Uwe Sunde
email: sunde@iza.org.

Heterogeneity and Performance in Tournaments: A Test for Incentive Effects using Professional Tennis Data

Keywords: Human Capital, Wage Models, Internal Labor Markets < J4 -Particular Labor Markets < Elimination Tournaments, Incentives in Tournaments, Tennis JEL-classification: J4, J3, M5 Elimination Tournaments, Uneven Tournaments, Incentives in Tournaments, Tennis

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Introduction

Firms frequently use promotion tournaments to provide their workers with incentives for effort exertion by posting a prize to be won by the relatively best performing contestant. Theory predicts that the total amount of effort to win the prize exerted in the course of a tournament is highest when the contestants are homogeneous (see [START_REF] Lazear | Rank Order Tournaments as Optimal Labor Contracts[END_REF][START_REF] Rosen | Prizes and Incentives in Elimination Tournaments[END_REF][START_REF] Prendergast | The Provision of Incentives in Firms[END_REF]. Uneven tournaments, where one contestant is ex ante stronger than the other(s), result in lower effort exertion of the contestants. Intuitively, the larger the initial disadvantage of the underdog, the more costly it is for him in terms of effort to compensate for the handicap. On the other hand, the favorite knows about his advantage and can reduce his efforts without endangering his prospects of success. The assumption of performance incentives through prizes has received some empirical support, see e.g. [START_REF] Ehrenberg | Do Tournaments Have Incentive Effects?[END_REF]Bognanno (1990a, 1990b) for evidence from golf tournaments. 1 However, the negative incentive effect of heterogeneity of contestants has so far not been tested with non-experimental data because of the difficulties associated with measuring heterogeneity and effort. 2 This paper is a first attempt to close this obvious gap in the literature. Testing the incentive effects of heterogeneity with real world data is important in view of the widespread use of tournaments and the fact that heterogeneity of contestants is the rule rather than the exception. I use data from the final two rounds of the most prestigious tennis tournaments for professionals from the Association of Tennis Professionals (ATP) to test the prediction that higher ex ante heterogeneity reduces incentives to provide effort. Tennis data are particularly well suited for this, since they perfectly replicate the simple setting of a two-person tournament, allowing for the clearest theoretical predictions. The structure of information available in tennis data closely resembles the structure and requirements of information in theoretical models of one-shot tournaments. At the same time, tennis contests resemble the environment of a corporate tournament, and involve considerable stakes for the players. Prizes are known for each tournament and at each stage. The strength of players can be inferred from their position in the world ranking prior to the respective tournament, allowing to construct a measure of the heterogeneity of the contestants that is generally not available in other data. Rankings and prizes in the data are predetermined and common knowledge before the match begins, which 1 Several theoretical predictions concerning incentives through prizes have recently been confirmed in empirical studies using firm-level data, see [START_REF] Eriksson | Executive Compensation and Tournament Theory: Empirical Tests on Danish Data[END_REF] and [START_REF] Bognanno | Corporate Tournaments[END_REF]. 2 [START_REF] Abrevaya | Ladder Tournaments and Underdogs: Lessons from Professional Bowling[END_REF] examines the determinants of winning probabilities of ex ante unequal players in professional bowling but does not consider effort exertion. Experimental studies dealing with uneven tournaments include [START_REF] Bull | Tournaments and Piece Rates: An Experimental Study[END_REF], Schotter and Weigelt (1992), and more recently [START_REF] Harbring | Separating the Wheat from the Chaff: An Experimental Study on the Sorting Function of Tournaments[END_REF]. None of these directly tests the effect of varying heterogeneity on effort. meets the requirements of the theory and avoids problems of endogeneity. Moreover, unlike in the context of intermediate-level firm data, tennis tournaments end with the final. Thus, prize money for reaching and/or winning the final is a good proxy for the prize of winning the elimination tournament implied by matches on the two last rounds of tennis tournaments.

Similar to settings in firms, effort exerted by tennis players is not directly observable by the principal or the audience (including the econometrician), which constitutes the major empirical problem in testing for the existence of an incentive effect caused by player heterogeneity. On the basis of a simple structural model of a tennis match, this paper derives testable implications that can be tested even if effort is not directly observable. The identification strategy rests on the observation that the omitted variable bias caused by unobservable effort has different effects for favorites and underdogs in a match. The empirical results suggest that the hypothesis that heterogeneity has no effect on incentives and effort exertion can be rejected.

The paper proceeds as follows. Section 2 describes the data, section 3 presents a simple statistical model of a tennis match that can be used to derive behavioral predictions, and that serves as basis for the empirical implementation. The section ends by describing the identification strategy used in the application. Section 4 presents the empirical results, and section 5 concludes.

Data from Professional Tennis Tournaments

The data are provided by the ATP and comprise information for 156 tournaments of the most and second most prestigious categories, Grand Slam and Master Series, for male tennis professionals for the years 1990 until 2002. 3 In particular, the data stem from the four Grand Slam of opponents and avoid effects of initial seedings, only data for the last two rounds, that is for semifinals and finals, are used. 5 Eliminating matches with missing information on games won or suspended matches due to injury leaves a total of 460 matches comprising 920 observations for individual players. Table 1 provides some summary statistics.

Empirical Model

A Simple Structural Model of a Tennis Match

Consider the following simple tournament model: two players, a favorite f and an underdog u compete for the (stage) prize $ that accrues to the winner of a tennis match. The two players differ with respect to their ability a i , i = f, u, to play, where a f > a u . Both players can choose their effort levels, e f ≥ 0 and e u ≥ 0, respectively, in order to affect their probability to win the match. The match itself consists of a certain number of balls, and, abstracting from the precise rules for winning a tennis match, the likelihood of winning the match increases in the number of balls won.6 Let the strength of play of a player during a given ball b be given by S f b and S ub , respectively. This strength of play depends on the players' respective effort levels e ib , their ability a i and a vector of other characteristics X, including factors like the stage price, the type of soil, season etc. Let us assume for simplicity that this relationship is linear, such that

S ib = αa i + βe ib + γ X i + ε ib , i = f, u .
(1)

The last term, ε ib , reflects a random component affecting the strength of play during a given ball b. This component of luck can be interpreted as a propensity to commit a mistake, or over the year. equally to hit an ingenious ball. Given that there are many ways to commit a mistake and loose or win a ball, it is natural to assume that ε ib is distributed according to a type-1 extreme value distribution, given that it is the first failure in one of many dimensions that leads to a mistake and the end of the ball. For convenience, we assume that ε f b and ε ub are distributed independently, which can be justified by the fact that players commit their mistakes individually and independently from their opponent on the other side of the net (assuming technical fouls are forbidden). The favorite wins a given ball b if

S f b ≥ S ub ⇔ αa f + βe f b + γ X f -αa u + βe ub + γ X u > ε ub -ε f b , ( 2
)
while the underdog wins it otherwise. Given this structure, ε ub -ε f b are independent draws from a log Weibull distribution. 7 The probability to win a given ball b is therefore given by a standard logit model, such that Pr(favorite wins ball) = e α(a f -au)+β(e f b -e ub )+γ (X f -Xu) 1 + e α(a f -a u )+β(e f b -e ub )+γ (X f -X u ) ,

(

and the associated likelihood given information on an entire match could be estimated as

L f = Π (# balls f ) e α(a f -a u )+β(e f -e u )+γ (X f -X u ) 1 + e α(a f -a u )+β(e f -e u )+γ (X f -X u ) • Π (# balls u) 1 1 + e α(a f -a u )+β(e f -e u )+γ (X f -X u ) .
(4)

The log odds ratio of the favorite rather than the underdog winning a given ball is consequently given by ln Pr(favorite wins ball)

Pr(underdog wins ball) = α(a f -a u ) + β(e f b -e ub ) + γ (X f -X u ) , (5)
which is a function of the relative strength of play of both players, S f b -S ub .

However, in the application to ATP data, we do not have information on every single ball played, or the associated effort levels, so that this model is not directly applicable. As is explained in the data section, the data are aggregated on the level of an entire match, including the total numbers of games and sets won by each player. Moreover, the odds ratio is of little use in tennis, where the rules of the game not necessarily imply that the player winning the most games wins the match. The model so far does neglect issues like the rules of winning a tennis match, in particular the endogeneity of the duration of a match. For the purpose of estimating a model of effort choice, it is important to control for the inherent endogeneity of the number of sets played, or the duration of the match in terms of the total number of games played. Hence, applying the odds ratio in this context does not seem appropriate, in particular since it is merely instrumental and does not allow to use all individual observations independently for the estimation. Consequently, in what follows we modify the framework in order to be applicable to the data at hand. Given the proportions data on the match level, we concentrate on average effort exertion during a match, rather than on the effort exerted to win a given ball. Optimal strategies of a player refer to the effort level for the entire match. This assumption neglects any dynamics of effort exertion during a match. On the other hand, it reflects the usual notion of a one-shot decision prevalent in the (theoretical and empirical) tournament literature (see e.g. [START_REF] Lazear | Rank Order Tournaments as Optimal Labor Contracts[END_REF]. The probability to win a match increases monotonically in the proportion of games won by a player, rather than the odds ratio of a given point. A sensible implementation has to control for the endogeneity of the duration of the match. This can be done by normalizing the number of games won by a player using information on the number of sets played, and estimating the model separately for favorites and underdogs, allowing for dependencies in the error structure. Such a specification takes into account that, in order to increase their chance to win the tennis match, players have to win as many games as possible, rather than the majority of games. With data on players i, j = f, u meeting in match m in tournament k, a corresponding specification of the relative strength of play during the match that can be directly estimated by OLS is

ln P imk = α(a imk -a jmk ) + β(e imk -e jmk ) + γ X mk + u imk , ( 6
)
where P imk is the average number of games per set (or match) that player i wins, which is directly related to the average differential in the strength of play, S f -S u during the match under observation. This reflects a modified version of a linear probability model to deal with proportions data that uses observations on both players. 8 Potential dependencies between the opponents in a given match are taken into account by allowing the error components imj to be heteroskedastic and correlated between the players of a given match m.

