Some recent studies have used hybrid models (e.g., Case and Quigley 1991, Case, Pollakowski and Wachter 1991, Knight, Dombrow and Sirmans 1995, Clapp and Giaccotto 1998), which apply the repeat sale technique on multiple-sale properties and also include single-sale properties whose housing characteristics are controlled for using the hedonic methodology. An additional stream of literature studies the qualitative differences of dwellings that are on and off the market. Empirical evidence shows that more frequently traded residential properties are of lower quality. Therefore, houses with singlesale transactions, or those off the market and therefore not part of estimated HPIs, are of higher quality. Costello and Watkins (2002) and Bourassa, Hoesli and Sun (2006) find that houses that are sold more frequently tend to be smaller, cheaper, and in poorer condition, underrepresenting the aggregate housing stock. Englund, Quigley and Redfearn (1999) found that the repeat sales price index in the Swedish residential housing market tends to track smaller, more modest homes that transact more often. Jansen, de Vries, Coolen, Lamain and Boelhouwer (2008) document the bias in residential property transactions in the Netherlands: 30% of the apartments (i.e., low quality) were sold at least twice during the period of study, while the proportion of detached homes (i.e., high quality) sold at least twice was at a mere 7%. Along the same lines, Goodman and Thibodeau (1995, 1998) evidence that home improvements are more extensive and more likely to be done by the dwelling’s seller who has occupied the property for a longer period, causing the value of less frequently transacting dwellings to be higher. In the empirical studies summarized above, selection bias is explained ex post using observable characteristics of houses and the time elapsed between transactions. The contribution of our model is that it explains the formation of the bias by buyers’ different ex ante expected holding horizons. Several empirical studies have looked at the effect of buyer ex ante holding period on their housing choice. Ermisch, Findlay and Gibb (1996) showed that age, income and household structure are the key determinants of tenure choice. Clark and Onaka (1983) showed that tenure determines the type of housing a

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homebuyer picks. The life-cycle theory of housing demand has linked tenure to buyer characteristics such as household age, mobility, marital status and presence of children; see, for example, McCarthy (1976) and Myers and Pitkin (1995). Myers and Pitkin (1995, p. 501) found that “high-turnover homes are likely to be less pleasing to their occupants and may command lower values” and are owned by high-mobility households, while older households have lower mobility and are more likely to occupy for longer periods homes with special attributes. Closely related to our work are a number of theoretical studies on pricing models of housing transactions. Krainer and LeRoy (2002) develop a simple searchand-match model with homogeneous houses and households, in which housing liquidity and selling price are jointly determined; see also Krainer (1999, 2001). Williams (1995) develops a continuous-time theoretical model of valuation of real assets based on bilateral bargaining between buyers and sellers. In his model houses and buyers are ex ante homogeneous and there is no information asymmetry between buyers and sellers prior to bargaining—assumptions that we relax in our model. Wheaton’s (1990) bargaining model focuses on vacancy rate determination. He has two types of households—singles and families— and two types of houses—small and large. In his model, singles always reside in small houses and families always occupy large homes. Hence, his clienteles are not endogenously determined, which is in contrast to our model. In this work, we have two classes of households and two types of houses. While we do not allow bargaining in our model, housing choices, prices, expected time to buy and time on the market are simultaneously determined in a simple Nash equilibrium setting, producing endogenous clientele effects. The Model The model developed in this paper is a theoretical model of residential real estate transactions. Our model builds on the sequential search-and-match model proposed in Krainer and LeRoy (2002). All households either already own a house and are enjoying the housing services (i.e., are matched) or are currently searching for one (i.e., are unmatched). As such, this is a model for house choice rather than for a homeownership decision. We begin with listing seven key assumptions and then proceed to describing the model in detail. Assumption 1. The economy consists of an equal number of houses and households.2 2 Households consume two goods: housing services and a background consumption good that serves as a numeraire. In this model, the selling price is expressed in the units of the background good rather than in some currency units.

Is Selection Bias Inherent in Housing Transactions? 893

Assumption 2. Households are of two distinct classes, long-tenure and shorttenure, equally represented in the economy. Assumption 3. The housing market consists of two types of houses, good and bad, equally represented in the economy. Assumption 4. Prospective buyer visits two houses randomly. Assumption 5. In every period, a household can buy only one house. Assumption 6. Households are risk-neutral. Assumption 7. A seller offers her or his house for sale for a take-it-or-leave-it price. Heterogeneous Households and Houses The economy consists of an equal number of houses and households (Assumption 1). Heterogeneity in households is observed in two ways. First, we assume that the population of households is dominated by two distinct classes: i = S, L, short-tenure (S) and long-tenure (L), respectively (Assumption 2). The two classes are equally represented in the population. Switching between tenure classes is allowed as long as it happens at the same rate for both classes.3 Short-tenure households expect to stay in a house for a short period of time and long-tenure households expect to remain in a house for a longer period of time. Examples of short-tenure households would be singles and professionals, and examples of long-tenure households would be families and retirees. The difference in expected housing tenure can be expressed using probabilities of preserving the match with the housing services during each period of homeownership, with the probability for long-tenure households being higher than for short-tenure households: π L > π S.

(1)

These probabilities are independent across periods. One can view the parameter π as a reparameterization of buyers’ planned occupancy period at the time of the search, known at the time of the house search, i.e., expected tenure E T = 1/(1 − π ). Of course, ex post the actual occupancy period may be altered by unexpected events, such as an income shock or employment change. However, the events that are expected at the time house search is taking place are already incorporated in the parameter π .

3 This rate is already indirectly accounted for by the expected holding horizons. When switching rates are unequal, there would be only one class of buyers remaining at steady state, thus eliminating the opportunity to observe clientele effects.

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In this model, housing options are non-homogeneous. We assume that the housing market is dominated by two types of houses, higher-quality and lowerquality (Assumption 3). We will refer to them as “good” and “bad” and denote them by H j , j = G, B, respectively. Each house type is equally represented in the economy. The second source of heterogeneity of households is present in the fits that they observe. These fits, εij , for j-houses observed by i-agents are assumed to be distributed uniformly on a certain interval (see Krainer and LeRoy 2002) reflecting heterogeneity in households’ preferences towards housing services. Fit is specific to the pair—house, agent—and stays constant during each period of homeownership. Houses differ in the maximum level of fit that each can provide to prospective homeowners. The assumption of two types of houses allows for their price levels to be different at equilibrium. Without loss of generality, we rescale these fits: we take the minimum to be zero for both house types but the maximum fit for good houses to be one and for bad houses to be θ, 0 < θ ≤ 1. The parameter θ is exogenous. We have:4 εGi ∼ U [0, 1],

i = S, L ,

(2a)

εiB ∼ U [0, θ ],

i = S, L .

(2b)

The supports of the two distributions overlap for the following two reasons. First, this satisfies the model assumption that the fit drops to zero if the match with housing services during homeownership is lost, regardless of the house type, as will be discussed next. Second, it allows for the possibility that, when both houses are visited, certain features of a bad house make it more preferable than a good house.

Buyer’s Problem A household owns its house for as long as the match with the housing services continues. If it is lost (due to job relocation, change in family size, change in income stream, etc.), the fit drops down to zero and the household starts looking for a new house. At this point, the homeowner becomes unmatched. He or she becomes simultaneously a seller (of the old house) and a buyer (of a new house). While being matched with a house, searching for another house

4 The choice of the uniform distribution not only reflects the high heterogeneity among households but also allows tractable equilibrium equations to be obtained later on.

Is Selection Bias Inherent in Housing Transactions? 895

in the hope of finding a higher fit is not allowed, thus implying a certain level of irreversibility of the buying decision. Upon visiting a house, a household considers it for purchase if the fit is at least as high as the reservation fit: εij ≥ ε¯ ij . From the uniform distribution it follows that the probability of this event is (1 − ε¯ ij ). The reservation fit depends positively on the house price offered by the seller. We assume that in every period, a prospective buyer visits two houses randomly (Assumption 4) and can buy at most one (Assumption 5). Visiting two homes allows for choices. We impose randomness due to two considerations. The first consideration is that randomization ensures that the clientele effect is not present ex ante. One may argue that house visits cannot be random if, for example, a household has preferences towards detached houses (good homes) rather than attached houses (bad homes). However, first, if there are, say, 10% detached and 90% attached houses on the market, then the household is likely to include the latter into their search since they dominate the market. Therefore, visiting houses at random is equivalent to saying that house buyers are aware of the ex ante quality of houses in the market but visit each house type with the probability equal to their relative presence in the market. Second, and more importantly, assigning buyer classes to certain property types would not allow for a joint determination of equilibrium values of variables for each house type (such as house price, time on the market, time until purchase and other endogenous variables), i.e., it would not allow transaction bias to be modeled ex post. The second consideration is that prospective buyers are not fully informed about the true home values, as residential real estate markets generally lack transparency. In residential real estate markets, buyers face two broad types of informational asymmetry5 (Garmaise and Moskowitz 2004): (a) sellers have superior information about local market conditions, including economic and social dynamics, local government regulations and environment, and (b) sellers have more accurate information about the condition of the property than do the buyers.6 In our economy, households are risk-neutral (Assumption 6). The common discount factor is β (0 < β ≤ 1). For any observed fit εij by an i-agent in a

5 Our model does not preclude property brokers; empirical evidence suggests that real estate brokers do not necessarily help in reducing this information asymmetry (Garmaise and Moskowitz 2004). 6 Dolde and Tirtiroglu (1997) and Garmaise and Moskowitz (2004) use geographical distance between buyers and properties to explain some asymmetries in information.

