Adverse selection, seller effort, and selection bias.

Chezum, Brian

1. Introduction

Researchers use several approaches to identify adverse selection. (1) Genesove (1993) tests the proposition that, in a lemons market, prices inversely relate with observable seller characteristics that correlate with seller incentives to select goods adversely. Genesove examines the proposition in used automobile auctions. Chezum and Wimmer (1997) examine the proposition in thoroughbred racehorse markets, arguing that sellers with a high propensity to race horses should receive, on average, lower prices. Both Genesove (1993) and Chezum and Wimmer (1997) use data limited to market transactions. Wimmer and Chezum (2003) model adverse selection as a case of Heckman's (1979) sample selection bias and examine the correlation between errors in participation and price equations to study how third party certification could alleviate the effect of adverse selection in thoroughbred auctions. (2)

In this paper, we extend the work of Genesove (1993) and Chezum and Wimmer (1997) and Wimmer and Chezum (2003) by characterizing adverse selection as a sample selection problem in a setting in which sellers possess (1) an informational advantage over buyers and (2) characteristics that correlate with both seller incentives to select goods adversely and the quality of goods produced. In such a setting, the relationship between prices and seller characteristics proves ambiguous, and researchers cannot easily disentangle adverse selection from quality effects. We show that Heckman's sample selection framework disentangles the correlation between a seller's characteristic and the effect of the selection decision on price from the correlation between a seller's characteristic and the quality of goods produced by a seller.

The notion that the decision to sell goods relates to the quality of goods produced is not new. For example, Kim (1985) developed an adverse selection model in which car owners' maintenance and upkeep decisions affected the quality of used cars. Kim showed that allowing owners to affect the quality of goods could, under certain conditions, lead to an equilibrium in which the expected quality of used cars sold exceeds the expected quality of cars not sold. Similarly, we develop a theoretical model that extends a standard adverse selection specification by allowing owners (potential sellers) to improve a good's quality by expending unobserved effort. We show that owner effort only affects the adverse selection equilibrium when owners choose effort before they make the sell-retain decision.

Owners decide whether to sell or retain a good once they observe their good's innate quality. Because buyers do not observe owner effort, the owner's dominant strategy is to expend zero effort on goods they will surely sell. Essentially, the model collapses to a standard moral hazard model, such as a fixed wage contract, in which employers cannot observe employee effort, and employees exert the necessary minimum effort to avoid dismissal. Because equilibrium effort equals zero, quality effects do not alter seller selection decisions and, therefore, the adverse selection equilibrium.

Owners who must exert effort before they observe innate quality know only the probability that they will sell their goods and that the expected return to effort increases in the probability of retaining those goods. Because the probability of retention increases in seller incentives to select goods adversely, sellers who more likely select goods adversely also exert more effort. In this version of the model, no clear relationship exists between the expected quality of goods sold and the seller characteristics that Genesove (1993), and Chezum and Wimmer (1997) use to measure adverse selection. The model shows that uncertainty about whether a good will be sold partially solves the hidden action problem, in which owners underprovide effort and lessen the effect of adverse selection on markets.

The notion that both adverse selection and moral hazard are important in markets affected by asymmetric information is well understood. Stewart (1994) modeled competitive insurance markets characterized by both adverse selection and moral hazard, showing that the two problems could partially offset one another. Jullien, Salanie, and Salanie (1999) and de Meza and Webb (2001) showed, theoretically, that standard adverse selection results might not hold when risk preferences affect both the policy selected and prevention activities. In an empirical study, Bradley (2002) showed that both moral hazard and adverse selection played important roles in health insurance markets. In related work, Abbring, Chiappori, and Pinquet (2003) used dynamic panel data to separate the effects of adverse selection from moral hazard, whereas Edelberg (2004) examined similar issues using consumer loan data. This paper extends this literature by examining the potential effect hidden action concerns can impose on adverse selection equilibrium in a goods market.

We test the theoretical model using data that include goods retained and sold by their original owners. We estimate a price equation using Heckman's standard correction for self-selection, separating the adverse selection effect on price from the effect of potential effort by including a proxy for each seller's preference for the goods in both the selection and price equations. Evidence of adverse selection exists when our proxy for the seller's preference for the good reduces the probability that a good is sold, and the inverse Mills ratio receives a negative coefficient. A positive coefficient on our proxy for a seller's preference for the good in the price equation, holding the selection effect constant, implies the presence of an effort effect.

Empirically, we compare standard ordinary least squares (OLS) regressions with the Heckman specification. Our OLS results contradict the findings of Chezum and Wimmer (1997), who used data from a single thoroughbred sale, and found that sellers who participate more intensively in the racing end of the business, on average, receive lower prices. With the use of data from a random sample of sales, and a similar specification, we find no significant relationship between price and a seller's racing intensity in standard OLS regressions. Correcting for sample selection, the data support the hypothesis that adverse selection plays an important role in the market for thoroughbred racehorse prospects. Our findings also support the prediction that sellers who are more likely to retain goods exert more effort.

The remainder of the paper is outlined as follows. Section 2 constructs a theoretical model that accounts for both hidden actions and adverse selection, showing that sellers for whom adverse selection is more severe may produce higher quality goods. Section 3 illustrates how Heckman's selection bias model can separate the effects of adverse selection from hidden seller actions. Section 4 discusses the data. Section 5 presents the results. We find that both adverse selection and our measure of hidden effort produce statistically significant effects on prices in the market for thoroughbred racehorse prospects. Section 6 offers concluding remarks.

2. Theoretical Framework

This section develops a three-period model that examines a goods market with asymmetric information. The market considered consists of m heterogeneous owners (potential sellers) and n identical buyers (n > m). Owners and buyers are risk-neutral expected utility maximizers. In period 1, owners are randomly allocated a single unit of a good with innate quality q + g([??]), where [??] is a vector of observable mean shifters, g'([??]) > 0 and q is stochastic. (3) The stochastic component of innate quality, q, is drawn from the cumulative distribution F(q) with support [[q.sub.L], [q.sub.H]], where 0 < [q.sub.L] < [q.sub.H] < [infinity] has continuous density f(q), and mean [mu]. (4) In period two, owners expend effort e, where e is nonnegative, finite, and bounded from above. We assume the cost of effort, given by c(e), is increasing and convex in effort--c'(e) > 0, c"(e) > 0, with c(0) = c'(0) = 0--and is common knowledge. The realized quality of goods at the end of period 2 equals the sum of innate quality and owner effort (i.e., [q.sup.R] = q + g[[??]] + e). (5) In period 3, owners observe realized quality, and the market opens. (6) The market is a standard lemons market, in which owners possess an informational advantage over buyers.

Following Genesove (1993), heterogeneous owners differ in the utility they receive from retained goods. At the time of sale, buyers cannot observe the realized quality of goods, but observe owner characteristics, and the distribution of innate quality. Owners choose the level of effort and whether to retain the good. Buyers choose whether to bid for a unit of the good and the amount they will bid.