Derivation of Testable Hypotheses

That stage prizes provide stronger incentives for effort exertion, while effort provision decreases with the heterogeneity of players is a well-known result in the tournament literature (see e.g. [START_REF] Lazear | Rank Order Tournaments as Optimal Labor Contracts[END_REF][START_REF] Rosen | Prizes and Incentives in Elimination Tournaments[END_REF][START_REF] Prendergast | The Provision of Incentives in Firms[END_REF]. This subsection derives these results within the framework laid out before, following the treatment of [START_REF] Prendergast | The Provision of Incentives in Firms[END_REF].

Consider the optimal strategies of a player, given that he can only choose one effort level for the entire match, and concentrate on pure strategy equilibria. 9 For simplicity, abstract from 8 See Greene (1997, pp. 894-896) and Kennedy (2003, pp. 264-266) for formal derivations.

9 These assumptions constitute a substantial simplification. In fact, [START_REF] Klaassen | Are Points in Tennis Independent and Identically Distributed? Evidence from a Dynamic Binary Panel Data Model[END_REF] show that the probability of winning a point is not independent from who won the previous point, and that some points and notation for match and tournament, and let the utility of a player be additively separable in the utility derived from winning the stage price and the disutility of exerting effort,

u i = w($) -c(e i ) , (7)
where as before $ reflects the stage prize, and w(•) is a monotonically increasing function. The cost of exerting effort, c(•), is an increasing and convex function of effort e i .

The probability of winning the match, and thus the stage prize, is a function π(•) that corresponds to the relative strength of play of both players at the match level, analogous to the strength of play during a ball in condition (2). Then, the player's problem is to choose effort in order to maximize expected utility, max

e i Eu i = max e i w($)π i (a i -a j , e i -e j , X)) -c(e i ) , (8)
where π i is defined as the probability of player i winning the match analogously to (3). As before, this probability can be shown to depend on the relative strength of play during the match. Hence, π(•) is a monotone transformation of P imk as expressed in condition (6). 10 Also note that, analogously to (3), the probability π(•) is characterized by a logistic distribution. The optimal effort choice e * i is characterized by the first order condition satisfying

∂Eu i e i ! = 0 ⇔ w($) ∂π(•) ∂e i ! = ∂c(e i ) ∂e i . ( 9
)
Existence and uniqueness is ensured by the properties of the cost function c(•), and the function governing the winning probability π(•) of player i. The equilibrium effort allocation is given by the effort of player i, e * i , and the corresponding effort level of the competitor player, e * j , such that neither player has an incentive to deviate from his choice given the other player's choice and the environment, implying

e * i = c -1 w($) ∂π(•) ∂e i . ( 10
)
Condition ( 9) directly delivers several testable hypotheses concerning the optimal effort choice. Consider first the favorite's effort choice, e * f : Everything else equal, the equilibrium effort of the favorite player, e f , is higher, the higher the stage prize $; and the lower the ability difference to the underdog, (a f -a u ) . 11 Concerning the underdog's effort choice, a similar set games are more important to players than others. Similarly Walker and Wooders (2001) provide evidence from ten tennis matches that the observed serving behavior is consistent with an equilibrium in mixed strategies. 10 In particular, the probability is strictly monotonically increasing and concave in the average number of games per set the player wins. 11 The first result follows directly from condition (9); the second result follows from the fact that the marginal change in the probability of winning ∂π(•)/∂e i is equal to the density in the distribution of uf evaluated at the respective ability differential a f -au. Since the error distribution is logistic and the density of the logistic has a mode at 0, equilibrium effort is highest for equally strong opponents with a f = a u who then exert the same effort in equilibrium. of results can be derived: Everything else equal, the equilibrium effort of the underdog player, e u , is higher, the higher the stage prize $; and the lower his (relative) ability (a f -a u ).12

Empirical Implementation and Identification

The simple framework illustrates how the incentive effects arising from the stage prize and the player heterogeneity rest on the assumption that players can affect their winning probability, and thus their probability of winning the prize, by choosing the effort with which they play. In most previous empirical studies using e.g. data on golf tournaments, effort is observed directly in terms of individual scores, and the tournament winner is determined by who requires the smallest number of hits for all rounds. Most importantly, effort is easily observed for each player independently of other players, and a measure of ability in terms of a ranking is available.

In tennis, on the other hand, as in corporate tournaments, a player's effort is hardly observable independently from the opponent, posing a particular conceptual difficulty for testing effects of prizes and heterogeneity in ability on absolute rather than relative performance.

However, it is intuitive that not only relative, but also absolute efforts of both players affect the observed outcomes in terms of games. While a player affects his probability to win the match by his (absolute) effort choice, the probability is essentially a function of relative effort, as is illustrated by the simple framework presented before.

Testing the hypothesis about the incentive effects of stage prizes is straightforward, since the only way that stage prizes can enter the optimization is through effort exertion, illustrated by condition (9). Testing the hypothesis about heterogeneity leading to lower effort provision in uneven tournaments is a little more intricate than testing for incentive effects of prizes. Note that heterogeneity in ability a f mk -a umk can be measured by the difference of the players' ATP ranking before the match. Recalling the empirical specification of strength of play given by ( 6)

and the behavioral model presented before, ability differences enter the individual optimization problem in two ways. The total effect of players' heterogeneity is given by

d ln P imk d(a f mk -a umk ) = ∂ ln P imk (•) ∂(a f mk -a umk ) + ∂ ln P imk (•) ∂e imk ∂e imk ∂(a f mk -a umk ) . ( 11
)
A higher superiority of the favorite over the underdog in terms of ability necessarily leads to a larger differential in the strength of play between the two, and therefore to quicker and clearer decisions, that is more games won by the favorite and fewer by the underdog. This is the case even if the efforts of the players are not affected by heterogeneity in their ability. This plain capability effect is given by ∂ ln P imk /∂(a imk -a jmk ) = α, the first term in condition (11). However, as was shown in the last section, both players exert less effort when the chances of winning are more unequal, as exemplified by a higher superiority of the favorite over the underdog, (a f mk -a umk ). Using the number of games won per set as outcome variable, agency theory would predict that the total number of games decreases with the heterogeneity of the two players. This is the incentive effect of the heterogeneity as predicted by tournament theory, the second term on the right hand side of condition (11). 13 Hence, heterogeneity affects the observable outcome variable through a direct capability effect, the first term on the right hand side of equation ( 11), and an indirect effort effect, the second term.

Distinguishing capability and incentive effects is important. From the tournament designer's view, the incentive effect is crucial, since it can be affected by the appropriate incentive scheme, while capability is a datum. From a methodological point of view, testing tournament theory essentially means identifying the incentive effect from the plain capability effect, which is only of secondary interest. The main empirical problem arises from the fact that effort, or the difference in effort exertion e imk -e jmk , is unobservable in the data. In order to identify whether a potential effect of heterogeneity on the outcome variable merely captures a plain capability effect, or whether it really involves an incentive effect, one can exploit the different impact of heterogeneity on the optimal decisions of favorite and underdog, as is shown now. In particular, the indirect effort effect, the second term of equation ( 11), is unambiguously negative for both favorite and underdog, given the results from the optimal effort decisions presented before:

Both players decide to exert less effort the larger the capability difference between favorite and underdog (i.e. the larger a f mk -a umk ). On the other hand, the direct capability effect tends to increase the relative strength of play of the favorite, ∂P f mk /∂(a f mk -a umk ) = α > 0, while it decreases the relative strength of play of the underdog, ∂P umk /∂(a f mk -a umk ) = -α < 0.

Suppose now the conjecture were true that the relative strength of play, reflected by the number of games per set a player wins, were entirely driven by capability differences, not by incentive effects. Then, larger heterogeneity would lead, everything else equal, to more games won by the favorite, and fewer games won by the ex ante weaker player, whose deficient capability reduces his winning probability for every single point. If incentive effects also played a role, the relative strength of play of the underdog would still be lower the larger the relative strength of the 13 Since the direct effect of prizes on observable outcomes, ∂ ln P imk /∂$ is zero, any effect of $ on P imk reflects an effort effect, which is supposedly positive, given the previous analysis: One expects (∂ ln P imk (•)/∂e imk )(∂e imk /∂$ mk ) > 0, since by assumption, effort increases the games won, ∂ ln P imk /∂e > 0, while a higher prize in turn increases effort. the favorite the overall effect on the relative strength of play is ambiguous: a larger capability difference would tend to increase relative performance, while an opposing incentive effect that leads to lower effort exertion decreases relative performance. Hence, by looking at favorites and underdogs separately, one can identify whether heterogeneity has an effect on incentives and effort exertion in tennis matches.

Econometrically, this identification strategy can be seen as consequence of the omission of unobserved relevant variables, namely effort, in the estimation. Suppose regressing the number of games per set won by player i, on the explanatory variables of interest, relative ability and relative effort, neglecting for the moment other relevant observable controls X, such that ln

P imk = α i (a imk -a jmk ) + βe imk + u imk , ( 12
)
where (ã f mk -ãumk ) is the observed heterogeneity in terms of the rank difference between favorite and underdog. Then it is easy to show that a regression taking only into account the observable relative ability difference, (a f mk -a umk ), yields biased and inconsistent estimates, with

E OLS (α i ) = α i + αCov ((a f mk -a umk ) , e imk ) V ar ((a f mk -a umk )) ,
as long as the incentive effect is present, i.e., Cov ((a f mk -a umk ) , e imk ) = 0. Note that according to the behavioral model presented before, this incentive effect is negative for favorites and underdogs, i = f, u. On the other hand, we have α f = -α u as consequence of the direct capability effect. Hence, the omitted variable bias provides an opportunity for a testable restriction imposed by the existence of an incentive effect: any asymmetry in the effects of heterogeneity between favorites and underdogs that blurs the pure capability effect can be interpreted as the consequence of an incentive effect at work.

Empirical Application and Results

A direct but somewhat restrictive way of testing for the existence of an incentive effect of player heterogeneity can be obtained by estimating the following empirical model

ln P imk = α + α 1 F av imk + α 2 HET imk + βHET imk * F av imk + γ 1 Z mk + γ 2 X k + u imk , ( 13
)
where P imk is the average number of games per set (or match) that player i won during match m in tournament k, measuring the strength of play of player i. and zero if he is the underdog. HET imk = (a imk -a jmk ) is a measure of heterogeneity of the relative strength of the contestants at the outset of the match. The baseline measure is the difference between the player's own ATP rank and that of his opponent. Note that favorites, that is players with a lower ranking than their opponent, are associated with negative values for HET , while underdogs exhibit positive values. Consequently, if the heterogeneity term exhibits different coefficients for favorites and underdogs the null that no incentive effect is present in the results can be rejected.