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Figure 1



Timeline for utility determination.

(ε)

(ε)

1

1

s*

s*

q

q

Note: This figure illustrates the determination of utility in time. Once a home is purchased in period 0, housing services are consumed beginning in period 1. If in period 1 the match is preserved with probability π , then housing services are consumed in period 2, and so on. If in period 1 the match is broken with probability (1 − π ) then the household assumes positions of home buyer, in which case it acquires the value of the search option s ∗ and of home seller, in which case it assigns its home a value of q. The same repeats in periods 2, 3 and so on.

j-house, the life-time utility from owning a house is        ν εij = βεij + β π i ν εij + (1 − π i ) s i∗ + q j .

(3)

The determination of ν(εij ) for an arbitrary home is illustrated in Figure 1. Equation (3) states that if a house is purchased, the matched homeowner i begins enjoying housing services in the amount of εij that is collected at the end of each period. If the match persists (with probability π i ), she or he maintains utility. Otherwise she or he becomes unmatched (with probability (1 − π i )). In this case she or he simultaneously assumes the roles of a buyer and a seller and owns the search option s i and the selling option q j that will be explained later.7

7

It is convenient to think of life-time utility as a perpetual stream of per-period values to the owner (akin to a stream of cash flows from holding a liquid financial asset). Expanding Equation (3),   βεij + β(1 − π i )(s i∗ + q j ) ν εij = , 1 − βπ i

i = S, L ,

j = G, B.

Is Selection Bias Inherent in Housing Transactions? 897

We say that an i-household has an option to search for a house during each period with value     i   i ε¯ G + 1 ε¯ B + θ i i i − pG + μ B ν − pB s = μG ν 2 2 (4)   + β 1 − μiG − μiB s i∗ , i = S, L . The variable p j denotes the respective house price. The asterisk in s i∗ denotes the optimal search option value in the next period. With probability μiG , a household buys a good house with the net expected utility as shown by the first summand of Equation (4).8 Similarly, with probability μiB , a household buys a bad house with the corresponding net expected utility as shown by the second summand. If neither house is selected for purchase, the search continues for another period with probability (1 − μiG − μiB ), at which time the search option is evaluated again. The chance of a delayed house purchase discounts the current value of the expected net gain by β, thus making house search costly. A buyer faces a trade-off between a longer (shorter) search and finding a house with a higher (lower) observed fit. Taking house prices as given, for each house type a buyer’s problem is to choose an optimal reservation fit that would maximize the expected net life-time utility, i.e., the search option value:

i∗ i∗  (5) ε¯ G , ε¯ B = arg max s i , i = S, L . i ε¯ G ,¯εiB

We now proceed to explain the buying probability. In a given period, the probability that an i-household buys a j-house is given by: μij = l ij a j ,

i = S, L ,

j = G, B.

(6)

The variable l ij denotes the probability that an i-household likes a j-house, and a j denotes the conditional probability that the j-house is available given that the prospective buyer liked it. The term a j is necessary because the purchase is not guaranteed in the event that more than one buyer wishes to buy a particular house. We refer to a j as the availability factor. We now elaborate on how the variable l ij is computed. It is computed as: l ij =

2

P(visited n H j )P i (liked H j |visited n H j ), i = S, L ,

j = G, B. (7)

n=0

  The expected life-time utility from owning a home equals ν (¯εGi + 1)/2 where the argument (¯εGi + 1)/2 is the expected fit conditional on it exceeding the reservation fit. This result follows from the assumptions of risk neutrality and the uniform distribution of fits. 8

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Here, P(visited n H j ) is the probability of visiting a j-house n times, where n = 0, 1 or 2. It is based on binomial trials with probabilities of “success” and “failure” denoted as λG and λ B , respectively, where λG is the probability of visiting HG and λ B is the probability of visiting H B . The variables λG and λ B are essentially the relative proportions of each house type on the market and are endogenous in this model.9 In equilibrium, these proportions are computed as:     (1−π S ) 1−π L +μ LB +μ SB 1−π L +μ LB −μGL μGS  ,   λG = (8a) (1−π S ) 1−π L +μGL +μ LB +(1 − π L ) 1−π S +μGS +μ SB

λ B = 1 − λG .

(8b)

The derivations are provided in the appendix. They rely on the assumptions regarding equal representations of two classes of households and two types of homes, and a Markov chain process of period-by-period dynamics of matched and unmatched households. The conditional probability P i (liked H j |visited n H j ) on the right-hand side of Equation (7) depends on the relation of the observed fits to the reservation fits for each of the two houses visited. We assume that if an agent visits both types of houses and their observed fits exceed the respective reservation fits, then the agent picks the house with the highest observed fit. The desired probabilities of liking each house type become:   2    ε¯ i + 2λG λ B 1 − ε¯ Gi B l Gi = λ2G 1 − ε¯ Gi θ   i   ε¯ 1 − B P i (HG  H B ), + 2λG λ B 1 − ε¯ Gi θ

l iB

=

2 

  ε¯ i + 2λG λ B ε¯ Gi 1 − B θ     ε¯ i + 2λG λ B 1 − ε¯ Gi 1 − B P i (H B  HG ). θ

λ2B



1−

ε¯ iB θ

(9a)

(9b)

The derivations are provided in the appendix. In Equation (9a), the term P i (HG  H B ) denotes the conditional probability that an i-household prefers 9 This is not to be confused with relative proportions of each house type in the entire housing stock, which are assumed equal.

Is Selection Bias Inherent in Housing Transactions? 899

a good home to a bad home when both are visited and both observed fits are in excess of the reservation fits:    P i (HG  H B ) := P εGi − εiB > 0εGi ≥ ε¯ Gi , εiB ≥ ε¯ iB . (10) The term P i (H B  HG ) in Equation (9b) is analogously defined. Its analytical expression is provided in the appendix. It involves integration of the joint density of the fits over all feasible values of εGi and εiB and the relation between ε¯ Gi , ε¯ iB and θ . Finally, the availability factor a j , j = G, B, in Equation (6) captures the role that competition plays on a prospective household’s ability to purchase a home. It is necessary to account for competition because real assets are scarce. The term a j is inversely related to the proportion of all households who visited a j-house during the same period and liked it. For example, suppose a household liked a house but another visitor also liked it; then we assume that there is a onehalf chance of the household in question actually buying it. Similarly, if two additional visitors liked it, then there is a one-third chance that the household in question will buy it, and so on. The competition outcome is determined at the end of the period.10 Thus, we have: aj =

1 − ex p(−ϕ j ) . ϕj

(11)

The proof is provided in the appendix. It relies on the Poisson distribution with intensity rate (i.e, the expected number of visitors to a home) equal to two and the binomial distribution of the number of those visitors who liked the home. In Equation (11), ϕ j denotes the market average probability of liking a house: ϕ j := ρ S l Sj + ρ L l Lj ,

(12)

where the weights, ρ S and ρ L , denote the relative proportions of short-term and long-term buyers. These proportions are determined endogenously and are functions of expected housing tenure and per-period probabilities of buying each house type. In equilibrium, they are computed as:   (1 − π S ) 1 − π L + μGL + μ LB   , (13a)   ρS = (1 − π S ) 1 − π L + μGL + μ LB + (1 − π L ) 1 − π S + μGS + μ SB

ρL = 1 − ρS.

10

We impose these assumptions for simplicity and better model tractability.