If the good is retained, owners endowed with quality q + g([??]) exerting effort e receive utility [U.sub.0] = v + s[q + g([??]) + e] - c(e), (7) where s is an owner's marginal rate of substitution of quality for other goods, and v is a numeraire good. (8) The utility that owners receive from retaining goods increases in the value of s. If the good is sold, the owner receives utility [U.sub.0] = v + P - c(e), where P is the market price. Owners maximize utility by choosing effort and whether to sell goods.

Buyers, who purchase a unit of the good at price P receive expected utility: [U.sub.B] = v + bE[q + g([??]) + e] - P, where the expectation of quality forms over the distribution of quality, whereas, as shown below, the expectation of e is conditional on seller type. Buyers receive [U.sub.B] = v if they do not buy a good. The parameter b is a buyer's marginal rate of substitution of quality for other goods. To ensure that all trades are mutually beneficial, we assume that b > s > 0 for all potential values of s. (9,10) Buyers maximize utility by submitting price bids to owners. Price bids reflect buyer expectations on the realized quality of goods sold, and depend on the distribution of innate quality and the owner's valuation s, which is common knowledge.

An owner's decision to sell a good depends on the realization of the stochastic component of innate quality. Owners sell when P [greater than or equal to] s[q + g([??]) + e], where P equals the highest price offered. Given P, g([??]), and e, owners are indifferent between selling and retaining a good when P= S[qm + g([??]) + e] or [q.sub.m] = P/s - g([??]) - e, where [q.sub.m] (marginal quality) equals the highest value of q an owner willingly sells. Expected innate quality of goods sold is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Buyers purchase a good when P [less than or equal to] bE[q + g([??]) + e]. Because buyers outnumber owners, sellers receive the entire surplus from trade and P = bE[q + g([??]) + e]. Equilibrium requires that buyer expectations be realized in equilibrium:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Because we are interested in the effect effort has on equilibrium in a lemons market, we assume that b > s > b{[[mu] + g([??])]/[[q.sub.H] + g(X)]} and s - b[d[q.sub.Ave]([q.sub.m])/d[q.sub.m]] > 0. (11) The first assumption guarantees that owners retain highest quality goods, q = [q.sub.H], whereas the second requires that owners value a one-unit increase in [q.sub.m] by more than buyers value the effect an increase in [q.sub.m] has on the expected value of goods sold. Because owners observe q, their marginal value of a one-unit increase in marginal quality equals s, whereas the buyer's marginal value is based on the effect an increase in marginal quality has on expected quality. As in standard models of adverse selection, we assume that an owner's marginal value of an increase in [q.sub.m] exceeds the buyer's expected marginal value.

Because buyers cannot observe owner choices, the sequential nature of owner decisions does not affect buyer expectations, and unobserved owner effort does not affect price bids, [differential]P/[differential]e = 0. As in standard hidden-action models, owners ignore the buyer's value for effort and effort is underprovided. Owner's effort choice does depend on whether owners observe innate quality at the time they choose effort.

First, consider the case in which owners observe the stochastic component of a good's innate quality (q) before they exert any effort. In this case, because the owner's effort does not affect price bids, an owner's dominant strategy is to provide zero effort for goods the owner sells. (12) Buyers, therefore, expect that all goods sold contain zero effort, (13) and equilibrium effort equals zero for goods sold. Owners who retain their goods exert the privately optimal level of effort [e.sub.p], s = c'([e.sub.p]). When owners observe innate quality before they choose effort, the equilibrium marginal quality, price, and effort for goods sold emerge from the following system of equations:

P - b[[q.sub.Ave]([q.sub.m]) + g(X)] = 0, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (1) e = 0.

These equations give the standard adverse selection result that market prices are inversely related to the value owners receive from retaining goods; [differential][q.sup.**.sub.m]/[differential]s < 0 and [differential][P.sup.**]/[differential]s < 0, (14) where [q.sup.**.sub.m] and [P.sup.**] are equilibrium marginal quality and price, respectively, when owners observe q before making effort decisions. Because sellers exert zero effort, equilibrium is consistent with a standard lemons model that assumes owners cannot provide effort. (15)

Now, consider the case where owners must choose effort before observing the stochastic component of innate quality, q. Because owners do not observe q, they must choose effort before they decide whether they will retain their goods, and effort decisions depend on the probability that a good is retained. At the time of sale, owners retain goods when q > [q.sub.m], where [q.sub.m] = P/s - g([??]) - e, and the probability that the good is retained is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Owners choose effort by maximizing expected utility,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

is the expected quality of retained goods. The term in braces captures the contribution to expected utility of a good with quality q + g([??]), which averages the payoff to retention, s[[??] + g([??]) + e], and the payoff to a sale, P, weighted by their respective probabilities. When choosing effort, owners must account for the effect effort has on the probability that a good is retained, 1 - F([q.sub.m]) = 1 - F[P/s - g([??]) - e]. A one-unit increase in effort, holding price constant, reduces marginal quality by one unit, [differential][q.sub.m]/[differntial]e = -1, and increases the probability the good is retained, [differential][1 - F([q.sub.m])]/[differential]e = f([q.sub.m]). The owner's first-order condition is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Substituting [differential][??]/ [differential][q.sub.m] = {f([q.sub.m])/[1 - F([q.sub.m])]}([??] - [q.sub.m]) and using the definition of marginal quality P = S[[q.sub.m] + g([??]) + e], the owner's first-order condition for effort reduces to [1 - F([q.sub.m])]S = c'(e).

An owner values an additional unit of effort at s, but only retains the good with probability [1 - F([q.sub.m])]. Because buyers do not observe effort, owners ignore the value buyers place on increases in realized quality. Sellers balance the private expected marginal benefit of increasing effort with its marginal cost. The uncertainty created by unobserved innate quality partially solves the moral hazard problem when 0 < [1 - F([q.sub.m])] < 1. As in the first case, effort equals zero for owners who know, with certainty, they will sell their goods and equals the privately optimal level for owners who know they will, with certainty, retain their goods, 1 - F([q.sub.m]) = 1.

The solution to the model for the second case is defined by the following three equations:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

The simultaneous solution of these three equations defines the owner's equilibrium choices of effort and marginal innate quality. In equilibrium, the market price depends on equilibrium marginal quality and equilibrium effort, [P.sup.*](s) = b{[q.sub.Ave][[q.sup.*.sub.m](s)] + g([??]) + [e.sup.*](s)}, where [P.sup.*](s), [[q.sup.*](s)], and [e.sup.*](s) are the equilibrium values of price, marginal quality, and effort, respectively.

We want to determine how a change in s affects equilibrium price through its effects on effort and marginal quality. The effect on the price is shown in Equation 3.