Since using the difference of ranks as heterogeneity neglects that quality differences are likely to be larger among low ranked players, i.e. higher among the top ten than among players ranked 91 to 100, I also construct the heterogeneity measure proposed by [START_REF] Klaassen | Are Points in Tennis Independent and Identically Distributed? Evidence from a Dynamic Binary Panel Data Model[END_REF], which takes the pyramid structure of quality prevalent in tennis explicitly into account.

The measure is based on the round of the tournament in which each player is expected to drop out, and transforms linear ranks into a logarithmic measure of relative strength of player i,

R i = K -log 2 (RAN K i ),
with K being the number of rounds played in the respective tournament k (i.e. 6 in Masters and 7 in Grand-Slams), and RAN K i being the ATP-ranking. Note that the better ranked a player is, the later he is expected to drop out of the tournament, implying a higher measure R i . The measure of relative strength of player i against opponent j in match m in tournament k is computed as HET KM imk = R j -R i . Given that higher R-measures indicate better players, HET KM imk , similar to the plain rank difference, should have a negative effect on individual effort, because it is negative if i is the favorite and positive if i is the underdog. The capability effect predicts a negative coefficient on HET for underdogs, but a positive coefficient for favorites. Z mk contains information on the level of matches, such an indicator for a final as opposed to a semi-final, and X k are tournament characteristics. In the estimations presented below, I use a full set of tournament and year dummies throughout. 14 Note that all regressors are predetermined before the respective tournament or match starts. Since the performance of the players of a particular match is interdependent, I allow the residuals u to be clustered on the level of a match, and report robust standard errors.

The hypothesis to be tested is that heterogeneity of players does not unfold an incentive effect, and that therefore any observable outcomes are exclusively driven by a capability effect. 14 To control for effects of the rules of tennis on the number of games, I also added W inner imj as an indicator whether player i eventually lost the match m in tournament k as additional control, even though this does not constitute a predetermined variable. The results are qualitatively unchanged. Experiments using data on characteristics such as soil surface, season, month, outdoor dummy, tournament size in terms of the number of participants, and tournament type (grand slam or not) instead of tournament dummies lead to essentially identical results. While also leading to similar results, I do not report findings when adding the respective rank of the player as additional explanatory variable, since it is not clear why the absolute rank in addition to relative strength should matter for the observed performance. Detailed results are available on request. A coefficient estimate of β that is significantly different from zero would allow to reject the hypothesis that no incentive effect is present.

The results presented in Table 2 provide support for this conjecture, regardless of whether one uses the games won per set or per match as dependent variable, and regardless of the measure for heterogeneity used: the interaction term is positive and significant. This indicates that, for the same level of heterogeneity, underdogs do significantly better than favorites, implying that a negative incentive effect moderates the positive effect of a stronger ability on the favorites' performance relative to the underdogs.

A less restrictive specification allows all coefficients to be different for favorites and underdogs. Consequently, we also estimate a model like,

ln P imk = α + βHET imk + γ 1 F inal mk + γ 2 Z mj + γ 3 X j + ε imk , (14)
The OLS estimation results displayed in the left panel of Table 3 show a negative effect of heterogeneity on the outcome variable for underdogs as expected from both capability and incentive effects.

In contrast, heterogeneity has smaller impact on favorites' performance, which is inconsistent with an explanation based exclusively on capability, but consistent with slacking-off due to reduced effort. A possible explanation for this would be that favorites happen to win most of their matches, while the rules of the game imply that the number of games won by the winner exhibits less variation than that won by the loser of a match. The effect of heterogeneity on the outcome variable should thus be downward biased for winners relative to that of losers. To test whether this explanation plays a role, I estimate the model separately for underdogs and favorites who lost their match, and whose number of scores is not constrained by the rules of the game. The results are depicted in the right panel of Table 3. Looking at losers who were underdogs, again the effect of individual relative ex ante weakness is significant and negative.

However, the empirical results show a positive but insignificant coefficient for heterogeneity in the favorites' sample, implying that the larger the predominance of a favorite, the fewer games he scores. This is only consistent with the incentive effect counteracting the capability effect, with both effects being apparently approximately equally strong. Moreover, simple t-tests show that the difference in the heterogeneity effects on the outcome variable between favorites and underdogs is significant. Results for the log of the share of games won per match rather than per set as outcome variable are qualitatively similar. 15

Conclusion

The empirical results presented in this paper support the theoretical implication that effort exertion in tournaments is highest for homogeneous contestants. In particular, the results provide evidence for an adverse incentive effect due to heterogeneity that counteracts the capability effect for favorites, while both effects reinforce each other for underdogs. While the application used data from professional tennis, the point made in this paper is more general. The widespread use of tournaments in corporations and organizations, the substantial heterogeneity of contestants in these tournaments, and the lack of reliable information about effort exertion emphasize the importance of devising identification strategies that allow the tournament designer to evaluate whether the intended incentives actually affect individual behavior. The approach shown above is a first step in this direction. In future research, it would be interesting to apply the model to (potentially more detailed) data from different sports, contexts and environments.

15 These results are not reported for brevity but available upon request.

F

  tournaments per year: Australian Open (Melbourne), Roland Garros (Paris), U.S. Open (New York), and Wimbledon (London). Additionally, the following eight Masters tournaments are included in the data: Cincinnati, Hamburg, Monte Carlo, Montreal, NASDAQ-100 (Miami), Pacific Life (Indian Wells), BNP Paribas (Paris), and Rome. For each tournament and year, the data contain detailed match information including the names of the respective players as well as their current ranking in the ATP Champion's Race, where players are ranked according to their performance in Grand Slam and Masters tournaments as well as their result in the Masters Cup and their five best results from International Series Tournaments in the year previous to the match, the numbers of games and sets won by each player in each respective match, and the identity of the winner of each match.4 To ensure sufficient random variation in the heterogeneity

  capability and incentive effects work in the same direction. Concerning

  standard errors are in brackets. * * * Coefficient significant at the 1-%-level. * * Coefficient significant at the 5-%-level. * Coefficient significant at the 10-%-level. Coefficients for constant terms are omitted for brevity.

Table 1 :

 1 Summary Statistics

	Observations (N )	Mean	Std. Dev.

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Table 2 :

 2 Evidence for Incentive and Capability Effects for different player categories: Interactions

			Dependent Variable:
		ln(Games Won per Set) ln(Games Won per Match)
	Favorite	0.084**	-0.008	0.084**	-0.008
		[0.034]	[0.047]	[0.034]	[0.047]
	Heterogeneity	-0.002***		-0.002***
	(own-opponent rank)	[0.001]		[0.001]
	Heterogeneity		-0.056***		-0.045***
	(Klaassen/Magnus, 2003)		[0.014]		[0.011]
	Interaction Term	0.001*	0.030**	0.001*	0.009*
		[0.001]	[0.013]	[0.000]	[0.005]
	Match is final	-0.006	-0.005	0.002	0.003
		[0.017]	[0.017]	[0.006]	[0.006]
	Tournament Dummies	Yes	Yes	Yes	Yes
	Year Dummies	Yes	Yes	Yes	Yes
	N	920	920	920	920
	R 2	0.08	0.09	0.08	0.1

Notes: Robust standard errors are in brackets. * * * Coefficient significant at the 1-%-level. * * Coefficient significant at the 5-%-level. * Coefficient significant at the 10-%-level. Coefficients for constant terms are omitted for brevity.

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Table 3 :

 3 Schotter, A., and K.Weigelt (1992): "Asymmetric Tournaments, Equal Opportunity Laws, and Affirmative Action: Some Experimental Results," Quarterly Journal of Economics, 107(2), 511-592.Walker, M., and J. Wooders (2001): "Minimax Play at Wimbledon," American Economic Evidence for Incentive and Capability Effects for different player categories

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Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK

Data and relevant links are available on the internet under http://www.atptennis.com.

Missing rankings are replaced by the last available ranking, possibly from the previous year. Note that the lower the rank, the better a player, i.e. the player ranked 1 had the best performance of all tennis professionals

This setting is conservative as any effects should be biased towards zero if the slacking-off due to heterogeneity is stronger in earlier rounds of the tournament.

As will become clear below, abstracting from the precise technicalities involved, like the crucial role of the number of sets won in determining the match winner, simplifies the analysis substantially without affecting the results too much.

See[START_REF] Kennedy | A Guide to Econometrics[END_REF], p. 275-276, for details.

Note that for the underdog, condition (9) has to be rewritten analogously, since the probability of winning the match is determined by the inverse strength of play. Again, the first result follows directly from this first order condition; the second result follows from a similar argument as before concerning the change in the winning probability, which corresponds to the logistic density function.

Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript

Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript

Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript


Michael Creel

Montserrat Farell

and Qualitative Choice Models < C2 -Econometric Methods: Single Equation Models < C -Mathematical and Quantitative Methods, I10 -General < I1 -Health < I -Health, Education, and Welfare

Keywords: C25 -Discrete Regression medical care , count data, maximum likelihood JEL classifications: C25, I10 medical care, count data, maximum likelihood

We explore the determinants of usage of six different types of health care services, using the Medical Expenditure Panel Survey data, years 1996-2000. We apply a number of models for univariate count data, including semiparametric, semi-nonparametric and finite mixture models. We find that the complexity of the model that is required to fit the data well depends upon the way in which the data is pooled across sexes and over time, and upon the characteristics of the usage measure. Pooling across time and sexes is almost always favored, but when more heterogeneous data is pooled it is often the case that a more complex statistical model is required.

Introduction

The demand for health care services may often be measured as the number of times that some event, for example, a doctor visit, occurs in a given time period. Such variables, defined on the natural numbers, are referred to as count data. There have been many recent advances in the econometric analysis of count data, especially the development of flexible density functions for univariate count data. In many cases, these papers include an empirical analysis of data on demand for health care. Examples of such contributions are Deb and Trivedi (1997), who investigate finite mixture models; Cameron and Johansson (1997), who adapt the polynomial reshaping technique of Gallant and Nychka (1987) to count data, and Gurmu (1997), who uses a flexible density to model latent heterogeneity.