(13b)

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The derivations are provided in the appendix. They rely on the assumptions regarding equal representations of two classes of agents and two types of homes, and a Markov chain process of the period-by-period dynamics of the matched and unmatched households. Seller’s Problem If a match with the current house is lost, the unmatched homeowner immediately puts his or her house on the market (and starts searching for another house) and becomes a seller. Following Krainer and LeRoy (2002), the seller offers his or her house for sale for a take-it-or-leave-it price (Assumption 7). A prospective buyer either accepts or rejects the price. Therefore, the seller has full Nash bargaining power. He or she draws randomly from the pool of unmatched households. Therefore, the seller does not observe the class of the prospective buyer nor is the observed fit revealed to the seller, which leads to information asymmetry. However, the seller knows the distribution of buyers across the two classes in this pool and the distribution of their fits. As a result, the seller of a good house offers the same price pG to all prospective buyers, and so does the seller of a bad house (price p B ). We also assume that neither buyers nor sellers have recollection of past interaction if those have taken place; because the pool is arbitrarily large, this possibility is near zero. Residential real estate markets do not clear, and so the value of having a house on the market, q j , is not equal to the selling price. Instead, it is expressed as the weighted average of the sales price, p j , and the discounted expected value of having the house on the market for another period (denoted with an asterisk). The weights are equal to the probability of selling the house, M j , and not selling the house, (1 − M j ), in this period. Thus, for a j-house, its value on the market equals: q j = M j p j + β(1 − M j )q ∗j ,

j = G, B.

(14)

When a transaction takes place, p is greater than q and the difference represents the seller’s surplus.11 A seller’s problem is to choose an optimal take-it-orleave-it price that would maximize the value of having her or his house for sale: p ∗j = arg max q j , pj

j = G, B.

(15)

If a price is set too high, the buyer’s reservation fit goes up and the buying probability falls.12 In addition, a change in price also has an indirect impact 11 12

In contrast, in liquid financial markets, the market clears with p = q.

It is possible for a household to be selling more than one house if the realized time on the market is longer than expected and overlaps with the holding period of the next house(s).

Is Selection Bias Inherent in Housing Transactions? 901

on the probability of buying the competitor house type. This is true because a household’s buying decision may require comparing two houses in each period: one good house and one bad house. As was shown in Equations (9a) and (9b), the probability of buying a house of a particular type depends on the reservation fits for the houses of both types. The first-order-condition is obtained by differentiating q j with respect to p j while holding q ∗j constant. The per-period selling probability is: M j = 1 − exp(−2ϕ j ),

j = G, B,

(16)

where the variable ϕ j was defined in Equation (12). The proof is provided in the appendix and relies on a Poisson arrival process of visitors to a house. Equilibrium In equilibrium, the buyers’ and sellers’ objective functions are simultaneously maximized. With respect to the buyer’s problem, the decision variables are ε¯ Gi and ε¯ iB , i = S, L. The maximization of the objective function s i (Equation (4)) yields two first-order-conditions. In performing this maximization, the next period’s search option s i∗ , the prices pG , p B , the house values to the sellers qG , q B , the market proportions of home types λG , λ B , and the market proportions of household classes ρ S , ρ L , are held constant. The derivations are provided in the appendix. After simplification, the first-order-conditions can be expressed as:   i∗ i i = S, L , j = G, B. (17) ν ε¯ i∗ j − p j = βs + C j , In essence, Equation (17) states that the net life-time utility from consuming a j-house is equal to the delayed search option value plus an additional term C ij . This term represents the value of choice when not one but two houses are visited in each period. Its sign is necessarily non-negative, reflecting the cost of choosing to delay the purchase of a house by another period by foregoing the chance of looking at a second house.13 With respect to the seller’s problem, the decision variable is p j , j = G, B. During the maximization of the objective function q j (Equation (15)), the relative proportions of the two classes of buyers are held constant, as an individual seller 13 In Krainer and LeRoy (2002) where there is only one buyer type and one house type, the first-order-condition is ν (¯ε∗ ) − p = βs ∗ . This means that, at the reservation fit, a homebuyer is indifferent between realizing the net gain and continuing to search for another period. However, with multiple house choices, the value of the next period’s search option falls below net utility from owning a home by the amount C ij that carries the meaning of the value of choice.

902 Chernobai and Chernobai

can make a marginal change to his or her own house price without affecting the entire market. Derivations are provided in the appendix. After simplification, the first-order-conditions can be expressed as:

 dl Sj dl Lj   ∗ S L (18) +ρ M j = −2 ρ exp(−2ϕ j ) p ∗j − βq ∗j , j = G, B. dpj dpj The term dl ij /d p j exists because a change in price affects the probability of liking a house through the reservation fit: dl ij dpj

=

∂l ij ∂ ε¯ ij ∂ ε¯ ij ∂ p j

.

(19)

All derivations and analytical expressions for these derivatives in Equation (19) are provided in the appendix. The resulting equilibrium is a Nash equilibrium. It is obtained by solving a system of equations that include the buyer’s and seller’s objective functions and the first-order-conditions. The model has four parameters: θ —the maximum fit in a bad house, π S and π L —the per-period probabilities of preserving a match that govern how long short- and long-tenure agents expect to stay in their current homes and β—the common discount factor. An analytical solution would require solving polynomial equations of degree higher than five. By Galois theory, an analytical solution for such polynomial equations does not exist (Rosen 1995 and Sturmfels 2002). Therefore, we solve our model numerically. Model Results In this section, we discuss our model results. For each result, we show numerical solutions of comparative statics. The discount factor is set to β = 0.95. We test the sensitivity of our results to the variance of the ex ante house quality by varying the parameter θ from 0.9 to 0.75, where the case of θ = 0.9 implies close similarity between the two house types and θ = 0.75 implies greater quality variance. For easier tractability of the results, we reparameterize the per-period probability of preserving a match with housing services and instead 1 14 We also test use expected tenure (ET) defined as E T i = 1−π i , i = S, L. the sensitivity of our results to the variance of households’ expected tenure. We fix ET S at 2, 2.5 or 3 (corresponding to π S = {0.5, 0.6, 0.67}) and vary E T L between 2 and 50 (corresponding to π L ∈ [0.5, 0.98]). For example, when E T S = E T L = 2 all households are of the same tenure class, and when E T S = 2 and E T L = 50 the population has two very distinct classes of households. 14 This follows from a property of the geometric distribution that assumes independence between periods.

Is Selection Bias Inherent in Housing Transactions? 903

Clientele Effect Figure 2a shows that a clientele effect occurs in the presence of heterogeneous houses and buyers. It illustrates the ratio of the buying probability of good houses to the buying probability of bad houses. Long-tenure buyers are more likely to purchase good houses than bad houses, as is indicated by the ratio being greater than one. Similarly, the ratio for short-tenure buyers is less than one, meaning that their purchase choices are predominantly bad houses. For example, if θ = 0.75 and the expected holding horizons of long- and shorttenure households are 35 periods (i.e., π L = 0.97) and two periods (i.e., π S = 0.5), respectively, then for long-tenure households μGL∗ /μ L∗ B = 1.6. This means that long-tenure buyers are 60% more likely to purchase a good home than a bad home. Similarly, for short-tenure households μGS∗ /μ S∗ B = 0.8, which means that they are 25% more likely to purchase a bad home than a good home. In addition, the clientele effect is magnified when the variance of the expected holding horizon is high. In the figures, this corresponds to alternative comparative statics with a longer holding horizon of long-tenure buyers (i.e., a movement along the horizontal axis) or a shorter holding horizon of short-tenure buyers (i.e., shifts of the lines). In addition, good houses are more expensive and less liquid, as is seen from the prices and time on the market15 in Figures 3a and 3b. One can think of an example of a town that has only detached and attached single-family residences. Since the ownership of a detached residence comes with such permanent advantages as the absence of adjacent residences and a wider range of home improvement possibilities compared to an attached residence, the former can exemplify a good house while the latter would be a house of the bad type. Other examples may include single-family houses versus condominiums, and houses with an ocean view versus houses with no such view. Thus, we expect the price differential in the housing market equilibrium to favor good houses. One can see that long-term buyers, who predominantly buy good houses, compensate for higher price and lower liquidity via higher net expected utility (Figure 2b) for good homes than for bad homes.16 While the same compensation takes 15 Time on the market (TOM) is a common measure of liquidity and is defined as the inverse of the per-period probability to sell, T O M j = M1 j , j = G, B. In our model, TOM can be reparameterized into the expected number of visitors until a house is sold. For that, TOM can be multiplied by the expected number of visitors per period which in our model equals two. Results of empirical studies of the relationship between selling price and expected time on the market are mixed. A positive relationship has been noted, for example, by Trippi (1997), Miller (1978) and Forgey, Rutherford and Springer (1996), while a negative relationship has been found by Turnbull and Sirmans (1993). 16 One may consider an alternative model specification in which the buyer, not the seller as assumed, sets the price. The differences in observed fits would result in a distribution

30

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S 2 S

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net net net net net net

v, v, v, v, v, v,

S

ET =2 ET S=2.5 ET S=3 S ET =2 ET S=2.5 S ET =3

S

ET =3

μ /μ , ET =2.5

L L G 2 L L μ /μ , G 2

μG/μ2, ET =2

L

μG/μ2, ET =3

S

μG/μ2, ET =2.5

S G S

μ /μ , ET =2

−p∗B , i = S, L

Expected Tenure of Long−Tenure Buyers (ET L)

0

ε¯Gi∗+1 2

5

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θ = 0.75

Expected Tenure of Long−Tenure Buyers (ET L )

0

i∗ μi∗ G /μB ,

Note: This figure illustrates the clientele effect in an economy with two types of houses and two classes of households. The common discount factor is β = 0.95. The maximum fit in a bad house is θ = 0.9 (left) or θ = 0.75 (right). Black and grey lines refer to long- and short-tenure buyers, respectively. Solid lines, dashed lines and dash-dotted lines are for expected tenure of short-tenure buyers E T S = {2, 2.5, 3}, respectively.