[differential][P.sup.*]/[differential]s = b([differential][e.sup.*]/[differential]s + d[q.sub.Ave]/d[q.sub.m] [differential][q.sup.*.sub.m]/[differential]s) (3)

Differentiating the equations contained in Equation 2 with respect to s and applying Cramer's rule gives the comparative static results for effort and marginal quality (Eqns. 3, 4). (16)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (5)

Equation 4 shows that equilibrium effort unambiguously increases in s when s - b d[q.sub.Ave]/d[q.sub.m] > 0. Equation 5 shows that owner-provided effort reduces problems of adverse selection, leading to an ambiguous relationship between [q.sup.*.sub.m] and s. The first term in Equation 5's numerator is negative, capturing the standard adverse selection result. The second term, which is positive, equals the gains to trade from effort, weighted by the probability that the good is retained, and is positive. Because buyers do not observe effort, goods sold contain less than the efficient level of effort, and effort only partially affects the adverse selection equilibrium. The ambiguous relationship between [q.sup.*.sub.m] and s exists because owner effort is increasing in s. (17)

Equation 3 shows that a change in s has two effects on equilibrium price. The first term shows that owner effort increases in s, (18) which raises the realized quality of goods sold. We refer to this as the "Effort Effect." The second term, which we label the "Selection Effect," shows that an increase in s potentially leads owners to retain a larger proportion of their goods; the expected innate quality of goods sold can decrease in s. This second effect captures the degree to which the expected quality of goods sold truncate endogenously in a lemons market. These competing effects indicate that no clear relationship exists between price and seller characteristics when seller effort affects the quality of goods sold.

Below, we apply the theoretical results using data that include both goods sold and retained by their original owners. Because Heckman's (1979) standard correction for self-selection isolates the effect endogenous truncation has on the expected value of goods sold; both the Effort and Selection effects can be estimated by including a proxy for a seller's preference for the good in both the selection and price equations.

3. Empirical Specification

To measure the effects of adverse selection and hidden actions on price, we must isolate the Selection Effect from the Effort Effect. Setting b = 1 (to simplify notation) and assuming innate quality is drawn from a normal distribution with population mean Ix (as defined above) and variance [[sigma].sup.2], the price from Equation 2 is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (6)

where [q.sup.*.sub.m] = [q.sup.*.sub.m](S) is the marginal quality sellers willingly take to market, [lambda]([q.sub.m]) = [phi]([q.sub.m])/[PHI]([q.sub.m]) is the inverse Mills ratio, [PHI] denotes the standard normal distribution function, and [PHI] denotes the corresponding density function. (19) In this setting, [sigma][lambda][[q.sup.*.sub.m](s)] measures the effect endogenous truncation has on the expected value of goods sold. Equation 6 shows that s affects price by shifting the mean of realized quality (Effort Effect) and by altering the degree of endogenous truncation (Selection Effect). (20)

Equation 6 implies an estimating equation of the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (7)

where we assume the deterministic component of quality is linear g([X.sub.i]) = [??]'[[??].sub.i], [alpha][s.sub.i] estimates the Effort Effect, and [[epsilon].sub.i] is a vector of unobservable factors. The term [[delta].sup.*][lambda][[q.sub.m]([s.sub.i])] estimates the effect of adverse selection on observed prices.

Estimating Equation 7 requires a measure of an owner's, generally unobservable, marginal quality choice. When we have data that include both goods sold and retained, we can observe whether an owner sells a particular good. An owner sells a good when the price exceeds the value received from retaining it. Let [Z.sub.i] be the owner of the ith good's net benefit from selling the good. That is,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where [W.sub.i] is a vector of observable characteristics (21) and [u.sub.i] is the random disturbance term. The good goes to market if [Z.sub.i] [greater than or equal to] 0 and does not go otherwise. Define [Z.sup.*.sub.i] to equal one for goods sold and zero otherwise; [Z.sub.i.sup.*] = 1 when [u.sub.i] [greater than or equal to] -[??]'[[??].sub.i] - [psi]'[s.sub.i], and the probability a good is offered for sale is Pr([u.sub.i] [greater than or equal to] -[??]'[[??].sub.i] - [psi]'[s.sub.i]). Assuming a normal distribution for [u.sub.i] gives the standard Probit model:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (8)

Equation 8 suitably estimates the participation Equation 2. In the model, [psi] < 0 implies that adverse selection dominates the effect of effort on marginal quality. Assuming that ([u.sub.i], [[epsilon].sub.i]) is distributed as bivariate normal (0, 0, 1, [[sigma].sub.[epsilon]], [rho]), Heckman (1979) shows that consistent estimates of the relationship between price and seller characteristics are obtained by estimating

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (9)

where [[??].sub.i] has a zero mean and constant variance. The information contained in the Mills ratio provides evidence about the presence of adverse selection. Differentiating Equation 9 with respect to s shows how a change in s affects the equilibrium price.

[differential]E[[P.sub.i]|[X.sub.i], [Z.sub.i] = 1]/[differential]s = [alpha] + [delta][[differential][lambda](*)/[differential]s]. (10)

The first term in Equation 10, [alpha], provides an estimate of the Effort Effect, showing how a change in s affects the unconditional expected quality of goods. The second term provides an estimate of the Selection Effect, showing how a change in s affects price through its effect on owner decisions to sell goods.

We estimate the standard Heckman model. This approach allows estimation of both the Selection and Effort Effects. Because the Selection Effect equals the product of the coefficient on the inverse Mills ratio, [delta], and [differential][lambda]/[differential]s, which is nonlinear, we use a standard bootstrap program to estimate the standard error of the Selection Effect.

Estimates of [delta] also provide a test for the presence of adverse selection. The coefficient on the inverse Mills ratio represents the covariance between errors from the selection equation and the price equation. Finding [??]] < 0 indicates that goods with unusually high price offers, given their observable characteristics, prove even more valuable to their owners, so owners do not sell, which suggests the presence of adverse selection.

Theoretically, the model predicts that factors that increase the probability of sale reduce effort, and all variables included in [??] should appear in [??]. (23) This facet of the model precludes the use of an exclusion restriction to identify the model, leaving only the functional form of the inverse Mills ratio to identify the price equation. As discussed by Vella (1998), the lack of an exclusion restriction to identify the model might introduce significant collinearity into the model, and researchers should view results with caution. Leung and Yu (1996) run Monte Carlo experiments to evaluate the conditions under which functional form effectively identifies Heckman models. Leung and Yu find that sufficient variation in at least one element of [??] is required to induce sufficient tail behavior in the inverse Mills ratio for functional form, alone, to identify the model. (24) Leung and Yu (1996) also find that relying on functional form to identify the model becomes less problematic as the percentage of censored observations decreases. To address the questions raised by the lack of an exclusion restriction, we run the model using a variety of specifications and a subsample of the data, in which censoring is less severe, to evaluate the stability of our estimates.