All of these approaches define densities that allow for modeling frequently observed features of the data, such as excess zeros and overdispersion, as well as more complicated departures from the behavior implied by standard models such as the Poisson and negative binomial.

At the same time, new sources of data have become available. One of these is the Medical Expenditure Panel Survey (MEPS). The MEPS data is a rich source of recent data on demand for health care, insurance coverage, and related topics. This paper applies many of the recently developed statistical models for univariate count data to the MEPS data, years 1996-2000. This allows comparison of models using a uniform, high quality data set.

From this we will be able to determine which models are most successful in capturing the features of six different measures of demand for health care services [START_REF] Cameron | Econometric models based on count data: comparisons and applications of some estimators and tests[END_REF] . Since the six measures exhibit substantially different character-F o r P e e r R e v i e w istics, they form an interesting test bed for the statistical models, at least within the general area of demand for health care. We seek to learn which of the available models seem most useful for analysis of data similar to the usage measures in the MEPS data. We also provide information about the usage measures in the MEPS data, upon which further research can build. Guo and Trivedi (2002) provide a similar, though somewhat less extensive comparison of models, using two data sets on counts of patents. Within the literature on demand for health care we are not aware of any papers that provide a similar comparison of models. Beyond the comparison of the statistical models, we also investigate the stability of parameters over time and across sexes, and we present brief estimation results for the most favored models.

Our criterion for comparison is the consistent Akaike information criterion (CAIC). The CAIC is a penalized goodness of fit criterion that is decreasing in the value of the likelihood function and increasing in the number of parameters. Lower CAIC values correspond to models that offer a good fit to the data without using an excessive number of parameters.

While one might be interested in other criteria such as out-of-sample fit, or marginal mean effects, we limit our attention to the CAIC. This is for two reasons. The first is that we believe that the CAIC provides more information, in a paper of limited length, than does any other criterion. Inclusion of other information such as out-of-sample fits would multiply the already large amount of tabular information contained in this paper. The likelihood criterion is probably the most parsimonious summary information about how well a model fits. A model that dominates another in terms of likelihood must on average fit the observed counts (0,1, etc.) better than the other model, in spite of the fact that the second model may better fit a par- ticular count, such as the observed number of zeros. If too many extra parameters are needed to improve fits to certain counts, the CAIC penalizes the likelihood value to control for this, thus avoiding overparameterized models. Features of interest such as conditional moments and their derivatives are functions of the estimated probabilities of counts. The CAIC picks the model that best assigns probabilities to the counts observed in the data, in a certain sense. The second reason that we focus on the CAIC and not on features such as fitted marginal effects is that we do not know the true marginal effects, so comparisons across models are difficult to evaluate. If we were doing Monte Carlo work, it would certainly be of interest to look at criteria of this sort. However, with real sample data as is used here, we believe that the fitted marginal effects of a model that is strongly dominated by another in terms of the CAIC are simply not of much interest, since the model is almost certainly very poorly specified. When a model is poorly specified, the fitted marginal effects will in general be biased and inconsistent. Our focus on the CAIC is admittedly narrow, but it is an attempt to provided as much useful information as is possible in a limited number of pages. Future work could explore other features of interest in a Monte Carlo context, focusing on the models that this paper identifies as the most promising according to the penalized goodness of fit criterion.

To summarize the main results, we find that some of the newer models are useful additions to the toolbox for analysis of health care usage, but others are almost always dominated. The complexity of the model that is favored depends upon the type of data that is analyzed. For variables that have relatively high means, significant overdispersion, and relatively few zeros, relatively complex models are needed to fit the data well. For other variables such as the number of inpatient hospitalizations, the simple neg- ative binomial density is perfectly adequate. Another result is that pooling data across time and sexes leads to a parsimonious model that still fits the data as well as separate models that allow all parameters to vary. Pooling should be done when the data allow it. When more heterogeneous data is pooled, it is more likely that a relatively complex statistical model will be required. With relatively homogeneous data, the simple negative binomial statistical model often fits well.

Statistical models for count data dependent variables

Data on health care demand often exhibits overdispersion, which means that the ratio of the conditional variance to the conditional mean is greater than one (Cameron and Trivedi, 1986; Pohlmeier and Uhlrich, 1995). Another common characteristic is that many zeros are observed, possibly more than can be accounted for by simple count densities (Pohlmeier and Uhlrich, 1995; Gerdtham, 1997). Factors such as latent variables or latent population groups could induce more marked departures from standard densities, leading to bimodality or especially fat right tails, for example. In this section we briefly survey some of the newer univariate count data models that can allow for such departures. Before surveying the recent models, we briefly discuss the more standard models upon which the newer approaches build.

Poisson (POISSON)

The Poisson density for a count random variable Y is ally not be suitable for analyzing demand for health care.

Negative binomial (NB)

If the Poisson mean contains a latent component, marginalization, under some assumptions, will lead to a negative binomial density (see for example Cameron and Trivedi, 1998, pp. 100-102) . The negative binomial density may be written as

f Y (y|φ) = Γ(y + ψ) Γ(y + 1)Γ(ψ) ψ ψ + λ ψ λ ψ + λ y (1)
where φ = {λ, ψ}, λ > 0 and ψ > 0.2 When ψ = λ/α we have the negative binomial-I model (NB-I), and ψ = 1/α gives the negative binomial-II (NB-II) model. Though other versions exist, we limit attention to these in this paper. The moment generating function of the NB density, which is needed below, is formulae for moment will give the conditional moments.

M Y (t) = ψ ψ λ -e t λ + ψ -ψ . ( 2

Hurdle negative binomial (HNB)

As noted by Pohlmeier and Uhlrich (1995) and Gerdtham (1997) (1986), who also presented "with-zeros" (also known as "zero-inflated") models. Here we present only the hurdle model, since it seems to have been used more widely than the "with-zeros" model for analysis of data on on usage of health care services. [START_REF] Cameron | Count data regression using series expansions: with applications[END_REF] We follow Deb and Trivedi (1997),

who use a NB model to parameterize the Bernoulli trial. For a NB random variable,

Pr(Y = 0) = f Y (0, φ h ) = ψ h ψ h + λ h ψ h Pr(Y > 0) = 1 -Pr(Y = 0),
where the parameter of the hurdle process is φ h = {λ h , ψ h }. To achieve identification one can set α h = 1 when parameterizing ψ h , which may be done as with the NB model. The above probabilities are used to estimate the binary 0/1 hurdle process. Then, for the observations where visits are [START_REF] Cameron | A microeconometric model of the demand for health insurance and health care in Australia[END_REF] Hurdle models are also known as "two-part" models. [START_REF] Cameron | Count data regression using series expansions: with applications[END_REF] Examples of applications of the HNB model to health care demand include Pohlmeier and Ulrich (1995), Gerdtham (1997), Deb and Trivedi (1997)

f Y (y, φ|y > 0) = f Y (y, φ) 1 -ψ ψ+λ ψ = Γ(y + ψ) Γ(y + 1)Γ(ψ) ψ ψ + λ ψ -1 -1 λ ψ + λ y
Since the hurdle and truncated components of the overall density for Y share no parameters, they may be estimated separately, which is computationally less burdensome than estimating the overall model. The expecta-

tion of Y is E(Y) = 1 - ψ h ψ h + λ h ψ h 1 - ψ ψ + λ ψ -1 λ.
NB-I and NB-II versions that allow for conditioning variables follow from the appropriate parameterizations of ψ h , ψ, λ h and λ.

The HNB model could possibly be considered the most sophisticated attempt to deal with the issues of excess zeros and overdispersion in the modeling of health care demand count data, up until 1996. Shortly after, the following models were introduced. All of these models can account for excess zeros and overdispersion, so they can deal with the issues the HNB model was designed to address. Some of the models are also more flexible than the HNB model, even though they may be more parsimonious.

A semiparametric approach (PSP, HPSP)

A semiparametric approach to modeling count data has been developed by Gurmu and Trivedi (1996), Gurmu (1997) and Gurmu et al. (1999). This approach introduces unobserved heterogeneity in a Poisson model, and al- This is conceptually similar to the way that a negative binomial model is obtained as a Poisson-gamma mixture density, but is more flexible in that the latent variable is not restricted to follow a one parameter gamma density. The semi-nonparametric density of the latent variable is closely related to that proposed by Gallant and Nychka (1987). The difference is that Laguerre polynomials are used instead of Hermite polynomials. Gurmu et al. (1999) show that, under weak assumptions, the Laguerre expansion density can consistently estimate densities of unknown form. As such, the mixture density is semiparametric, since the Poisson specification is parametric but the modelization of the heterogeneity is not. Gurmu and Trivedi (1996) found that the basic semiparametric approach of Gurmu et al. (1999) [START_REF] Cragg | Some statistical models for limited dependent variables with applications to the demand for durable goods[END_REF] did not fit data well -specifically, excess zeros were a problem. To overcome this problem, Gurmu (1997) proposed a hurdle version of the semiparametric model.

The original semiparametric model is based upon an infinite mixture of a Poisson random variable and an independent random variable V which captures unobserved heterogeneity. The assumption is that the Poisson mean is random, so that E(Y|V = v) = λv. Integrating out the heterogeneity, one obtains the marginal density:

f Y (y, λ, φ) = e -λv (λv) y y! g V (v, φ)dv (3) = λ y y! M y V (-λ) (4)
where M y V (-λ) is the yth order derivative of the moment generating function of V, evaluated at -λ. M 0 V (-λ) = M V (-λ), is the moment generating The gamma baseline density is

f (v, φ) = v α-1 β α Γ(α) e -βv
where φ = (α, β). The semi-nonparametric density for v is

g V (v|φ, γ) = h p (y, γ) 2 f (v|φ) η p (φ, γ)
where

h p (y, γ) = p ∑ k=0 γ k P k (v), (5)
γ = (1, γ 1 , γ 2 , ..., γ p ), and P k (v) is the kth order Laguerre polynomial. The term η p (φ, γ) = γ γ is the normalization factor that makes the density sum to one. The restriction γ 0 = 1 is used to achieve identification, since the density is homogeneous in γ. This density is semi-nonparametric in the sense that, under weak assumptions, there exist (φ, γ) such that a density of unknown form can be approximated arbitrarily well as p goes to infinity. Gurmu et al. (1999) provide the consistency proof, which is similar to that of Gallant and Nychka (1987).