Expected Tenure of Long−Tenure Buyers

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θ = 0.90

Equilibrium results: Clientele effect.

Expected Tenure of Long−Tenure Buyers (ET L)



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Figure 2 904 Chernobai and Chernobai

Is Selection Bias Inherent in Housing Transactions? 905

Figure 3



Equilibrium results: Liquidity measures and prices. θ = 0.90

θ = 0.75

(a) House prices: p∗j , j = G, B

8.5

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10 15 20 25 30 35 40 45 L Expected Tenure of Long−Tenure Buyers (ET )

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Note: This figure illustrates liquidity measures in an economy with two types of houses and two classes of households. The common discount factor is β = 0.95. The maximum fit in a bad house is θ = 0.9 (left) or θ = 0.75 (right). In panels (a) and (b), black and grey lines refer to good and bad houses on the market, respectively, and in panel (c), black and grey lines refer to long- and short-tenure buyers, respectively. Solid lines, dashed lines and dash-dotted lines are for expected tenure of short-tenure buyers E T S = {2, 2.5, 3}, respectively.

place for short-tenure buyers, its magnitude is much smaller. Since a higher net expected utility makes the purchase of a good house worthwhile for longtenure buyers, in equilibrium they spend more time on their house search17

of price offers received by the seller in each period of the selling process. This would in turn require an endogenous determination of the reservation price at which the seller would be indifferent between selling the house and keeping it for one more period. At equilibrium, however, we would expect to obtain the same qualitative results: a higher reservation price for sellers of good houses would imply a higher average selling price and a lower probability of selling when compared to bad houses. 17

Expected time to buy (TTB) measures liquidity from a buyer’s perspective. It is defined as the inverse of the per-period probability of buying a house, TTBij = μ1i , i = j

S, L , j = G, B. Like TOM, TTB can be reparameterized into the expected number of

906 Chernobai and Chernobai

(Figure 3c).18 On the seller side, this translates into a longer expected time on the market. The clientele effect described above is more pronounced in a market with a higher variance of ex ante house qualities. When θ is exogenously decreased from 0.9 to 0.75, better quality houses are even more expensive relative to lower quality houses, are less liquid and require a higher compensation via net expected utility and a longer search. Because of this, longer-horizon buyers further outbid shorter-horizon buyers in purchasing better homes. In summary, our equilibrium results support empirical evidence (summarized in our literature review section) of the clientele effect: high-mobility households occupy low-quality high-turnover houses and vice versa. The clientele effect is stronger with greater differences in house qualities and planned holding horizons of homebuyers. Sample Selection Bias Figure 4 shows sample selection bias produced at equilibrium. In particular, Figure 4a illustrates relative proportions of homes on the market at equilibrium. It shows that the market is always dominated by lower-quality homes. As we showed earlier, long-tenure buyers are more likely to find a match with good houses, thereby removing them from the market for long periods of time. At the same time, the buyer pool consists predominantly of buyers with shorter planned holding horizons (Figure 4b). Hence, bad houses will exhibit a faster turnover, giving rise to transaction bias. From Equations (13a) and (13b), ρS > ρL

if

μGS + μ SB 1 − πS > , 1 − πL μGL + μ LB

(20)

which means that the steady-state proportion of shorter-tenure buyers is greater than that of longer-tenure buyers if their relative proportion of losing the match exceeds their relative probability of buying a house in a given period.19 houses visited until a house is purchased. For that, TTB is multiplied by the number of houses visited per period which in our model equals two. 18 Indeed, the search option value is higher for long-tenure buyers than short-tenure buyers. Relevant illustrations are omitted here for brevity. 19 We tried an alternative model specification in which a household is restricted to visiting just one house per period which is randomly drawn from the pool of all houses on the market. The equilibrium results under this model specification remain qualitatively unchanged (we omit simulation results for brevity). The clientele effect is somewhat less pronounced and so is the transaction bias. We attribute this to the absence of choice pushing up the value of the search option. As a result, the probability of buying a house drops for both classes of buyers, especially so for good homes.

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λB, ET =2

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λG, ET =3

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λG, ET S=2.5

λG, ET S=2

Note: This figure illustrates market equilibrium proportions of the two types of houses on the market and the two classes of buyers. The common discount factor is β = 0.95. The maximum fit in a bad house is θ = 0.9 (left) or θ = 0.75 (right). In panel (a), black and grey lines refer to good and bad houses on the market, respectively, and in panel (b), black and grey lines refer to long- and short-tenure buyers, respectively. Solid lines, dashed lines and dash-dotted lines are for expected tenure of short-tenure buyers E T S = {2, 2.5, 3}, respectively.

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10 15 20 25 30 35 40 45 Expected Tenure of Long−Tenure Buyers (ET L)

λB , ET S=3

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(a) Market proportion of the two types of houses on the market: λ∗j , j = G, B

θ = 0.90

Equilibrium results: Market composition of unmatched houses and households.

1

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Figure 4

Is Selection Bias Inherent in Housing Transactions? 907

908 Chernobai and Chernobai

Figure 4a shows how the selection bias depends on the variance of house qualities and planned holding horizons of homebuyers. For houses of similar quality (θ = 0.9), when the expected tenure of short- and long-tenure households equal 2 and 10 periods, the higher- and lower-quality houses are available in the market in relative proportions of 48% and 52%, respectively. When quality variance is higher (θ = 0.75), the relative proportion of higher-quality houses drops to nearly 45% whereas the relative proportion of lower-quality houses increases to 55%. The same result can be achieved by creating more divergent classes of households, specifically, if the expected tenure of long-tenure households increases from 10 to 50 periods. In all scenarios, in equilibrium short-tenure buyers dominate long-term buyers in the buyer pool (Figure 4b).20 In summary, our equilibrium results support empirical evidence (summarized in our literature review section) of the sample selection bias: properties on the market are overrepresented by those of lower quality and higher turnover rates. The selection bias is stronger with greater differences in house qualities and planned holding horizons of homebuyers. In our comparative statics analysis, for low values of long-tenure households’ expected holding horizon the non-linearities observed for some variables are explained as follows.21 Long-tenure buyers are less sensitive to price changes than short-tenure buyers because they can amortize a higher price over a longer holding horizon. On the margin, a higher selling price has a smaller negative effect on long-tenure buyers’ expected time to buy. Holding short-tenure buyers’ holding horizon fixed, when longer-tenure buyers’ expected holding horizon is exogenously increased, sellers raise their price to take advantage of the buyer pool with the higher average planned holding horizon and thus lower price sensitivity. Following the price increase, expected time on the market increases. 20 This clientele effect and the sample selection bias can be understood intuitively by examining the model set-up. Initially, the houses are equally distributed among the different buyer types. As houses are offered for sale and repurchased, the difference in the buying criteria for the two buyer types leads to a shift in the house ownership distribution. Specifically, good houses will be more attractive to all buyers because of the higher variance of drawn fit distribution and will be offered at a higher price. This will make both buyer classes set similarly high reservation fits with housing services that would make a purchase worthwhile. However, for low-quality houses short-term buyers have a much lower reservation fit than do long-term buyers, and so they will purchase this type of house even at the low end of the quality spectrum, which long-term buyers would reject. In other words, an inexpensive house that is good enough for a short-term buyer is not acceptable for a long-term buyer. Therefore, over time, the lower quality houses will preferentially be owned by short-term buyers, leaving more higher-quality houses to be owned by long-term buyers. Relevant illustrations are omitted for brevity and are available upon request. 21 Similar non-linearities were observed in Chernobai (2008). In her model there is only one housing type.