4. Data

The data used in this study consist of a 10% random sample of all thoroughbreds born in the United States in 1993. We obtained these data from The Jockey Club's Foals of 1993 (1995; hereafter, Foal Book), an annual supplement to the American Stud Book that includes information on all thoroughbreds born and registered in the United States, Canada, and Puerto Rico. (25) The sample consists of every U.S.-born horse listed on every 10th page of the Foal Book, and includes 3374 horses. (26)

The market for thoroughbreds provides an ideal setting for examining the effect of asymmetric information on market outcomes because owners likely possess an information advantage over potential buyers and differ in their propensities to sell horses they breed. As discussed by Chezum and Wimmer (1997), the primary informational advantage relates to the horse's temperament and medical history. James Schenk (1997), a former trainer and now president of Hyperion Thoroughbred Consultants, notes that "most trainers (observe) that their best horses seem to get sick less often, have fewer minor injuries, and progress faster than their lesser horses. Better horses also are smarter than average horses; I have never been around a dumb good horse." (p. 1661)

Consistent with the theoretical model, breeders who own the foal at the time of its birth must make decisions regarding the care and treatment of their thoroughbreds before they observe innate quality. (27) These decisions include visits to the veterinarian and blacksmith, choice of feed and supplements, and overall general care that affect the quality of a racehorse but probably remain unobserved by potential buyers. Finally, and in contrast to other empirical studies of adverse selection, published data exist on both horses sold and those retained by their breeders.

We obtained sales data for each observation by matching data from the Foal Book to information contained in the Blood Horse's Auction Guide (Anonymous 1998), which includes the results of every public thoroughbred auction held in North America. These data allow the identification of the horses offered for sale at public auction and the price elicited. The empirical analysis follows Chezum and Wimmer (1997) and concentrates on horses offered for sale as yearlings or younger. (28) In the sample, 925 horses were offered for sale as yearlings or younger.

The variable of interest, Racing Intensity, captures differences in seller characteristics and proxies s, as discussed in the theoretical section. In the thoroughbred industry, breeders differ in the "intensity" of their racing operations. (29) Some breeders sell all horses they raise, whereas others retain a portion for racing purposes. Because some breeders also purchase horses for racing, Racing Intensity is not equal to the probability breeders retain horses they breed. We use Racing Intensity to proxy s because it is publicly available and captures differences in the value breeders receive from participating in the racing end of the business. We expect that the probability a horse is retained increases in the breeder's Racing Intensity.

For each racing season, the American Racing Manual (ARM) publishes the earnings of thoroughbred owners whose horses earned at least $50,000 and whose horses they bred earned at least $30,000 in that year. To measure Racing Intensity, we gathered data for each breeder on the number of races started by horses they owned at the time of the race: Racing Starts and the number of races that horses they bred started, Breeder Starts. (30) Racing Intensity equals the ratio of Racing Starts to (Total Starts + 1), where Total Starts is the sum of Racing and Breeding Starts. (31) To avoid potential endogeneity problems, we construct Racing Intensity using data from 1993, the year before the sale of horses in our sample.

The lower limits for inclusion in the ARM exclude relatively small breeding operations from the data. To account for relatively small operations, we include the variable, Unlisted Breeder, which equals one if both Racing Starts and Breeding Starts are zero and zero otherwise, in the regressions.

We expect the primary variable of interest, Racing Intensity, to affect prices through both owner effort (Effort Effect) and owner choice to sell a horse (Selection Effect). Selection Effect captures the degree to which racing-intensive breeders adversely select the horses they sell, resulting in an inverse relationship between Price and Racing Intensity ([[delta].sup.*][[differential][lambda]/[differential]RI] < 0, where [differential][lambda]/[differential]RI [congruent to] [differential][lambda]/[differential]s). Because racing-intensive breeders are more likely to retain horses they breed, the expected return from effort increases in Racing Intensity, and the Effort Effect predicts a positive relationship between Price and Racing Intensity ([alpha] > 0).

To isolate the effects seller characteristics impose on prices, we must control for each horse's observable characteristics. We obtained data on observable characteristics from the ARM and from the Bloodstock Research Information Service (2000) American Produce Records 1940-1999. We expect prices to increase in the presence of favorable characteristics.

To account for the quality of an observation's pedigree, we include variables on the quality of the sire (father) and dam (mother) that measure their successes as breeding stock and as racehorses. For dams, we include Stakes-Winning Siblings, which equals the number of a dam's offspring that won a stakes race. For the sire, we include Sire's Crop, which equals the number of foals produced by the sire in 1993, and Sire's Age. Successful stallions probably experience relatively long careers and the number of foals produced should increase in the demand for a sire's services. To account for the success of a horse's parents on the racetrack, we include the Dam Standard Starts Index and the Sire's Standard Starts Index. The Standard Starts Index provides a measure of a racehorse's success that allows a comparison across horses and time. The Standard Starts Index increases in the quality of a racehorse.

For each observation, we include the Age in Months, the age of the horse measured in months, on January 1, 1994 (the horses in the sample were born in 1993), in the selection equation. For the price equation, we include Age at Sale, also measured in months. We expect that older horses experience a higher probability of sale and receive higher prices. We include an indicator variable, Kentucky, which is equal to one for horses born in Kentucky and zero otherwise. The largest and most successful breeding operations and thoroughbred sales locate in Kentucky. Kentucky breeders can access facilities, veterinarians, and other professionals that breeders in other states cannot access. (32) Last, we control for the horse's gender by including the variable, Colt, which equals one for male horses and zero otherwise.

Table 1 reports summary statistics for the variables used in the analysis. The first three columns contain the summary statistics for the full sample, horses offered for sale, and horses retained, respectively. The final two columns provide similar data broken down by whether the breeder appears in the ARM. Overall, breeders offered 27% of the horses in the sample for sale. Not surprisingly, listed breeders offer horses for sale at twice the rate (42.6% compared with 19.3%) and receive an average price more than double the price received by unlisted breeders ($36,728 compared with $15,660). Because Leung and Yu (1996) argue that the use of functional form to identify the model proves less problematic as the percentage of censored observations falls, we include a specification that drops unlisted breeders from the regression.

The data show that Racing Intensity does not vary much between horses sold and horses retained by their breeders. Mean Racing Intensity is approximately 0.087 for both horses offered for sale and horses retained by their breeders. The data also show that the mean observable characteristics of horses offered for sale exceed those retained by their breeders, indicating that quality commands higher prices. Stakes-Winning Siblings, Sire's Crop, and both of the Standard Start indexes exhibit appreciably higher values for horses offered for sale compared with horses retained by their breeders. Finally, more than 42% of the horses offered for sale were bred in Kentucky.