Next, they are able to obtain a closed form for M y V (-λ), which upon substitution into equation 4 yields the semiparametric density for the count random variable Y. In estimation, a restriction is imposed such that E(V) = 1, which leads to E(Y) = λ. In the course of the empirical work reported below, we have found that the model is poorly identified without this restriction, and that it is very difficult to obtain convergence if it is not im- To extend this to the hurdle case, Gurmu (1997) allows a first PSP model to determine whether the zero/positive hurdle is crossed, and a second PSP model is used to model the positives. For the hurdle crossing process, the relevant probabilities are

Pr(Y = 0) = M V (-λ h ) Pr(Y > 0) = 1 -Pr(Y = 0).

The truncated version of the PSP density is

f Y (y|y > 0, λ, φ) = λ y y! M y V (-λ) 1 -M V (-λ) .
Just as in the case of the HNB model, the binary and truncated components of the hurdle Poisson semiparametric (HPSP) model may be estimated separately. Notationally, we will let PSP(k) or HPSP(k) refer to a model that uses a k-order expansion.

Semi-nonparametric approaches (PSNP, NBSNP)

Cameron and Johansson (1997 tions under which the density has nonparametric properties. [START_REF] Deb | Demand for medical care by the elderly: a finite mixture approach[END_REF] The PSP and HPSP models embed the semi-nonparametric density in a parametric density to obtain a semiparametric model, after marginalization of the latent variable. As such, one expects that the approach of Cameron and Johansson should be able to capture more extreme departures from the baseline model, though perhaps at the cost of needing to estimate many parameters.

)
For example, the PSNP model can accommodate bimodal densities, while the PSP density cannot.

The PSNP density is

f Y (y|λ, γ) = h p (y|γ) 2 η p (φ, γ) e -λ λ y y! , (6)
where

h p (y|γ) = p ∑ k=0 γ k y k , (7)
and η p (φ, γ) is a normalizing factor to make the density sum to one. The normalizing factor is

η p (λ, γ) = ∞ ∑ y=0 h p (y|γ) 2 e -λ λ y y! .
Cameron and Johansson show that this has the closed form

η p (λ, γ) = p ∑ k=0 p ∑ l=0 γ k γ l m k+l ( 8
)
where m r (λ) is the rth noncentral moment of the Poisson density. Because 3rd order polynomial expansion. To our knowledge, this is the first paper that applies this model to data on health care demand. Guo and Trivedi (2002) apply a version of this model based upon an NB-II base density to [START_REF] Dismuke | Has the caveat of case-mix based payment in¯uenced the quality of inpatient hospital care in Portugal?[END_REF] We used MuPAD version 2.51 to perform these calculations.

A finite mixture approach (MNB, CMNB)

The finite mixture approach to fitting data on health care demand was introduced by Deb and Trivedi (1997). The mixture approach can be interpreted as allowing for latent groups in the population. The data for each group may be characterized by a parameter vector. Since the group to which an individual belongs is not observed, a mixing probability is used to classify individuals probabilistically. There may be two or more latent groups. The mixture approach has been also applied by Gerdtham and Trivedi (2000), who find that it performs better than the HNB approach.

The mixture negative binomial (MNB) model has the virtue of being conceptually simple. The density is

f Y (y, φ 1 , ..., φ p , π 1 , ..., π p-1 ) = p-1 ∑ i=1 π i f (i) Y (y, φ i ) + π p f p Y (y, φ p ),
where

π i > 0, i = 1, 2, ..., p, π p = 1 -∑ p-1 i=1 π i , and ∑ p i=1 π i = 1. The f (i) Y (y, φ i ), φ i = {λ i , ψ i }
are p separate NB-I or NB-II densities, as in equation 1. Identification requires that the π i be ordered in some way. We follow Deb and Trivedi (1997)

by imposing π 1 ≥ π 2 ≥ • • • ≥ π p and φ i = φ j , i = j.

Variables

From these files we use six different measures of annual health care usage, for each of the five years. These are office-based doctor visits (OBDV), outpatient doctor visits (OPV), emergency room visits (ERV), inpatient hospital visits (IPV), dental visits (DV), and number of prescription drugs taken (RX).

The explanatory variables used are months of public insurance coverage during the year, divided by 12 (PUB), sex (SEX -coded as 0 for men and 1 for women), age (AGE), years of schooling (EDUC), and family income in thousands of dollars (INC). Since health care issues change considerably with age, we limit the sample to individuals between the ages of 40 and 65, inclusive. Work not reported here revealed that models that pool data for broader age groups often do not pass specification tests. Also, extremely few younger people have publicly provided insurance coverage. This fact causes problems in obtaining convergence of models that use data limited to that for younger people. We also suspect that women's and men's health issues are different enough to warrant the consideration of models that allow the form of the model and all parameters to differ by sex. We investigate the possibility of pooling the form of the model or some of the parameters across sexes.

We limit the sample to people who have private health insurance coverage during the entire year. Originally we used months of private coverage as an explanatory variable. This variable is very likely to be endogenous in a model for health care usage, since latent health status will likely simultaneously affect choices regarding health care usage and purchase of health care insurance. [START_REF] Geil | Economic incentives and hospitalization in Germany[END_REF] The econometric problem is to find convincing in- strumental variables for private coverage, that can reasonably be excluded from the equation that explains health care usage. Since we were not able to find such variables in the survey data, we prefer to simply estimate models conditional on full private insurance coverage, and avoid the issue of possible endogeneity entirely. The analysis is more limited, but the results are sharper and more reliable. Depending upon the year and the value of SEX, we loose between 20% and 35% of the available sample due to this decision. We include the measure of publicly provided insurance, PUB, to investigate the effects of double coverage. We believe that PUB and the other explanatory variables may be safely considered as exogenous, a priori.

All the variables with the exception of PUB and INC are directly available from the survey data. PUB is simply the sum of the monthly indicators of public health care coverage, divided by 12. Thus, it runs from 0 to 1, with 1 indicating that a person enjoys publicly provided insurance coverage during the full year. INC was constructed by summing the incomes of all members of the family. In the MEPS data, total personal income is the sum of many different sources of income, which may or may not be directly reported. Observations for which any source of any family member's income was "hot decked" were dropped, since hot decking introduces measurement error which leads to inconsistent estimation in the context of regression analysis. [START_REF] Gallant | Seminonparametric maximum likelihood estimation[END_REF][START_REF] Gerdtham | Equity in health care utilisation: further evidence based on hurdle models and Swedish macro data[END_REF] We do not use any information on health status, and instead treat health surance coverage is in fact a problem, especially for the OBDV and RX use variables. These results are not entirely reliable, however, due to the problem of poor instruments, discussed in the body of the paper, and thus we do not present them in detail. [START_REF] Gallant | Seminonparametric maximum likelihood estimation[END_REF] "Hot decking" is a term used in the MEPS documentation to describe a method of replacing missing data with conditional or unconditional sample means of the variable.

See the documentation at ¡ ¡ © £¢ # " § ¥ $ ¡¤ ¦¥ £ § ¤ ¦¦© £ ¥ £ £ £ § ¡ ! for more details. status as entirely latent. This is in contrast to many studies that have incorporated objective and/or subjective indicators of health status. The health status information in the MEPS data include measures of perceived health status as well as objective measures of limitations to activities. The recorded data is based upon one family member's assessment of all family members' healths. We have the problem that individual A and individual B may evaluate individual B's health very differently, which at a minimum implies that this data will be subject to measurement errors which can lead to inconsistency if not properly addressed (Windmeijer and Santos Silva, 1997). In the case of limitations to activities, many variables are recorded in the data sets. These include, for example, indicators of whether or not individuals have difficulty standing 20 minutes, or difficulty in reaching over the head, and a number of similar variables, and again, one family member reports for the entire family. These variables are likely to be highly collinear, and none of them seems suitable as a single measure of overall health status. Furthermore, it is not clear that results that are conditional on such measures of health status are directly useful for many sorts of economic analysis. Since an economic analysis would likely need to marginalize results that are conditional on these variables, and since the only means of marginalizing them is using the sample information itself, we prefer simply not to condition on them from the outset. Thus, we treat health status as a purely latent source of heterogeneity, and we model it as such. The primary concern in treating health status in this way is the possibility that latent health status might be correlated with conditioning variables such as private insurance coverage, which would induce problems of endogeneity.

Our solution, as noted above, is to condition on full private insurance coverage, so that its level disappears as a regressor. We think that the other conditioning variables may safely be assumed to be exogenous.

Descriptive Statistics

As noted above, we limit the data used in this paper to people between 40 and 65 years of age, and initially we estimate separate models for women and men. The sample sizes by year and sex are found in Table 1.

To obtain a first idea of the characteristics of the six measures of use, Tables 2 through 5 give descriptive statistics for women's and men's health care usage, for the the years 1996 and 2000 [START_REF] Gerdtham | Equity in Swedish health care reconsidered: new results based on the finite mixture model[END_REF] . Studying these tables, we can make a few observations:

• Women, on average, use all six forms of health care more frequently than do men. This result is very uniform and is stable over time. This suggests that models that pool across sexes will require a dummy variable for sex.

• Men are more likely than women never to use forms of care. The difference is especially notable in the cases of OBDV, DV, and RX, which are probably more elective forms of care than are the other three measures.

• There is considerable temporal stability of the statistics, for all six measures of usage. However, there are some temporal variations that are notable. For example, the mean of ERV for women is 50% higher in 1996 than in 2000. This suggests the use of dummy variables for years in models that pool across time.

• Most measures of health care use exhibit considerable unconditional overdispersion. The IPV and ERV measures are in some cases rea- [START_REF] Gerdtham | Equity in Swedish health care reconsidered: new results based on the finite mixture model[END_REF] Results for the other years are very similar and are omitted to save space. sonably close to unconditional equidispersion. For these measures it is possible that conditional equidispersion might hold, and that a Poisson model might be adequate. In the other cases the models that allow for overdispersion will likely be preferred.