Is Selection Bias Inherent in Housing Transactions? 909

For larger exogenous increases, however, the effect of relative proportions of buyers kicks in: the lower proportion of long-tenure buyers remaining in the pool in steady state forces sellers to revise their pricing downward targeting the majority of the buyers, which are short-tenure buyers with higher price sensitivity. As price falls, liquidity improves.22 Transaction Costs There are two sources of market friction in our model: asymmetric information and search costs. We have so far maintained the assumption that there are no direct transaction costs such as moving and out-of-pocket search costs, real estate broker fees, taxes, etc. These can involve lump-sum and proportional components. In this section we add a direct proportional transaction cost to the model and express it as a fraction t of the house price, t ∈ [0, 1]. When switching houses, a household incurs a “round-trip” financial loss associated with paying more money for a new house than receiving from the sale of an existing one. Without loss of generality, one can incorporate proportional cost on the seller’s side. This will change the seller’s objective function as follows: q j = M j p j (1 − t) + β(1 − M j )q ∗j ,

j = G, B.

(14 )

Equilibrium results remain qualitatively unchanged, and the clientele effect and the transaction bias both remain strong. A notable quantitative difference from the earlier equilibrium results is a drop in the price for both house types. In addition, the expected time to buy a house and the relative presence in the buyer pool decreased for long-tenure buyers, and the effects are reversed for short-tenure buyers. Intuitively, for sellers who also act as simultaneous buyers, lowering prices is a profit maximizing strategy, as this minimizes the absolute loss from the incurred “round-trip” transaction cost associated with selling one and buying another house. For each house type, this raises life-time utility for long-tenure buyers who can amortize the costs over a longer span of future home ownership, but lowers it for short-tenure buyers for whom the life-time burden of more frequent future transaction costs is higher. The net effect on the net life-time utility is positive for long-term buyers but is negative for short-term buyers. As a result, the equilibrium length of the search for a matching house decreases for long-term buyers but increases for short-term buyers. This lowers the relative presence of the former in the pool of prospective buyers. 22 To draw a parallel with financial assets, such as stocks and bonds, our results indicate that, while the “high liquidity—high price” trade-off observed in the context of shorterterm buyers of financial assets does not exist for residential real estate assets, the trade-off does appear to hold on the margin.

910 Chernobai and Chernobai

Conclusion In this paper, we developed a theoretical model for the valuation of residential housing transactions in an economy with houses that differ in their quality and households that differ in their planned holding horizon. House quality depends on the amount of the maximum potential housing services that could be consumed in each period, e.g., attached versus detached, condominium versus single-family residence, inland versus ocean-front property, and so on. Expected holding period may depend on a wide variety of factors, such as expectation of future income, job stability, family size, relocation plans, age, etc. Our model has two classes of buyers, those with a shorter and a longer planned holding horizon, and two types of houses, “good” and “bad.” In each period, buyers are allowed to visit randomly two houses and buy at most one. They choose an optimal reservation fit with housing services so as to maximize the value of their house search. Sellers receive on average two prospective buyers in each period, and they choose an optimal selling price that would maximize the value of their house for sale. The model yielded two major results. First, at equilibrium the clientele effect exists: buyers with longer ex ante expected holding horizons predominantly choose higher-quality houses. Such houses are priced higher and exhibit lower liquidity (i.e., a longer time on the market). Second, this clientele effect creates a sample selection bias: the properties that are on the market are predominantly of lower quality. Since these are the preferred choice of short-horizon buyers, they demonstrate a faster turnover. Both the clientele effect and the selection bias are more pronounced with an increase in the variance of ex ante house quality and in the variance of the planned holding horizon. Empirical literature documents that houses that are on the market are on average dominated by properties that are inferior to those that are off the market, giving rise to the sample selection bias. As a result, house price indices are biased downward. Many studies have attempted to build econometric models to correct the bias by accounting for ex post holding period in the sold properties. In support of the results in these studies, our theoretical model shows the existence of this selection bias in an equilibrium setting. In contrast to these studies, however, the model illustrates that the differences in the ex ante planned holding horizons determine this bias at the time of house purchase. We refer to this bias as an inherent selection bias, as we show it to be a natural attribute of a housing market with heterogeneous houses transacted by buyers with different planned holding horizons. Our results have implications for empirical research and may help predict selection bias in future house price indices based on currently observed buyer clienteles.

Is Selection Bias Inherent in Housing Transactions? 911

Our model was developed in the context of a residential real estate housing market. While our paper’s contributions add to the growing literature on valuation of residential real estate properties, the model can be extended to other types of housing markets: in the commercial housing market, for example, the meaning of fit with housing services in our model would be equivalent to rent. An appropriate model would require allowing buyers to purchase multiple properties. Additionally, because in the commercial real estate market the investment incentive is dominant, the number of properties purchased and their expected holding periods may be time-variant, subject to current housing market conditions and their expected duration. We would like to thank the Editor and two anonymous referees for constructive suggestions. We are indebted to Stephen LeRoy for extensive discussions on this topic. We are also grateful to Thomas Barkley, Jonathan Dombrow, Chris Leach, Thomas Thibodeau and the participants of the American Real Estate and Urban Economics Association 2010 International Annual Meeting and the European Real Estate Society 2010 Annual Meeting for helpful comments.

References Bailey, M., R. Muth and H. Nourse. 1963. A Regression Method for Real Estate Price Index Construction. Journal of the American Statistical Association 58: 933–942. Bourassa, S., D. Haurin, J. Haurin, M. Hoesli and J. Sun. 2009. House Price Changes and Idiosyncratic Risk: The Impact of Property Characteristics. Real Estate Economics 37: 259–278. Bourassa, S., M. Hoesli and J. Sun. 2006. A Simple Alternative House Price Index Method. Journal of Housing Economics 15: 80–97. Case, B., H. Pollakowski and S. Wachter. 1991. On Choosing Among House Price Index Methodologies. Journal of the American Real Estate and Urban Economics Association 19: 286–307. Case, B. and J. Quigley. 1991. The Dynamics of Real Estate Prices. Review of Economics and Statistics 73: 50–58. Case, K. and R. Shiller. 1987. Prices of Single-Family Homes Since 1970: New Indexes for Four Cities. New England Economic Review Sept./Oct.: 45–56. Case, K. and R. Shiller. 1989. The Efficiency of the Market for Single-Family Homes. American Economic Review 79: 125–137. Chernobai, E. 2008. When Does Mobility Decrease Liquidity? Working Paper, California State Polytechnic University, Pomona. Clapp, J. and C. Giaccotto. 1998. Price Indices Based on the Hedonic Repeat-Sale Method: Application to the Housing Market. Journal of Real Estate Finance and Economics 16: 5–26. Clark, W. and J. Onaka. 1983. Life Cycle and Housing Adjustment as Explanations of Residential Mobility. Urban Studies 20: 47–57. Costello, G. and C. Watkins. 2002. Towards a System of Local House Price Indices. Housing Studies 17: 857–873.

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Crone, T. and R. Voith. 1992. Estimating House Price Appreciation: A Comparison of Methods. Journal of Housing Economics 2: 339–357. Dolde, W. and D. Tirtiroglu. 1997. Temporal and Spatial Information Diffusion in Real Estate Price Changes and Variances. Real Estate Economics 25: 539–565. Dreiman, M. and A. Pennington-Cross. 2004. Alternative Methods of Increasing the Precision of Weighted Repeat Sales House Price Indices. Journal of Real Estate Finance and Economics 28: 299–317. Englund, P., J. Quigley and C. Redfearn. 1999. The Choice of Methodology for Computing Housing Price Indexes: Comparisons of Temporal Aggregation and Sample Definition. Journal of Real Estate Finance and Economics 19: 91–112. Ermisch, J., J. Findlay and K. Gibb. 1996. The Price Elasticity of Housing Demand in Britain: Issues of Sample Selection. Journal of Housing Economics 5: 64–86. Forgey, F., F. Rutherford and T. Springer. 1996. Search and Liquidity in Single-Family Housing. Real Estate Economics 24: 273–292. Garmaise, M. and T. Moskowitz. 2004. Confronting Information Asymmetries: Evidence from Real Estate Markets. Review of Financial Studies 17: 405–437. Gatzlaff, D. and D. Ling. 1994. Measuring Changes in Local House Prices: An Empirical Investigation of Alternative Methodologies. Journal of Urban Economics 35: 221–224. Gelfand, A., M. Ecker, J. Knight and C. Sirmans. 2004. The Dynamics of Location in Home Price. Journal of Real Estate Finance and Economics 29: 149–166. Goetzmann, W. 1992. The Accuracy of Real Estate Indices: Repeat Sales Estimators. Journal of Real Estate Finance and Economics 5: 5–53. Goodman, A. and T. Thibodeau. 1995. Age-Related Heteroskedasticity in Hedonic House Price Equations. Journal of Housing Research 6: 25–42. Goodman, A. and T. Thibodeau. 1998. Dwelling Age Heteroskedasticity in Repeat Sales House Price Equations. Real Estate Economics 26: 151–171. Haurin, D. 1998. The Duration of Marketing Time of Residential Housing. Journal of the American Real Estate and Urban Economics Association 16: 396–410. Jansen, S., P. de Vries, H. Coolen, C. Lamain and P. Boelhouwer. 2008. Developing a House Price Index for the Netherlands: A Practical Application of Weighted Repeat Sales. Journal of Real Estate Finance and Economics 37: 163–186. Knight, J., J. Dombrow and C. Sirmans. 1995. A Varying Parameters Approach to Constructing House Price Indices. Real Estate Economics 23: 187–205. Krainer, J. 1999. Real Estate Liquidity. Federal Reserve Bank of San Francisco Economic Review 3: 14–26. Krainer, J. 2001. A Theory of Liquidity in Residential Real Estate Markets. Journal of Urban Economics 49: 32–53. Krainer, J. and S. LeRoy. 2002. Equilibrium Valuation of Illiquid Assets. Economic Theory 19: 223–242. Mark, J. and M. Goldberg. 1984. Alternative Housing Price Indices: An Evaluation. Journal of the American Real Estate and Urban Economics Association 12: 30–49. McCarthy, K. 1976. The Household Life Cycle and Housing Choices. Papers of the Regional Science Association 37: 55–80. Miller, N. 1978. Time on the Market and Selling Price. Journal of the American Real Estate and Urban Economics Association 6: 161–174. Musgrave, J. 1969. The Measurement of Price Changes in Construction. Journal of the American Statistical Association 64: 771–786. Rosen, M. 1995. Niels Hendrik Abel and Equations of the Fifth Degree. American Mathematical Monthly 102: 495–505.