5. Empirical Results

Table 2 reports the results of our empirical analysis. Columns 1, 2, and 4 through 6 give results for the price equations, where the natural logarithm of Price equals the dependent variable. (33) Column 3 provides the selection equation's marginal effects, evaluated at the mean.

Column 1 provides results from an OLS regression. In the OLS regression, the coefficient on Racing Intensity proves insignificant, suggesting that asymmetric information does not affect the market. (34) Column 2 contains the results from the Heckman specification, which allows separate estimation of the Effort and Selection Effects. After correcting for sample selection bias, the positive and significant coefficient on Racing Intensity is consistent with the presence of an Effort Effect, indicating that the average quality of horses produced by owners increases in Racing Intensity. The Heckman model also provides strong support for the hypothesis that breeders adversely select the horses they sell, as demonstrated by the negative and significant estimated Selection Effect. (35) The negative and statistically significant coefficient on the Mills Ratio indicates that unobservable factors that increase the probability that a horse sells inversely correlate with unobservable factors that increase price, consistent with the presence of adverse selection.

Column 3 contains the marginal effects from the selection equation, estimated jointly with the price equation from the full-sample Heckman model. These results are generally consistent with our priors. The data indicate that the probability a breeder sells a horse decreases in Racing Intensity. Racing Intensity's marginal effect is negative and significant. Overall, the coefficients in columns 2 and 3 have the expected signs, with the majority significant at the 1% level. Somewhat surprisingly, Unlisted Breeder receives a positive and insignificant coefficient in the price equation, whereas the selection equation indicates that unlisted breeders offer horses for sale 16.5% less often than listed breeders. The marginal effect of Unlisted Breeder proves statistically significant at the 1% level in the selection equation.

To determine whether functional form adequately identifies the price equation, we report the characteristic number from the regression and find that it exceeds twice the magnitude of the acceptable level suggested by Besley, Kuh, and Welsch (1980), indicating that the results should be viewed with caution. Several additional specifications test the robustness of our findings.

Column 4 reports the results of a more parsimonious specification, which we label "Limited Variables," for which we drop the Standard Start Index variables and Sire's Age. As an alternative to relying on functional form, column 6 gives results with the use of Unlisted Breeder as the excluded exogenous variable to identify the price equation. Although the underlying theory precludes the use of exclusion restrictions, the finding that Unlisted Breeder is insignificant in the price equation, but significant in the selection equation, suggests that it meets the criteria for a suitable exclusion restriction. The final column provides the results for which we limit the sample to horses from listed breeders.

The results of the additional specifications produce estimates of the Selection and Effort Effects consistent with the results found in column 2. All estimates are of the expected signs and exhibit statistical significance at standard levels. With the exception of the limited variables regression, the magnitude of the estimated Selection Effect falls within a range of -0.62 to -0.66. Similar results emerge for the Effort Effect, although the estimate of the Effort Effect increases by about 20% in the more parsimonious specification. For the limited variables and listed breeder regressions, the characteristic numbers continue to exceed suggested levels, whereas the specification with an exclusion restriction falls to Besley, Kuh, and Welsch's cutoff of 30. (36,37) The results presented prove robust to changes in specification and data. The estimated coefficients are relatively precise. Nearly every coefficient in the price equations receives the predicted sign and is statistically significant at the 1% level.

In sum, our results show that the unconditional expected quality of horses owned by racing-intensive breeders exceeds the quality of horses produced by less racing intensive breeders, suggesting that breeders who retain more horses exert greater effort than breeders who primarily sell their horses. The results concerning breeder selection decisions clearly show that breeders adversely select the horses they sell as shown by the negative and significant Mills Ratio coefficient. Estimates of the Selection Effect indicate adverse selection proves more severe for racing-intensive breeders.

6. Conclusion

Standard models of adverse selection assume that each seller draws goods from an identical quality distribution and that differences in owner decisions to sell particular goods leads to differences in the expected quality of goods sold. In this paper, we examine a setting in which sellers affect quality by expending effort before offering goods for sale. The analysis predicts that sellers who more likely adversely select the goods they sell exert more effort, complicating researcher efforts to uncover evidence of adverse selection.

We test the propositions using data that contain a random sample of all horses born in the United States in 1993, including horses retained by their breeders. Treating adverse selection as a case of selection bias, we show that racing-intensive breeders, on average, produce higher quality horses. The data also show that sellers adversely select the horses they sell. The negative and statistically significant coefficient on the inverse Mills Ratio indicates that unobservable factors that increase the probability a horse sells negatively correlate with unobservable factors that increase prices. We also estimate a Selection Effect that shows adverse selection exhibits more severity for racing-intensive breeders. Although the Heckman specification supports the hypothesis that asymmetric information affects the market for thoroughbred yearlings, a simple OLS regression provides no support for that hypothesis. The empirical work shows that the assumption that sellers draw goods from an identical quality distribution could prove problematic in some settings.

In this study, buyers cannot observe the owner's quality-enhancing efforts and owners are uncertain whether they will sell a particular good. This study generates uncertainty surrounding the sell-retain decision by requiring owners to exert effort before observing innate quality. The analysis presented could apply to a variety of settings. In labor markets, worker decisions to remain in the labor force depend, in part, on random unobservable shocks at the time workers invest in human capital. Workers more likely to exit the market, invest in less human capital. (38) Additionally, firms that intend to go public might invest less effort in projects that affect future earnings if the market cannot observe such effort. Finally, recent work in insurance and other markets shows that both moral hazard and adverse selection affect market outcomes. Failure to account for potential moral hazard problems in a variety of settings could lead researchers to conclude that adverse selection does not affect outcomes when both hidden actions and hidden information play important roles.

Appendix A

This appendix provides the details of the comparative static analysis presented in the paper.

In the first case, in which owners observe innate quality before choosing effort, market equilibrium effort equals zero. The simultaneous solution of Equations l defines equilibrium price and marginal quality. We want to know how changes in s affect equilibrium price and marginal quality. Totally differentiating the Equations 1 and setting d[??] = 0 gives, in matrix notation,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where [P.sup.**] and [q.sup.**.sub.m] are the equilibrium values of price and marginal quality for this specification. Applying Cramer's rule, the comparative static results are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

These signs occur because we assume that s - b([differential][q.sub.Ave]/[differential][q.sub.m]) > 0. This condition must hold for the equilibrium to reflect adverse selection.

In the second version of the model, owners invest in effort before observing innate quality. The simultaneous solution of Equations 2 defines equilibrium price, marginal quality, and effort. To simplify the comparative static analysis, we substitute price out of Equations 2 and obtain Equations A.1.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (A.1)

The first equation is the difference between the value owners and buyers place on the marginal good. The second equation is the owner's first-order condition for effort. The total differential of Equations A.1, setting d[??] = 0, is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where [q.sup.*.sub.m] and [e.sup.*] are the equilibrium values of marginal quality and effort, respectively. The determinant of the Jacobian is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Applying Cramer's role, the comparative static results are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Substituting the equilibrium values of marginal quality and effort into the first of Equations 2, from the text, gives

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Differentiating with respect to s gives

[differential][P.sup.*]/[differential]s = b([[dq.sub.Ave]/[dq.sub.m]][[dq.sup.*.sub.m]/[differential]s] + [[differential][e.sup.*]/[differential]s]).