• The percentage of zeros for the OPV, IPV and ERV measures is usually around 90% or higher. The OBDV, DV, and RX measures have positive usage by a much larger proportion of the sample.

Next, to obtain an idea of the characteristics of the explanatory variables, Tables 6 and7 present descriptive statistics for the four conditioning variables, for women and men. We present these statistics only for the year 2000 data, since the other years are substantially similar. Highlights include:

• The means of AGE and EDUC are quite similar across sexes.

• There is a notable difference in the mean of INC, which presumably is due to a sex differential in the incomes of single people (recall that INC is defined as family income). The fact that maximum values of INC are the same is because INC was top-coded during the execution of the survey.

• Only a small part of the population has access to publicly-provided health care insurance.

Model selection

We have under consideration 6 measures of health care usage, 2 sex groups, and 5 years of data. For each of these 60 data sets we wish to determine which of a number of statistical models is most appropriate. Some (e.g., whether to use an NB-I or NB-II base model, or the degree of the polynomial expansion for the PSNP, PSP, HPSP and NBSNP models). In the face of so many comparisons to make we use an information criterion approach, concretely the consistent Akaike information criterion (CAIC).

The CAIC is defined as CAIC = -2lnL + p(lnn + 1), where lnL is the loglikelihood value, p is the number of parameters of the model, and n is the sample size. The CAIC is a penalized goodness of fit criterion. Additional parameters usually allow for better fit, in terms of the log-likelihood value, but the penalty term prevents selection of overparameterized models. The CAIC is a consistent model selector, in the sense that the correct model in a set of models will have the lowest CAIC value, as the sample size tends to infinity (Sin and White, 1996) [START_REF] Guo | Flexible parametric models for long-tailed patent count distributions[END_REF] . The simple Akaike information criterion (AIC), which has been used in some of the related literature, is not consistent, in that it can favor overparameterized models. The Bayes (or Schwartz) information criterion (BIC) that also appears in the literature can be expressed as BIC = CAICp. This criterion is also consistent. It may favor a somewhat more highly parameterized model than the CAIC. The BIC can be calculated using the information we provide in our results, but we do not report it here, to save space.

We report CAIC values starting with the models that allow parameters to vary by both sex and year, then we report results where parameters are constant across sexes (except for the coefficient of a dummy variable for sex) but vary by year, and finally we report results that pool both across sexes and time. The pooling across time is only for the years 1996, 1998 and 2000, so that no individual enters the sample in more than one year. In this [START_REF] Guo | Flexible parametric models for long-tailed patent count distributions[END_REF] If none of the models is the correct model, then the model that is closest to the correct model in the sense of the Kullback-Leibler information criterion will have the smallest CAIC value, asymptotically. It may be shown that, for two models that share no parameters and use disjoint data sets, the overall CAIC value is simply the sum of the CAIC values of the two models. Thus, one can compare the sum of the CAIC values for separate models for men and women in a given year with the CAIC value of a model that pools across sexes using the data of the same year.

If the CAIC of the pooled model is lower, pooling is supported, otherwise, separate models are favored. Likewise, we can compare models that pool across time with analogous models that allow coefficients to vary by time.

In this way we can determine what level of pooling is supported by the data, for each of the 6 measures of use.

With regard to estimation details, some of the models lead to a loglikelihood function that may have local maxima. For the models that do not have a globally concave log-likelihood function, we used simulated annealing to find a rough maximizer which satisfied convergence of the log-likelihood function out to 2 decimal places, then iterated to convergence using a BFGS maximizer. For the other models we used the BFGS maximizer directly. All estimation routines were programmed using GNU Octave (¢¢ §' ¡ ¡ ) §' ! #% ) and are available from the authors.

Separate models by sex and year

We begin with the CAIC values of the various models, for the year 2000 data. For the other years we only report (below) the results for the favored models, to avoid overwhelming the reader with details. Tables 8 and9 • Some use measures exhibit considerable variation over time in the models that are favored (e.g., OBDV and DV). The CAIC-favored model for these use measures has "close competitors" in Tables 8 and9. The favored model is very stable over time for the IPV and RX use measures.

• One result that stands out is that the simple Poisson-style specification of the conditional mean, E(y|x) = e xβ , is used by the favored model in 52 of 60 cases (86.6%).

With relatively homogeneous data that are for single sex groups and single years, simple models work well in the great majority of cases. the sum of the CAIC values of the models that allow all parameters to vary by sex, relative to the CAIC value of the favored model. We do not present such detailed results for the other years, but instead only report the favored models for this level of pooling, for each of the five years, in Table 13. In this table we can observe that:

Pooling across sexes

• Pooling across sexes is favored in all cases except for ERV in 1998. In all other cases, use of a dummy variable and a common model and slope coefficients is favored. Pooling across years We have seen that pooling across sexes is almost always favored. Next we present CAIC results for models that pool across the years 1996, 1998 and 2000, adding dummy variables that allow the constant to vary by both sex and year. We do not use the data from 1997 and 1999 so that a given individual appears only once in the sample, and thus the data consists of independent observations. [START_REF] Gurmu | Semi-parametric estimation of hurdle regression models with an application to medicaid utilization[END_REF] Table 14 presents the results. We note that • Pooled models are always favored. Time-wise heterogeneity seems to be adequately captured by a dummy variable.

• Relatively complicated, newer models (PSP, NBSNP, CMNB) are favored in 5 of 6 cases. However, for the ERV data where the PSP model is favored, the NB-I and NB-II models have only slightly higher CAIC values.

• The simple Poisson-style mean function E(y|x) = e xβ is favored in only 2 of 6 cases (for IPV and ERV). Again, as we pool more heterogeneous data, more complicated densities are required to fit the data well. These more complicated densities imply more complicated conditional moments. Note that the cases where the simple mean function is accepted are those where the unconditional mean of the dependent variable is lowest, and the percentage of zeros is highest, and the mean/variance ratio is closest to 1 (see Tables 2345).

The overall conclusion is that pooling by age and sex is almost always favored, when data is available to make it possible. Simpler models often work well when the data is relatively homogeneous (for example, separate [START_REF] Gurmu | Semi-parametric estimation of hurdle regression models with an application to medicaid utilization[END_REF] If we were to include the data from 1997 and 1999, but still treat the observations as independent, the estimators would not truly be maximum likelihood estimators, and thus the use of the CAIC to compare models would not be valid. models by sexes, for a single year) and more complex models are often required when more heterogeneous data is pooled. Of the statistical models compared in this paper, some (the Poisson, PSNP, HNB and MNB) are always dominated, and the HPSP model is likely too highly parameterized for all but exceptional cases. Of the more complicated newer models, the NBSNP, PSP and CMNB models are found to be useful contributions for analysis of this sort of data and probably deserve consideration in future work.

Estimation results

Though a detailed economic analysis of estimation results for the favored models is beyond the scope of this paper, Table 15 presents estimation results for the CAIC-favored models for the pooled 1996-1998-2000 data, with pooling across sexes. The Table contains results for all six use measures.

The models are the favored models that appear in Table 14.

Examining the estimation results, we can make several notes:

• With respect to time trends, DV usage has declined significantly over the 1996-2000 period. Consumption of prescription drugs (RX) has increased significantly. No other trends are clear.

• For all usage measures except DV, holding publicly-provided health care insurance (PUB) has a positive and strongly significant effect on usage levels.

• The dummy variable that indicates that the individual is a woman is positive in all cases, and is highly significant except for the IPV and ERV measures. with events such as serious illness or accidents that are in a large part beyond the control of the individual.

• Age always has a positive coefficient, and is highly significant in all cases except the ERV usage measure.

• Income is negative and significant for the IPV, ERV and RX use measures. It is positive and significant for the DV measure.

• Education has a positive and significant effect upon the OBDV and DV measures, and it has a negative and significant impact upon the IPV and ERV measures.

• There is evidence that low-income, low-education individuals use IPV and ERV services more than the average individual. They make less use of dental care visits than average. Other effects are not so clear.

• The CMNB model used for the OBDV, OPV and RX use measures is characterized by mixing two NB densities, both of which are overdispersed, and at least one of which is highly overdispersed. The mix (π) parameter is estimated with poor precision in all three cases. The constant shifter for the second NB density is highly significant.

• The α and γ parameters of the PSP(1) density for the ERV usage measure are estimated imprecisely. It appears that they are not well identified separately for this data set, but that their joint impact is important (since the model had the best CAIC score).

• The α and one of theγ parameters are significant for the NBSNP model used for the DV use measure. There appears to be a problem of poor separate identification similar to that of the PSP( 1 Of the forms of health care under consideration, OBDV and DV are those that are most likely to include preventive visits such as checkups. We can see that more educated, and in the case of DV, higher income individuals, use these two forms of care more frequently than average. Likewise, IPV and ERV may be used more than average by people who have not taken care of their health through preventive care, or who are seeking to use emergency room visits in place of ordinary doctor visits in an attempt to avoid insurance copayments. The fact that poorer, less educated people use these forms of care more frequently than average might be explained by such factors.

Conclusions

This paper has surveyed a number of statistical models for univariate count data and has applied them data on health care usage from the Medical Expenditure Panel Survey, years 1996-2000. The objective of the paper has been to attempt to determine which models are most appropriate for this sort of data. A secondary objective has been to determine which level of pooling across time and sexes is supported by the data.