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Myers, D. and J. Pitkin. 1995. Evaluation of Price Indices by a Cohort Method. Journal of Housing Research 6: 497–518. Shiller, R. 1991. Arithmetic Repeat Sales Price Estimators. Journal of Housing Economics 1: 110–126. Sturmfels, B. 2002. Solving Systems of Polynomial Equations. Proceedings of CBMS Regional Conference Series in Mathematics, Paper Number 97. The American Mathematical Society: Providence, RI. Trippi, R. 1997. Estimating the Relationship Between Price and Time to Sale for Investment Property. Management Science 23: 838–842. Turnbull, G. and C. Sirmans. 1993. Information, Search and House Prices. Regional Science and Urban Economics 23: 545–557. Wang, F. and P. Zorn. 1997. Estimating House Price Growth with Repeat Sales Data: What’s the Aim of the Game? Journal of Housing Economics 6: 93-118. Wheaton, W. 1990. Vacancy, Search and Prices in a Housing Market Matching Model. Journal of Political Economy 98: 1270–1292. Williams, J. 1995. Pricing Real Assets with Costly Search. Review of Financial Studies 8: 55–90.

Appendix Derivation of Market Proportions of Unmatched Assets, λG , λ B In equilibrium, the relative proportions of each house type on the market are determined as:     (1 − π S ) 1 − π L + μ LB + μ SB 1 − π L + μ LB − μGL μGS  ,   λG = (1 − π S ) 1 − π L + μGL + μ LB + (1 − π L ) 1 − π S + μGS + μ SB λ B = 1 − λG . Proof. The relative proportion of a house type placed on the market is the conditional probability that the house is of type j given that it is currently offered for sale by its unmatched owner. We denote by HG the event that a house is good, by H B that a house is bad, and by N the event that a household (or house) is unmatched. Then, λG := P(HG |N )

and

λ B := P(H B |N ).

(A.1)

Using Bayes’ formula, λG =

P(NG ) P(NG ) = P(N ) P(NG ) + P(N B )

λ B = 1 − λG ,

(A.2a)

(A.2b)

914 Chernobai and Chernobai

where P(NG ) and P(N B ) indicate the joint probabilities that an unmatched house is type HG or H B , respectively. To clarify, in our notations, subscripts denote a joint relation. Thus, P(NG ) = P(N ∩ HG ) and P(N B ) = P(N ∩ H B ). While it is assumed in this model that P(HG ) = P(H B ) = 0.5, the proportions of unmatched (N ) and matched households are determined endogenously. Using our assumptions P(S) = P(L) = 0.5 and P(HG ) = P(H B ) = 0.5, we expand Equation (A.2a) as follows: P(HG ) − P(YG ) P(HG ) − P(YG ) + P(H B ) − P(Y B )     P(HG ) − P YGS P(S) − P YGL P(L)       = P(HG )− P YGS P(S)− P YGL P(L)+ P(H B )− P Y BS P(S)− P(Y BL )P(L)     1 − P YGS − P YGL       = . (A.3a) 1 − P YGS − P YGL + 1 − P Y BS − P(Y BL )

λG =

In the above equation, we use superscripts to denote a condition. P(Y jS ) and P(Y jL ) denote the conditional probabilities that short-tenure and long-tenure households are matched with a j-house: P(Y jS ) = P(Y j |S) and P(Y jL ) = P(Y j |L). Similarly, we can define P(N S ) and P(N L ) to denote the conditional probabilities that short-tenure and long-tenure households are unmatched: P(N S ) = P(N |S) and P(N L ) = P(N |L). Thus, for each household type and each period, our economy is represented by three states of the world: unmatched, matched with a good house, and matched with a bad house. For these three states, we can represent the dynamics of the Markov chain by the transition probabilities matrix: ⎛

1 − μiG − μiB 1 − πi P=⎝ 1 − πi

μiG πi 0

⎞ μiB 0 ⎠. πi

(A.4)

Let T = (P(N i ) P(YGi ) P(Y Bi )) be the vector of the  corresponding stationary probabilities. The solution satisfies: T = T P and k Tk = 1. Using P(N i ) = 1 − P(YGi ) − P(Y Bi ) and solving

Is Selection Bias Inherent in Housing Transactions? 915



    P(N i ) P YGi P Y Bi 



 i



 i

= P(N i ) P YG P Y B

⎛

1 − μiG − μiB ⎝ 1 − πi 1 − πi

μiG πi 0

⎞ μiB 0 ⎠ πi

yields the solutions to the stationary probabilities for household class i = S, L: P(N i ) =

1 − πi , 1 − π i + μiG + μiB

(A.5a)

  P YGi =

μiG , 1 − π i + μiG + μiB

(A.5b)

  P Y Bi =

μiB . 1 − π i + μiG + μiB

(A.5c)

Substituting these into Equation (A.3a) yields the expressions for λG and λ B .  Derivation of the Probability that a Household Likes a House, l Gi , l iB , i = S, L The probability that an i-agent likes a j-house is computed as:   2    ε¯ i l Gi = λ2G 1 − ε¯ Gi + 2λG λ B 1 − ε¯ Gi B θ     ε¯ iB i i P (HG  H B ), + 2λG λ B 1 − ε¯ G 1 − θ

 i 2    ε¯ B ε¯ iB 2 i i l B = λB 1 − + 2λG λ B ε¯ G 1 − θ θ  i    ε ¯ + 2λG λ B 1 − ε¯ Gi 1 − B P i (H B  HG ). θ Proof. In every period, each prospective buyer randomly visits two houses. He or she is willing to buy one if the observed fit is at least as high as the reservation fit: εij ≥ ε¯ ij , i = S, L , j = G, B. A fit has a uniform distribution (Equations (2a) and (2b)). In the first summand of the l Gi equation, λ2G is the probability that both visited homes are good. In this case, with probability (1 − (¯εGi )2 ) the agent’s observed

916 Chernobai and Chernobai

fit will exceed his or her reservation fit at least for one of the two homes. In the second and third summands, 2λG λ B is the probability that the two homes are of different types. In this case, (1 − ε¯ Gi ) is the probability that the observed fit will exceed the reservation fit for the good home. Then, either the observed fit will fall below the reservation fit for the bad home with probability ε¯ iB /θ (in the second summand) or the observed fit will also exceed the reservation fit for the bad home with probability (1 − ε¯ iB /θ ) (in the third summand) and so the agent would prefer the good home with probability P i (HG  H B ). The last term is the conditional probability that a random variable εGi distributed uniformly on support [0, 1] exceeds a random variable εiB distributed uniformly on support [0, θ ], i.e., εGi > εiB , given that εGi ≥ ε¯ Gi and εiB ≥ ε¯ iB :

  P (HG  H B ) := P εGi − εiB > 0|εGi ≥ ε¯ Gi , εiB ≥ ε¯ iB ⎧  θ 1 1 ⎪ ⎪   dxdy if ε¯ Gi ≥ θ ⎪ ⎪ i i i i ⎪ 1 − ε ¯ θ − ε ¯ ε ¯ ε ¯ ⎪ G B ⎪ 