Totally differentiating Equations A.1, setting ds = 0, gives

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

The Jacobian is unchanged, and it follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Because b > s, any factor that increases the underlying mean of innate quality increases the equilibrium probability a good is sold and reduces equilibrium effort.

Appendix B: Existence of Unique Equilibrium

In this appendix, we show that the assumptions

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

guarantee the existence of a unique adverse selection equilibrium. Equilibrium is defined by Equations B.1 and B.2.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (B.1)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (B.2)

We use the intermediate value theorem to show the conditions under which a unique adverse selection equilibrium exists. The intermediate value theorem is applied by collapsing the system of equations in Equations B.1 and B.2 into a single equation.

We begin by showing that Equation B.2 can be used to express effort as a well-defined function of [q.sub.m], provided [??]([q.sub.L], [e.sub.0]) > 0, [??]([q.sub.H], [e.sub.0]) < 0, and [??]'([q.sub.m], [e.sub.0]) < 0, for an arbitrary fixed level of effort [e.sub.0]. In the model, effort ranges from 0 to [e.sub.P], where [e.s is the owner's privately optimal effort (defined by s - c'[[e.sub.P]] = 0). We examine Equation B.2 at fixed [e.sub.0], such that [e.sub.0] E (0, [e.sub.P]).

At [q.sub.L],

[??]([q.sub.L],[e.sub.0]) = S[1 -- F([q.sub.L])] -- c'([e.sub.0]) = s - c'([e.sub.0]) > 0, (B.3)

because s - c'([e.sub.P]) = 0 for [e.sub.0] < ep, s - c'([e.sub.0]) > 0.

Evaluating Equation B.2 at [q.sub.H],

[??]([q.sub.H], [e.sub.0]) =s[1 - F([q.sub.H])] - c'([e.sub.0]) = c'([e.sub.0]) < 0. (B.4)

The results contained and Equations B.3 and B.4, coupled with the observation that Equation B.2 strictly decreases in [q.sub.m], allow us to write Equation B.2 to express e as a well-defined function of [q.sub.m],

e([q.sub.m]) = [(c'(e)).sup.-1] s[1 - F([q.sub.m])], (B.5)

where [(c'(e)).sup.-1] is the inverse function of marginal cost.

Substituting Equation B.5 into Equation B.1,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (B.6)

allows us to use the intermediate value theorem to find the conditions under which a unique adverse selection equilibrium exists.

Evaluating Equation B.6 at [q.sub.m] = [q.sub.L] gives

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

because b > s and [q.sub.Ave]([q.sub.L]) = [q.sub.L].

At [q.sub.m] = [q.sub.H], Equation B.5 shows e([q.sub.H]) = 0 and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where [mu] is the mean of population q. As required by the intermediate value theorem, [PSI]([q.sub.H]) < 0 when s > b[[mu] + g([??])]/[q.sub.H] + g([??])], or s must be large enough to ensure that owners retain the highest quality goods allocated, which is necessary for the existence of an adverse selection equilibrium with market failure.

According to the intermediate value theorem, equilibrium exists when [PSI]([q.sub.L]) > 0 and [PSI]([q.sub.H]) < 0. The equilibrium is unique if [PSI]([q.sub.m]) monotonically decreases in [q.sub.m].

Differentiating Equation B.6 with respect to [q.sub.m] gives

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (B.7)

where de([q.sub.m])/[dq.sub.m] = -sf([q.sub.m])]c". [PSI]'([q.sub.m]) is unambiguously negative when [b [dq.sub.Ave]([q.sub.m])/[dq.sub.m]] - s < 0. The result contained in Equation B.7 shows that our assumption is stronger than necessary for the existence of a unique equilibrium. If effort is removed from the model, this assumption is necessary for a unique equilibrium.

Rearranging Equation B.7, the condition for a unique equilibrium shows that the Jacobian presented in Appendix A is positive:

J = c"(s -b[[dq.sub.Ave]/[dq.sub.m]) + sf([q.sub.m])(b - s) > 0.

We thank Stephen M. Miller, Paul Thistle, Dan Gagliardi, Dan Black, Scott Savage, Alan Schlottmann, John Garen, David Richardson, and two anonymous referees for helpful comments. We also thank Dotti Britt and Tian Han for competent research assistance. All mistakes, of course, are ours alone.

Received May 2004; accepted October 2005.

References

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Ausubel, Lawrence M. 1999. Adverse selection in the credit card market. Unpublished Paper, University of Maryland.

Bagnoli, Mark, and Ted Bergstrom. 2005. Log-concave probability and its applications. Economic Theory 26:445-59.

Besley, David A., Edwin Kuh, and Roy E. Welsch. 1980. Regression diagnostics: Identifying influential data and sources of collinearity. New York: Wiley.

Bloodstock Research Information Systems. 2000. American produce records 1940-1999. Lexington, KY: Bloodstock Research Information Systems Inc.

Bond, Eric W. 1982. A direct test of the lemons model: The market for used pickup trucks. American Economic Review 72: 836-40.

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Bradley, Ralph. 2002. Is adverse selection simply moral hazard? Evidence from the 1987 Medical Expenditure survey. U.S. Department of Labor, Bureau of Labor Statistics Working Paper No. 360.

Chezum, Brian, and Bradley S. Wimmer. 1997. Roses or lemons: Adverse selection in the market for thoroughbred yearlings. Review of Economics and Statistics 79:521-6.

Chiappori, Pierre-Andre, and Bernard Salanie. 2000. Testing for asymmetric information in insurance markets. Journal of Political Economy 108:56-78.

Daily Racing Form. 1995. American racing manual: 1994. Lexington, KY: Daily Racing Form.

de Meza, David, and David C. Webb. 2001. Advantageous selection in insurance. RAND Journal of Economics 32:249-62.

Edelberg, Wendy. 2004. Testing for adverse selection and moral hazard in consumer loan markets. Board of Governors of the Federals Reserve System Working Paper.

Genesove, David. 1993. Adverse selection in the wholesale used car market. Journal of Political Economy 101:644-65.

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Greene, William H. 1997. Econometric analysis, 3rd edition. Upper Saddle River, NJ: Prentice Hall.

Heckman, James J. 1979. Sample selection as a specification error. Econometrica 47:153-62.

Jockey Club, The. 1995. American stud book, foals of 1993. New York: The Jockey Club.

Jullien, Bruno, Bernard Salanie, and Francois Salanie. 1999. Should more risk-averse agents exert more effort? The Geneva Papers on Risk and Insurance Theory 24:19-28.