We have found that some of the newer models are quite useful and warrant serious consideration when undertaking empirical work with this sort of data. In particular, depending upon the usage measures and the level of pooling, the NBSNP, PSP, and CMNB models are found to fit the data better than more traditional models such as the NB and especially the HNB. Other newer models such as the MNB and HPSP are found to be Another result is that pooling the data, both across time and across sexes, is almost always favored. There is enough parameter stability so that dummy variables can be used to capture the important variations in a simple and parsimonious way, without imposing overly strong restrictions on the model. As more heterogeneous data is pooled, more complex statistical models become necessary so that the assumption of parameter constancy (except changes in the constant captured by dummy variables) can be maintained. The basic finding of the paper is that it is more parsimonious to use a relatively complex statistical model with parameter constancy than to use simple statistical models with parameters that vary across data groups. The degree of complexity of the statistical mode required for adequate fit to the data depends upon the usage measure under consideration. Factors that lead to more complicated models being needed are a high mean, low proportion of zeros, and overdispersion. This paper has not focused upon estimation results or economic analysis of the such results. Nevertheless, we have presented some limited results using the pooled by time and sex data, which is the favored approach in all cases. We have seen that the coefficients of the variables have signs that can be given a plausible economic interpretation. However, the discussion has not been deep, since this sort of analysis is not the focus of this paper. Some directions for further work are quite clear. Given that pooling across time has been found to be desirable, it would be useful to develop models that allow for dependent observations, so that the entire data set for all years could be used. This will require explicit modeling of the de-

4 Page

 4

  To allow for covariates, λ is usually parameterized as λ = e xβ . The Poisson density implies that the conditional mean and the conditional variance of y are both equal to λ. Since data on health care demand usually exhibit overdispersion and possibly excess zeros, the basic Poisson model will usu-

)

  For the NB-I density, V(Y) = λ + αλ. In the case of the NB-II model, we have V(Y) = λ + αλ 2 . For both forms, E(Y) = λ. Thus, both forms capture overdispersion, with the NB-II model allowing for a more extreme form.As with the Poisson models, the usual means of incorporating conditioning variables is the parameterization λ = e xβ . When this is done, the previous

8

 8

  heterogeneity to follow a semi-nonparametric density.

  density g V (v, φ) flexibly, Gurmu et al. (1999) use a normalized Laguerre polynomial expansion around a gamma baseline density.

10 Page

 10 results we report always impose the restriction. We will refer to this model as the Poisson semiparametric model (PSP). To incorporate conditioning variables, the Poisson-style parameterization λ = e xβ is used, so that E(Y|x) = exp(xβ).

11 Page

 11 directly adaptGallant and Nychka's (1987) semi-nonparametric density to the count data case. They reshape a Poisson baseline density using a squared polynomial, and then normalize the result to sum to one. We shall refer to this as the Poisson semi-nonparametric (PSNP) approach, though there has been no formal proof of the condi-

2 η 2 η

 22 p (λ, γ) that reshapes the baseline density in equation 6 is a homogeneous function of γ, it is necessary to impose a normalization to achieve identification: γ 0 is set to 1. The moments of Y may be calculated using the closed form expression in Cameron and Johansson's equation 4. The typical Poisson-style parameterization of the mean is used to incorporate conditioning variables. Since the NB model usually fits health care data dramatically better than does the Poisson model, using only one more parameter, one might suspect that changing the baseline model to the NB might allow the model to fit well using fewer parameters. What we shall refer to as the negative binomial semi-nonparametric (NBSNP) model is obtained by making this change. The density is f Y (y|φ, γ) = h p (y|γ) p (φ, γ) Γ(y + ψ) Γ(y + 1)Γ(ψ) (y|γ) and η p (φ, γ) are defined as in equations 7 and 8, respectively, and the raw moments m r (λ, ψ) are obtained from equation 2. The moments of Y are again obtained from Cameron and Johannson's equation 4, substituting the NB raw moments for those of the Poisson density. 7 The model can use either the NB-I or the NB-II as the baseline model. We investigate both versions in what follows. Notationally, let NBSNP-I(3), for example, indicate the NBSNP model using a NB-I baseline density, and a

14 Page 15 Page

 1415 This is simple to accomplish post-estimation by rearrangement of the component densities. Another issue is how to consistently estimate the number of component densities, supposing that the true density is in fact a mixture density (seeJames, et al. 2001, for example). We skirt this issue by considering only the possibility of 2 component densities.The properties of the mixture density follow in a straightforward way from those of the components. In particular, the moment generating func-is the same mixture of the moment generating functions of the component densities, whenceE(Y) = ∑ p i=1 π i λ i .The MNB density may suffer from overparameterization, since the total number of parameters grows rapidly with the number of component densities. To address this problem, Deb and Trivedi propose a constrained mixture negative binomial model (CMNB) which restricts all the slope parameters in λ j = e xβ j to be the same across all component densities. The constants and the overdispersion parameters α j are allowed to differ.3 The MEPS data3.1 Data SourcesThe Medical Expenditure Panel Survey composed of four surveys of individuals, nursing homes, health care providers, and employers in the United States. We use only the Household Component, which is a survey of a nationally representative sample of households. The Household Component uses an overlapping panel design where individuals are interviewed five times over the course of 2.5 years, such that complete data for two calendar years is collected. Each year, a new series of contacts is initiated. Thus, data for a given individual will appear in the data files for two consecutive years, and the samples for consecutive years are not independent. The raw MEPS data files are available at the site ¢¡ ¢¡ ¢£ ¥¤ §¦ ¢¦ ©¢¢ £ § ¢! #" $ &% (' ) .The data files used are the Household Component Full-Year files for years 1996-2000, which are files HC-012, HC-020, HC-028, HC-038 and HC-050, respectively. These data files collect responses to many questions related to health care usage, health, insurance coverage, income, etc.

Page

  of the statistical models require determination of the specific parameterization 20

  observations are independent of one another. MLE estimation of models with dependent observations would require specifying the nature of the dependence, which is a step we prefer to avoid in this work.

2 .

 2 report the relative CAIC values, for women and men, respectively, for the statistical models that were discussed 22 For the model with the minimum CAIC value, the tables report this value. For the other models the tables report the CAIC value relative to that of the favored model, to facilitate comparisons. For the OBDV, DV and RX measures of use (which are those with higher sample means), we can see that there are a number of models that reach a CAIC within 1% of that of the favored model, while for the OPV, IPV and ERV measures of use, the distances between the favored model and the other models are often larger. Tables10 and 11report which are the CAIC-favored models for each of the five years, for women and men, respectively. Some points to note:• The numbers of times models are favored are: NB -44 times; PSP -8 times; NBSNP -6 times; and CMNB and HPSP, one time each. The other models are never favored.

•

  Only four models (apart from parameterization details) are ever favored: they are NB (14 times), NBSNP (8 times), PSP (6 times), and CMNB (3 times).• The simple Poisson-style specification of the conditional mean E(y|x) = e xβ is implied by the favored model in 20 of 31 cases (the NB and PSP models) which is 64.5% of the cases, down from the 86.6% for separate models by sexes. Pooling is supported in all cases but 1 out of 30 (ERV, 1998), but pooling seems to require more flexible densities to capture the greater heterogeneity of the data.• There is considerable stability over time. For example, the NBSNP model is favored in 4 of 5 years for the DV use measure, and the PSP model is favored 4 of 5 times for the OPV use measure.• The Poisson model and the more highly parameterized hurdle and mixture models (HPSP, MNB) are never favored.• When the NBSNP model is favored, it is always the version that uses a NB-I base density.

  The IPV and ERV measures are often associated 26

Page

  ) model for the ERV 27 problem was noted byCameron and Johansson (1997) for the PSNP model.

  the usage measures in the MEPS data used here, according to the CAIC criterion.

  measures over time, which will lead to the consideration of multivariate count data densities and issues of estimation of such nonlinear models with panel data. Another direction for work would to be to try to tackle the endogeneity of private health care insurance in a convincing way. This may not be possible using the MEPS data due to lack of good instruments, but with other data sets it could be undertaken.

  Pohlmeier, W. and V. Ulrich (1995), An econometric model of the twopart decision-making process in the demand for medical care, Journal of Human Resources, 30, 339-61. [19] Sin, Chor-Yiu, and H. White (1996), Information criteria for selecting possibly misspecified parametric models, Journal of Econometrics, 71, 207-225.[20] Windmeijer, F. and J. Santos Silva (1997), Endogeneity in count data models: an application to demand for health care, Journal of Applied Econometrics, 12, 281-94.[21] Yoshida, A. and Y.-S. Kim (2008), Sharing health risk and income risk within households: evidence from Japanese data, Applied Economics,

  , health care demand may exhibit excess zeros with respect to what a NB model can accommodate. This leads us to consider the hurdle version of the NB model. The hurdle 3 negative binomial model first models the zero vs. posi-

tive outcome using a probit or similar model. Then, conditional on positive visits, the count follows a zero-truncated negative binomial density. Different parameter vectors are associated with the binary and truncated densities. Hurdle count models were introduced by Cragg (1971) and Mullahy

7 Page 7 of 40 Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript

  and Yoshida and Kim (2008).

	positive, a truncated NB density, with a different parameter φ = {λ, ψ} is
	estimated. This density is
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[START_REF] Geil | Economic incentives and hospitalization in Germany[END_REF]

Exploratory work with Hausman-type tests suggested that endogeneity of private in-16 Page

  Table 12 reports the CAIC values for models that pool the coefficients across sexes, and add a dummy variable that allows the constant to vary by sex, for the year 2000. In the last row we present

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Table 1 :

 1

				Sample Sizes
			1996 1997 1998 1999 2000
		Women 737 1205 477	830	817
		Men	680 1104 478	802	800
	Table 2: Descriptive Statistics, Use Variables, Women, 1996
		OBDV	OPV	IPV ERV	DV	RX
	mean	4.970	0.288 0.117 0.156	1.640	13.323
	st. dev.	6.965	1.597 0.423 0.474