B i G ⎪ ⎪  θ 1 ε¯ G 1 ⎪ 2  ⎪ 1 ⎪ i ⎨   dxdy + dxdy + 0.5 θ − ε¯ G i i 1 − ε¯ Gi θ − ε¯ iB ε¯ iB ε¯ G ε¯ G θ = ⎪ ⎪ ⎪ if θ > ε¯ Gi ≥ ε¯ iB ⎪ ⎪ ⎪ 

⎪   ⎪ θ 1 ⎪ (A.6) 2  1 ⎪ ⎪ ⎪   dxdy +0.5 θ − ε¯ iB if θ > ε¯ iB ≥ ε¯ Gi ⎩ i i i 1− ε¯ G θ − ε¯ B ε¯ B θ ⎧ 1 if ε¯ Gi ≥ θ ⎪ ⎪ ⎪   ⎪ ⎪ i i 2 i i 2 ⎪ ⎨ θ − 0.5θ − ε¯ G − 0.5 ε¯ G + ε¯ G ε¯ B    if θ > ε¯ Gi ≥ ε¯ iB i i = 1 − ε ¯ θ − ε ¯ G B ⎪ ⎪ ⎪ ⎪ 1 − 0.5θ − 0.5¯εiB ⎪ ⎪ ⎩ if θ > ε¯ iB ≥ ε¯ Gi , i = S, L . 1 − ε¯ Gi

The three possible scenarios for the conditional probability in Equation (A.6) are illustrated in Figure A.1. The first multiplicative term in each of the three cases is the joint density that is uniform on a rectangle with sides (1 − ε¯ Gi ) and (θ − ε¯ iB ). The second term is the shaded area that represents the support on which the joint density is defined. The expression for l iB is analogously derived.



Is Selection Bias Inherent in Housing Transactions? 917

Figure A1  i-Buyer’s house choice when observed fits for good and bad houses exceed reservation fits. Scenario 1: ε¯iG ≥ θ

θ

Scenario 2: θ > ε¯iG ≥ ε¯iB

θ

Scenario 3: θ > ε¯iB ≥ ε¯iG

θ

Note: This figure illustrates the three scenarios associated with the probability of preferring a good home over a bad home when both observed fits exceed the reservation fits. The probability i i i is defined as P(HG  H B ) := P(εG − εiB > 0|εG ≥ ε¯ G , εiB ≥ ε¯ iB ), i = S, L. The three scenarios i i differ by the relation between ε¯ G , ε¯ B that are endogenously determined, and θ . For each scenario, the desired probability is computed by multiplying the corresponding shadowed area by the joint i density ((1 − ε¯ G )(θ − ε¯ iB ))−1 .

Derivation of Market Proportions of Buyers, ρ S , ρ L In equilibrium, the market proportions of buyers are determined as:

ρS =

  (1 − π S ) 1 − π L + μGL + μ LB  ,   (1 − π S ) 1 − π L + μGL + μ LB + (1 − π L ) 1 − π S + μGS + μ SB

ρL = 1 − ρS. Proof. The relative proportions of buyers of each class are the conditional probabilities that a household is of S- or L-class given that it is unmatched:23 ρ S := P(S|N )

and

ρ L := P(L|N ).

(A.7)

23 We are slightly abusing notations: the superscripts in ρ do not signify a condition as in our other notations. The relative proportions of short- and long-tenure households are always conditional on the household being unmatched.

918 Chernobai and Chernobai

Using Bayes’ formula and the assumption that household classes are equally represented in the population (i.e., P(S) = P(L) = 0.5), we obtain: P(N S )P(S) P(N ) P(N S )P(S) = P(N S )P(S) + P(N L )P(L) P(N S ) . = P(N S ) + P(N L )

ρS =

(A.8)

Using Equation (A.5a), we obtain the desired expression for ρ S . Then, ρ L = 1 − ρS. 

Derivation of Availability Factors, aG , a B In a competitive environment, finding a house for purchase does not necessarily secure the transaction. Therefore, the probability that a household actually purchases it is computed as:

aj =

1 − exp(−ϕ j ) , ϕj

j = G, B,

where ϕ j is as defined in Equation (12).

Proof. Since in every period a prospective buyer visits two houses, the arrival process of visitors to any house follows a Poisson distribution with intensity 2. Hence, for every prospective buyer who has visited a home, the conditional arrival process of additional visitors V has a standard Poisson distribution, i.e., V ∼ Poisson(1).

The subset X j of V represents the number of those additional visitors who liked the j-house. It is a binomial random variable with probability of success ϕ j and total number of trials V = n, i.e., X j ∼ Binomial(n, ϕ j ). Let a j = P(A j ) denote the probability that a visited j-house is available. aj =

∞ x=0

P(A j |X j = x)P(X j = x).

(A.9)

Is Selection Bias Inherent in Housing Transactions? 919

We assume that if more than one agent likes an asset, then each gets the asset with the probability inversely proportional to the number of agents: P(A j |X j = x) = (x + 1)−1 for x = 0, 1, . . . For example, a house is available with probability one when X j = 0, with probability one half when X j = 1, and so on. Then, P(X j = x) = ∞ n=x P(X j = x|V = n)P(V = n). Finally, using V ∼ Poisson(1) and X j ∼ Binomial(n, ϕ j ), Equation (A.9) becomes: aj =

∞ x=0

∞

1 n! exp(−1)1n ϕ xj (1 − ϕ j )n−x , (x + 1) n=x x!(n − x)! n!

j = G, B. (A.10)

Rearranging and simplifying, we obtain: a j = exp(−ϕ j )

∞ x=0

= exp(−ϕ j )

∞ x=0

∞

exp(−(1 − ϕ j )) 1 n! ϕ xj (1 − ϕ j )n−x (x + 1) n=x x!(n − x)! n! ∞ exp(−(1 − ϕ j ))(1 − ϕ j )m 1 ϕ xj , (x + 1)! m=0 m!    =1

where m = n − x = ϕ −1 j

∞ exp(−ϕ j )ϕ mj m=1

 =

(A.11)

m! 

=1−P(V =0)

1 − exp(−ϕ j ) , ϕj

,

where m = x + 1

 j = G, B.



Derivation of Buyer’s First Order Conditions During each period, a buyer of class i has an option to search for a house, defined as:   i   i   ε¯ + 1 ε¯ B + θ s i = μiG ν G − pG + μiB ν − pB 2 2 i i i∗ + β(1 − μG − μ B )s , i = S, L . A buyers’s objective is to find optimal reservation fits ε¯ Gi and ε¯ iB , i = S, L, in order to maximize the value of the search option. To obtain the first-orderconditions, we differentiate s i with respect to each of the reservation fits, while

920 Chernobai and Chernobai

holding constant the other class’s reservation fit, the next period’s search option s i∗ , the prices pG , p B , the house values to the sellers qG , q B , market proportions of unmatched houses λG , λ B , and unmatched households ρ S , ρ L : ⎡

∂s i ⎢ ∂ ε¯ i ⎢ ∇s i = ⎢ Gi ⎣ ∂s ∂ ε¯ iB ⎡

⎤ ⎥ ⎥ ⎥ ⎦ 

ε¯ Gi + 1 2 i d¯εG

⎤

dν ⎢ dμi   ε¯ i + 1  ⎢ G G i∗ i − pG − βs + μG ⎢ i ν ⎢ d¯εG 2 ⎢   i  ⎢ ε¯ B + θ dμi ⎢ − p B − βs i∗ ⎢ + iB ν ⎢ 2 d¯εG =⎢   i ⎢ ε¯ B + θ ⎢  dν ⎢ i   i ⎢ dμG ε¯ G + 1 2 i∗ i ⎢ ν + μ − p − βs G B i ⎢ d¯εi 2 d¯ε B ⎢ B  ⎢ i   i ε¯ B + θ dμ ⎣ − p B − βs i∗ + iB ν 2 d¯ε B

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥. ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(A.12)

Then, setting expressions in (A.12) to zero produces the following general expression:   i∗ i ν ε¯ i∗ j − p j = βs + C j ,

i = S, L ,

j = G, B.