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Pagan, Adrian, and Frank Vella. 1989. Diagnostic tests for models based on individual data: A survey. Journal of Applied Econometrics 4:S29-S59.

Puelz, Robert, and Arthur Snow. 1994. Evidence on adverse selection: Equilibrium signaling and cross-subsidization in the insurance market. Journal of Political Economy 102:236-57.

Schenk, James. 1997. Thoughts from a professional. The Blood Horse 22 March: 1661.

Stewart, Jay. 1994. The welfare implications of moral hazard and adverse selection in competitive insurance markets. Economic Inquiry 32:193-208.

Vella, Francis. 1998. Estimating models with sample selection bias: A survey. Journal of Human Resources 33:127-69.

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Wimmer, Bradley S., and Brian Chezum. 2003. An empirical examination of quality certification in a 'lemons' market. Economic Inquiry 41:279-91.

(1) Several authors attempt to identify the presence of adverse selection in a variety of markets. Examples include, but are not limited to, credit markets (e.g., Ausubel 1999), insurance markets (e.g., Puelz and Snow 1994; Chiappori and Salanie 2000), automobile markets (e.g., Bond 1982; Genesove 1993), and the market for initial public offerings (e.g., Booth and Smith 1986; Gompers and Lerner 1999).

(2) Chiappori and Salanie (2000), who examined the relationship between unobserved factors from participation and performance equations in automobile insurance markets, used a similar approach.

(3) We assume g([??]) is nonnegative, finite, and bounded from above.

(4) In the empirical specification, we assume that q has a normal distribution.

(5) This specification is not without loss of generality. A more general production function introduces ambiguities because the level of innate quality affects the marginal product of effort. The conclusion that seller effort clouds the relationship between seller characteristics and adverse selection holds with a general production function.

(6) Below, we examine the case in which owners exert effort after observing innate quality.

(7) Our specification is in the spirit of Wilson (1980), except we introduce a first stage in which owners can expend effort.

(8) We assume that the innate quality of a good allocated to an owner is independent of the owner's type, s.

(9) In Appendix B, we show that a unique internal equilibrium (i.e., [q.sub.m] [member of] [[q.sub.L], [q.sub.H]]) exists for s [member of] b[[mu] + g([??])]/[[q.sub.H] + g([??]), b], where [mu] is q's population mean.

(10) As discussed by Genesove (1993), reasons other than owners exploiting their informational advantage over buyers must be present for trade to take place in a lemons market. Any individual owner with s > b will not participate in the market.

(11) Assuming that an innate quality's distribution is log-concave ensures that [differential][q.sub.Ave]/[differential][q.sub.m] < 1 (Bagnoli and Bergstrom 2005). Without further restrictions on s/b, log-concavity does not guarantee the existence of a unique adverse selection equilibrium. Appendix B provides details of why the assumptions lead to a unique adverse selection equilibrium.

(12) This result is akin to the moral hazard that occurs in labor contracts when employers cannot verify employee effort. In a fixed-wage contract, employees provide the minimum effort necessary to avoid dismissal.

(13) Consider the alternative, in which buyers expect positive effort and bid accordingly. The owner's dominant strategy still provides zero effort, but the increase in buyer bids increases the marginal quality of goods sold. Because [differential][q.sub.Ave]/[differential][q.sub.m] < s/b < 1 and [differential][q.sup.R]/[differential]e = 1, the effect an increase in marginal quality has on expected realized quality is less than the effect an increase in effort has on expected realized quality, and buyers do better by expecting zero effort.

(14) These results are derived in Appendix A.

(15) This version of the model leads to the result that a seller's propensity to select goods adversely increases in s, which, as in Genesove (1993), results in an inverse relationship between observed prices and s.

(16) Appendix A contains the formal derivation of these results.

(17) The result shows that as the marginal cost curve becomes steeper, the adverse selection effect becomes more dominant. When the marginal cost of effort increases quickly, effort becomes less responsive to changes in s.

(18) Thus, we also note that the probability of retention provides a credible signal about owner effort.

(19) Greene (1997, p. 951) reports that a truncated normal distribution with mean [mu], standard deviation [sigma], and truncation point [alpha] has an expected value of [mu] + [sigma][lambda]([alpha]), where [lambda] is the inverse Mills ratio as defined above.

(20) Although the theoretical model proposes that owner effort shifts the distribution of quality, the empirical model captures the effect that any factor that correlates with the unconditional expected quality of goods and seller characteristics imposes on prices.

(21) We discuss the relationship between [[??].sub.i] and [[??].sub.i] below.

(22) Other distributional assumptions could be equally valid. Our results show that the data fit the probit model relatively well.

(23) In Appendix A, we show that [q.sub.m] increases in [??], or goods with relatively high observable quality are more likely to be sold.

(24) Citing Besley, Kuh, and Welsch (1980), Leung and Yu (1996) recommend using the characteristic number to test for sufficient variation in elements of X, finding that results with characteristic numbers over 20 might prove problematic. Besley, Kuh, and Welsch (1980) use 30 as the cutoff. We report the characteristic number for each regression presented.

(25) The Jockey Club reports that 36,455 registered horses were born in 1993, of which 33,174 were born in the United States.

(26) The Foal Book lists horses in alphabetic order by the name of their dam (mother).

(27) Decisions regarding the care of the foal's dam (mother) affect the quality of the young horse as well.

(28) In the thoroughbred industry, thoroughbreds become yearlings on January first of the year following their birth. This classification continues for all ages. The analysis includes a small number of horses sold in November of their first year and are, therefore, not yet yearlings.

(29) In the thoroughbred industry, breeders can differ in their propensity to sell horses they raise for a variety of reasons. Chezum and Wimmer (1997) follow Genesove (1993) and argue that some breeders might experience a capacity constraint and must sell a portion of the horses they raise.

(30) For horses bred, the breeder may or may not own the horse at the time the horse races.

(31) Adding one to the denominator avoids division by zero.

(32) In regressions not reported, we included indicator variables for other states with large concentrations of horse farms. The character of the results does not differ from the results reported below.

(33) Specifically, we use the "hammer price," the last bid received.

(34) These results differ from Chezum and Wimmer (1997), who found that Racing Intensity significantly negatively affects prices. Chezum and Wimmer used data from a single sale with fewer observations. Because a single sale draws a smaller pool of sellers, differences in seller effort might not vary as much in a single sale as they vary in a random sample of sales. In any event, these results call into question the robustness of an approach that does not account for seller selection decisions to measure the effect adverse selection imposes on market outcomes.

(35) Following Chezum and Wimmer (1997), we test for the presence of scale effects in the data and find that scale does not affect these results.

(36) In addition to these regressions, we used the variable Racer, which equals one if the breeder had experienced at least one racing start and zero otherwise, as the variable of interest. The results from this regression prove consistent with those reported. We also estimate the model with a variety of randomly selected subsamples, and the results remain consistent.

(37) As part of our analysis, we follow Pagan and Vella (1989) and test the assumption that normality provides the appropriate functional form for the selection correction term and cannot reject the null of normality.

(38) Mincer and Ofek (1982) argue that job market interruptions are endogenous, in which human capital investment decreases in the probability of a career interruption. In this setting, workers with high nonmarket opportunity costs invest less in human capital and experience more career interruptions.

Bradley S. Wimmer * and Brian Chezum ([dagger])

* Department of Economics, University of Nevada Las Vegas, Las Vegas, NV 89154-6005, USA; E-mail [email protected]; corresponding author.

([dagger]) Department of Economics, St. Lawrence University, Canton, NY 13617, USA; E-mail [email protected].

Table 1. Summary Statistics (a)

 Offered Breeder
Variable Full Sample for Sale Retained

Price -- 27,048 --
 (55,371)
ln(Price) -- 9.197 --
 (1.408)
Offered for sale 0.274 1 0
 (0.446)
Racing intensity 0.0870 0.0866 0.0872
 (0.226) (0.214) (0.231)
Stakes-winning 0.167 0.280 0.125
 siblings (0.485) (0.594) (0.429)
Sire's crop size 23.286 36.406 18.330
 (20.217) (21.236) (17.417)
Sire's age 13.242 13.056 13.312
 (4.425) (4.580) (4.364)
Sire standard 17.262 28.496 13.020
 start index (28.052) (34.892) (23.644)
Dam standard 1.556 2.292 1.278
 start index (6.816) (5.293) (7.290)
Age (months) 8.449 8.608 8.389
 (1.228) (1.211) (1.230)
Age at sale -- 14.974 --
 (4.299)
Kentucky 0.206 0.427 0.122
 (0.404) (0.495) (0.327)
Colt 0.504 0.522 0.497
 (0.500) (0.500) (0.500)
Unlisted breeder 0.652 0.459 0.725
 (0.476) (0.499) (0.447)
Observations 3374 925 2449

 Listed Unlisted
Variable Breeders Breeders

Price 36,728 (b) 15,660 (c)
 (70,513) (24,276)
ln(Price) 9.533 (b) 8.803 (c)
 (1.397) (1.317)
Offered for sale 0.426 0.193
 (0.495) (0.395)
Racing intensity 0.250 0
 (0.326)
Stakes-winning 0.321 0.085
 siblings (0.666) (0.324)
Sire's crop size 33.430 17.880
 (20.128) (18.076)
Sire's age 13.150 13.291
 (4.465) (4.403)
Sire standard 27.024 12.060
 start index (37.128) (19.856)
Dam standard 2.648 0.974
 start index (10.851) (2.744)
Age (months) 8.600 8.368
 (1.237 (1.216)
Age at sale 15.298 (b) 14.593 (c)
 (4.104) (4.493)
Kentucky 0.356 0.125
 (0.479) (0.331)
Colt 0.517 0.497
 (0.500) (0.500)
Unlisted breeder 0 1

Observations 1173 2201

(a) Standard deviations in parentheses.

(b) Summary statistics for 500 horses offered for sale.

(c) Summary statistics for 425 horses offered for sale.

Table 2. Price and Selection Regressions (a)

 Heckman Selection
 OLS Specification Equation (b)

Racing intensity 0.061 0.641 ** -0.294 **
 (effort effect) (0.370) (3.15) (6.49)
Selection effect -- -0.660 ** --
 ([[beta].sub.[lambda]] (5.34)
 [differential][lambda]/
 [differential]RI)
Stakes-winning 0.55 ** 0.531 ** 0.008
 siblings (10.14) (8.91) (0.51)
Sire's crop size 0.023 ** 0.013 ** 0.006 **
 (12.44) (5.30) (11.84)
Sire's age 0.035 ** 0.032 ** 0.001
 (4.56) (3.79) (0.60)
Sire's standard 0.006 ** 0.004 ** 0.001 **
 start index (4.98) (3.25) (2.87)
Dam standard 0.037 ** 0.041 ** 0.000
 start index (3.03) (2.85) (0.44)
Age at sale (months) 0.036 ** 0.035 ** 0.021 **
 (4.66) (4.70) (3.53)
Kentucky 0.667 ** 0.392 ** 0.150 **
 (8.75) (4.35) (6.31)
Colt 0.134 ** 0.105 0.025
 (2.03) (1.47) (1.61)
Unlisted breeder -0.250 ** 0.076 -0.165 **
 (3.43) (0.83) (7.76)
Inverse mills -- -0.917 ** --
 ratio ([lambda]) (8.90)
[rho] -- -0.74 ** --
 (14.10)
Constant 6.739 ** 8.034 ** --
 (35.94) (34.41)
[R.sup.2] (log likelihood) 0.504 -1453 -1453
Characteristic
 number -- 69 --
Observations 925 3374 3374

 Limited Exclusion Listed
 Variables Restriction Breeders

Racing intensity 0.763 ** 0.766 ** 0.735 **
 (effort effect) (3.47) (3.35) (3.76)
Selection effect -0.635 ** -0.620 ** -0.550 *
 ([[beta].sub.[lambda] (4.96) (5.65) (2.02)
 [differential][lambda]/
 [differential]RI)
Stakes-winning 0.573 ** 0.632 ** 0.571 **
 siblings (9.29) (8.35) (9.31)
Sire's crop size 0.014 ** 0.016 ** 0.014 **
 (6.16) (4.92) (6.14)
Sire's age -- -- --
Sire's standard -- -- --
 start index
Dam standard -- -- --
 start index
Age at sale (months) 0.039 ** 0.051 ** 0.038 **
 (4.99) (4.42) (4.98)
Kentucky 0.514 ** 0.648 ** 0.517 **
 (5.82) (5.67) (5.86)
Colt 0.103 0.083 0.103
 (1.40) (0.79) (1.40)
Unlisted breeder 0.026 -- --
 (0.28)
Inverse mills -0.893 ** -0.877 ** -0.855 **
 ratio ([lambda]) (8.87) (17.53) (5.03)
[rho] -0.709 ** -0.70 ** -0.682 **
 (13.69) (14.01) (7.24)
Constant 8.495 ** 8.131 ** 8.491 **
 (43.99) (30.81) (42.96)
[R.sup.2] (log likelihood) -2958 -2958 -1454
Characteristic
 number 57 30 109
Observations 3374 3374 1173

(a) Absolute value of White-corrected t or z statistics in parentheses.

(b) Marginal Effects, and appropriate z statistics, evaluated at means
from Full Information Maximum Likelihood (FIML) Full-Sample Heckman
model selection equation.

* Significant at 0.05 level.

** Significant at 0.01 level.