(A.13)

The term C ij depends on the relation between the reservation fits through Equations (6), (9a) and (9b). This relation was presented in Equation (A.6). Grouping with all relevant terms and simplifying, we obtain the expressions for C ij : Scenario 1: ε¯ Gi ≥ θ, i = S, L . ε¯ i 1− B    a B θ C Gi = −λ B βs i∗ + p B − ν ε¯ iB i aG ε¯ G λG + λ B (A.14a)

 2  λG 1 − ε¯ Gi aB λB  1 β i 2   + θ − ε¯ B + , 1 − βπ i 2 ε¯ Gi λG + λ B 2 aG θ C Bi

β = λB 1 − βπ i



θ − ε¯ iB 2

2

1 . λ B ε¯ iB + λG ε¯ Gi θ

(A.14b)

Is Selection Bias Inherent in Housing Transactions? 921

Scenario 2: θ > ε¯ Gi ≥ ε¯ iB , i = S, L .   θ 1 a B λ B ε¯ Gi − ε¯ iB  i∗ β βs + p B − ν ε¯ iB + i i i aG ε¯ G λ B + λG θ 1 − βπ 4¯εG λ B + λG θ

  ε¯ Gi 2 − ε¯ Gi × λG + λ B (2 − θ ) − (λ B + λG θ ) θ    a B λ B θ − ε¯ iB ε¯ Gi − ε¯ iB β + , (A.15a) 1 − βπ i aG 2¯εGi λ B + λG θ

C Gi = −

C Bi =

 2  i β 1 1 ε¯ B λ B + ε¯ Gi λG θ i i i 4 1 − βπ ε¯ B λ B + ε¯ G λG θ   − 2¯εiB λ B θ + θ 2 λ B + λG θ − 2¯εGi λG .

Scenario 3: θ > ε¯ iB ≥ ε¯ Gi , i = S, L .  2    1 1 − ε¯ Gi λG θ − λ B θ − ε¯ iB θ + ε¯ iB − 2 β i , CG = 1 − βπ i 4 ε¯ iB λ B + ε¯ Gi λG θ

(A.15b)

(A.16a)

  aG λG θ ε¯ iB − ε¯ Gi  i∗ βs + pG − ν ε¯ Gi (A.16b) i a B ε¯ B λ B + λG θ

    2 θ 2 + ε¯ iB aG 1 β λG θ ε¯ iB − ε¯ Gi 1 − ε¯ Gi β i + − ε¯ B θ − . 1 − βπ i 2¯εiB 2 a B 1 − βπ i 2¯εiB λ B + λG θ C Bi = −

Derivation of Probability to Sell, MG , M B For each house type, the per-period selling probability is computed as the probability that at least one of the visitors to the house this period wants to buy it. It is given by: M j = 1 − exp(−2ϕ j ),

j = G, B,

where ϕ j is as defined in Equation (12). Proof. On average, any given house will be visited by two agents. Variable W counts the number of visitors to an arbitrary house and follows a Poisson distribution with mean 2, i.e., W ∼ Poisson(2). Conditional on its realization, the number of visitors who liked the house, L j , follows a binomial distribution,

922 Chernobai and Chernobai

i.e., L j ∼ Binomial(n, ϕ j ). Then the selling probability becomes: Mj = =

∞ n=0 ∞

P(L j ≥ 1|W = n)P(W = n) (1 − (1 − ϕ j )n )P(W = n)

n=0

= 1−

∞

exp(−2)2n n! ∞ exp(−2(1 − ϕ j ))(2(1 − ϕ j ))n

(1 − ϕ j )n

n=0

= 1 − exp(−2ϕ j )

n=0

= 1 − exp(−2ϕ j ),

(A.17)

n! j = G, B.



Derivation of Seller’s First-Order-Conditions During each period, a seller owns a j-type house with value q j = M j p j + β(1 − M j )q ∗j ,

j = G, B.

Differentiation of q j with respect to p j , while holding the relative proportions of the buyer classes constant, produces:

 dl Sj dl Lj dq j S L = Mj + ρ +ρ 2exp(−2ϕ j )( p j − βq ∗j ), j = G, B. (A.18) dpj dpj dpj Setting the derivative to zero and rearranging produces the first-order-conditions as follows:

 dl Sj dl Lj S L +ρ M j = −2 ρ exp(−2ϕ j )( p ∗j − βq ∗j ), j = G, B. (A.19) dpj dpj Any change in the sales price has a marginal effect on a buyer’s reservation fit for that house type, which in turn affects the buyer’s probability of buying the house. Consequently, holding competition factors and the price of a house of the competitor type constant, the derivatives dl Sj /d p j and dl Lj /d p j in Equation (A.19) must be computed as follows: dl ij dpj

=

∂l ij ∂ ε¯ ij ∂ ε¯ ij ∂ p j

,

i = S, L ,

j = G, B.

(A.20)

First, to compute ∂l ij /∂ ε¯ ij , we differentiate l ij in Equation (7) with respect to reservation fit for each scenario as shown in Equation (A.6). These partial derivatives are as shown below.

Is Selection Bias Inherent in Housing Transactions? 923

ε¯ Gi ≥ θ, i = S, L .   ∂l Gi /∂ ε¯ Gi = −λG 2 ε¯ Gi λG + λ B , Scenario 1:

∂l iB /∂ ε¯ iB = −λ B Scenario 2:

 2 i ε¯ B λ B + ε¯ Gi λG θ . 2 θ

(A.21a)

(A.21b)

θ > ε¯ Gi ≥ ε¯ iB , i = S, L .

∂l Gi /∂ ε¯ Gi = −λG

2¯εGi (λ B + λG θ ), θ

(A.22a)

∂l iB /∂ ε¯ iB = −λ B

 2 i ε¯ B λ B + ε¯ Gi λG θ . 2 θ

(A.22b)

Scenario 3:

θ > ε¯ iB ≥ ε¯ Gi , i = S, L .

∂l Gi /∂ ε¯ Gi = −λG

 2 i ε¯ B λ B + ε¯ Gi λG θ , θ

(A.23a)

∂l iB /∂ ε¯ iB = −λ B

2¯εiB (λ B + λG θ ). θ2

(A.23b)

Second, to compute ∂ ε¯ ij /∂ p j , we compute the inverse of ∂ p j /∂ ε¯ ij . We rearrange buyer’s first-order-conditions (Equations (A.13), (A.14a), (A.14b), (A.15a), (A.15b), (A.16a) and (A.16b)) as price expressions, which we then differentiate with respect to ε¯ ij , j = G, B. Scenario 1: ε¯ Gi ≥ θ, i = S, L .

   λG 1 − ε¯ Gi β i  ∂ pG /∂ ε¯ G = 1+  i 1 − βπ i 2 ε¯ G λG + ε¯ iB

 2  2  λG 1 − ε¯ Gi a B λ B θ − ε¯ iB β λG + +   1 − βπ i 2 ε¯ i λG + ε¯ i 2 2 aG θ G B   ε¯ i λG λ B 1 − B   aB θ  i∗ βs + p B − ν ε¯ iB , (A.24a) −  i  2 aG ε¯ λG + ε¯ i G B ∂ p B /∂ ε¯ iB

β = 1 − βπ i

     λ B θ − ε¯ iB 2¯εGi λG θ + λ B ε¯ iB + θ 1+ . (A.24b)  2 4 ε¯ iB λ B + ε¯ Gi λG θ

924 Chernobai and Chernobai

Scenario 2: θ > ε¯ Gi ≥ ε¯ iB , i = S, L .

θ λG + λ B (2 − θ ) β 3 i ∂ pG /∂ ε¯ G = +  2 i 1 − βπ 4 4 ε¯ i λ B + λG θ G    a B λ B ε¯ iB θ − ε¯ iB −   aG 2 ε¯ i 2 (λ B + λG θ ) G

+

 i∗   aB βs + p B − ν ε¯ iB ,  i 2 aG ε¯ (λ B + λG θ ) G λ B ε¯ iB

(A.25a)

β 1 − βπ i

 2  2   λ B ε¯ iB λ B + 2¯εGi ε¯ iB λG θ − θ ε¯ Gi λG + θ (λ B + λG θ ) × 1− .  2 4 ε¯ iB λ B + ε¯ Gi λG θ ∂ p B /∂ ε¯ iB =

(A.25b) Scenario 3: θ >

ε¯ iB

≥

∂ pG /∂ ε¯ Gi

β = 1 − βπ i

∂ p B /∂ ε¯ iB

β = 1 − βπ i +

ε¯ Gi , i



= S, L .

  2  6¯εGi ε¯ iB λG λ B θ + λG θ 2 λG + 3 ε¯ Gi λG + λ B (2 − θ )  2 4 ε¯ iB λ B + ε¯ Gi λG θ  (¯εiB )2 λ B (4λ B + λG θ ) + , (A.26a) 4(¯εiB λ B + ε¯ Gi λG θ )2

3 1 +  2 4 4 ε¯ i B

   aG 2θ λG ε¯ Gi 1 − ε¯ Gi θ − (A.26b) aB λ B + λG θ 2

 i∗   ε¯ Gi λG θ aG βs + pG − ν ε¯ Gi .  i 2 a B ε¯ (λ B + λG θ ) B