Table 8

 8

			: CAIC Values, Women, 2000
		OBDV	OPV	IPV	ERV	DV	RX
	POISSON	1.6320	1.8007	1.0103	1.0094	1.1131	2.8494
	PSNP(1)	1.5060	1.7249	1.0227	1.0189	1.1162	2.5116
	PSNP(2)	1.3932	1.5585	1.0066	1.0126	1.0222	2.2547
	PSNP(3)	1.2782	1.5037	1.0230	1.0250	1.0250	2.2560
	PSP(1)	1.0019	1.0234	1.0119	1.0129	1.0032	1.0017
	PSP(2)	1.0080	1.0582	1.0774	1.0927	1.0097	1.0069
	PSP(3)	4332.1958	1.0126	1.0262	1.0253	1.0060	1.0030
	HPSP(1)	1.0103	1.0638	1.1030	1.1175	1.0152	1.0092
	HPSP(2)	1.0005	1.0096	1.0415	1.0372	1.0085	1.0028
	HPSP(3)	1.0139	1.0732	1.1290	1.1407	1.0197	1.0118
	NB-I	1.0031	1.0530	1.0008 622.1461 2752.9661 5844.7338
	NB-II	1.0027	1.0514 471.1805	1.0021	1.0051	1.0012
	HNB-I	1.0047	1.0864	1.0598	1.0551	1.0042	1.0032
	HNB-II	1.0040	1.0616	1.0539	1.0574	1.0034	1.0032
	MNB-I	1.0115	1.0289	1.0990	1.0729	1.0158	1.0048
	MNB-II	1.0125	1.0234	1.1045	1.0841	1.0169	1.0069
	CMNB-I	1.0014	1.0007	1.0400	1.0357	1.0053	1.0008
	CMNB-II	1.0015 911.6805	1.0400	1.0367	1.0090	1.0033
	NBSNP-I(1)	1.0057	1.0128	1.0200	1.0139	1.0027	1.0023
	NBSNP-I(2)	1.0014	1.0212	1.0236	1.0258	1.0006	1.0036
	NBSNP-I(3)	1.0031	1.0010	1.0456	1.0367	1.0032	1.0038
	NBSNP-II(1)	1.0056	1.0139	1.0175	1.0155	1.0028	1.0062
	NBSNP-II(2)	1.0031	1.0040	1.0236	1.0277	1.0045	1.0039
	NBSNP-II(3)	1.0049	1.0125	1.0482	1.0367	1.0067	1.0033

Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK

Table 9

 9

				: CAIC Values, Men, 2000
			OBDV		OPV	IPV	ERV	DV	RX
	POISSON		1.4931	1.2675	1.0317	1.1393	1.1277	3.1952
	PSNP(1)		1.4693	1.2760	1.0439	1.1486	1.1307	2.7935
	PSNP(2)		1.2857	1.0421	1.0143	1.0342	1.0254	2.4597
	PSNP(3)		1.1948	1.0532	1.0309	1.0463	1.0287	2.4614
	PSP(1)		1.0071	1.0054	1.0218	1.0005	1.0130	1.0065
	PSP(2)		1.0118	1.0621	1.1195	1.0710	1.0242	1.0122
	PSP(3)		1.0051	1.0164	1.0372	1.0099	1.0153	1.0071
	HPSP(1)		1.0140	1.0795	1.1526	1.0908	1.0308	1.0136
	HPSP(2)		1.0033	1.0274	1.0528	1.0221	1.0170	1.0074
	HPSP(3)		1.0153	1.1010	1.1856	1.1144	1.0364	1.0168
	NB-I	3564.8823 693.6498 463.0455	1.0027 2298.0098 4589.4350
	NB-II		1.0051	1.0025	1.0064 634.1129	1.0097	1.0057
	HNB-I		1.0056	1.0439	1.0748	1.0512	1.0129	1.0066
	HNB-II		1.0083	1.0490	1.0706	1.0532	1.0141	1.0077
	MNB-I		1.0099	1.0710	1.1045	1.0677	1.0170	1.0102
	MNB-II		1.0138	1.0678	1.1128	1.0695	1.0313	1.0161
	CMNB-I		1.0018	1.0261	1.0477	1.0244	1.0096	1.0032
	CMNB-II		1.0032	1.0275	1.0545	1.0230	1.0185	1.0083
	NBSNP-I(1)	1.0018	1.0125	1.0167	1.0167	1.0036	1.0022
	NBSNP-I(2)	1.0005	1.0186	1.0324	1.0140	1.0066	1.0036
	NBSNP-I(3)	1.0027	1.0293	1.0489	1.0260	1.0099	1.0049
	NBSNP-II(1)	1.0069	1.0140	1.0233	1.0106	1.0127	1.0066
	NBSNP-II(2)	1.0079	1.0239	1.0398	1.0195	1.0159	1.0081
	NBSNP-II(3)	1.0100	1.0340	1.0553	1.0300	1.0192	1.0097
		Table 10: CAIC-Favored Models, Women, 1996-2000
		OBDV		OPV	IPV	ERV	DV	RX
	1996	PSP

Table 12

 12

	Page 35 of 40	Submitted Manuscript
		: CAIC Values, Pooling Across Sexes, 2000
		OBDV	OPV	IPV	ERV	DV	RX
	POISSON	1.5736	1.5941	1.0273	1.0767	1.1222	3.0133
	PSNP(1)	1.5727	1.5688	1.0346	1.0821	1.1240	2.6808
	PSNP(2)	1.3665	1.3916	1.0085	1.0143	1.0231	2.6816
	PSNP(3)	1.2445	1.3309	1.0177	1.0211	1.0247	2.3243
	PSP(1)	1.0036	1.0100	1.0087 1233.0104	1.0094	1.0083
	PSP(2)	1.0053	1.0329	1.0729	1.0452	1.0096	1.0064
	PSP(3)	1.0001	1.0052	1.0179	1.0048	1.0100	1.0091
	HPSP(1)	1.0048	1.0376	1.0905	1.0588	1.0129	1.0069
	HPSP(2)	1.0012	1.0041	1.0270	1.0116	1.0087	1.0099
	HPSP(3)	1.0062	1.0478	1.1089	1.0650	1.0150	1.0085
	NB-I	1.0026	1.0274 912.9064	1.0025 5019.8251 10405.3000
	NB-II	1.0046	1.0297	1.0004	1.0009	1.0078	1.0084
	HNB-I	1.0034	1.0503	1.0469	1.0369	1.0044	1.0024
	HNB-II	1.0028	1.0387	1.0452	1.0308	1.0052	1.0040
	MNB-I	1.0096	1.0276	1.0613	1.0403	1.0103	1.0035
	MNB-II	1.0091	1.0233	1.0646	1.0393	1.0157	1.0130
	CMNB-I	7865.6354 1581.2384	1.0270	1.0121	1.0038	1.0002
	CMNB-II	1.0009	1.0013	1.0268	1.0121	1.0110	1.0085
	NBSNP-I(1)	1.0042	1.0331	1.0102	1.0104	1.0009	1.0014
	NBSNP-I(2)	1.0003	1.0061	1.0191	1.0061	1.0025	1.0022
	NBSNP-I(3)	1.0013	1.0027	1.0270	1.0129	1.0033	1.0023
	NBSNP-II(1)	1.0062	1.0091	1.0101	1.0047	1.0093	1.0104
	NBSNP-II(2)	1.0056	1.0144	1.0193	1.0113	1.0070	1.0110
	NBSNP-II(3)	1.0066	1.0087	1.0284	1.0181	1.0085	1.0117
	Separate Models	1.0040	1.0152	1.0234	1.0189	1.0062	1.0028
			35

Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK

Table 13 :

 13 CAIC-Favored Models, Pooling Across Sexes, 1996-2000

	OBDV	OPV	IPV	ERV	DV	RX
	1996 NBSNP-I

Table 14 :

 14 CAIC Values, Pooling across 1996-1998-2000 and Sexes

		OBDV	OPV	IPV	ERV	DV	RX
	POISSON	1.6126	1.5517	1.0515	1.0583	1.1692	2.9997
	PSNP(1)	1.5255	1.5465	1.0547	1.0608	1.1699	2.6795
	PSNP(2)	1.4060	1.3059	1.0033	1.0076	1.0537	2.3916
	PSNP(3)	1.2867	1.1573	1.0073	1.0106	1.0544	2.3920
	PSP(1)	1.0047	1.0042	1.0050 3051.3255	1.0074	1.0057
	PSP(2)	1.0055	1.0158	1.0390	1.0283	1.0008	1.0029
	PSP(3)	1.0045	1.0005	1.0076	1.0019	1.0076	1.0051
	HPSP(1)	1.0020	1.0203	1.0469	1.0344	1.0021	1.0024
	HPSP(2)	1.0037	1.0000	1.0115	1.0050	1.0083	1.0055
	HPSP(3)	1.0030	1.0241	1.0548	1.0399	1.0033	1.0032
	NB-I	1.0043	1.0217 2338.3952	1.0003	1.0010	1.0004
	NB-II	1.0061	1.0231	1.0003	1.0001	1.0081	1.0053
	HNB-I	1.0040	1.0285	1.0272	1.0212	1.0031	1.0014
	HNB-II	1.0042	1.0247	1.0274	1.0209	1.0030	1.0019
	MNB-I	1.0077	1.0183	1.0350	1.0272	1.0046	1.0022
	MNB-II	1.0044	1.0211	1.0362	1.0240	1.0097	1.0070
	CMNB-I	19541.3171 4029.5548	1.0099	1.0052	1.0005 25190.7329
	CMNB-II	1.0004	1.0002	1.0115	1.0052	1.0059	1.0050
	NBSNP-I(1)	1.0050	1.0242	1.0043	1.0037	1.0016	1.0010
	NBSNP-I(2)	1.0000	1.0112	1.0083	1.0027 13100.3787	1.0010
	NBSNP-I(3)	1.0005	1.0054	1.0121	1.0057	1.0005	1.0011
	NBSNP-II(1)	1.0068	1.0221	1.0042	1.0033	1.0088	1.0059
	NBSNP-II(2)	1.0054	1.0117	1.0080	1.0044	1.0085	1.0060
	NBSNP-II(3)	1.0058	1.0140	1.0117	1.0072	1.0092	1.0064
	Separate Models	1.0057	1.0192	1.0354	1.0125	1.0039	1.0029
			36

Table 15 :

 15 Estimation Results, Overall CAIC-Favored Models, 1996-1998-

	2000	OBDV	OPV	IPV	ERV	DV	RX
	Const.	0.2098 -3.8315 -2.6857 -1.7703 -1.6691 -0.1809
		(0.210)	(0.673)	(0.549)	(0.561)	(0.200)	(0.172)
		[0.318]	[0.000]	[0.000]	[0.002]	[0.000]	[0.294]
	1998	0.0215	0.0038 -0.2639 -0.0865 -0.0112	0.0498
		(0.044)	(0.127)	(0.168)	(0.218)	(0.049)	(0.049)
		[0.627]	[0.976]	[0.115]	[0.692]	[0.818]	[0.306]
	2000	-0.0163 -0.0603 -0.1818 -0.0635 -0.1732	0.1048
		(0.038)	(0.110)	(0.138)	(0.133)	(0.044)	(0.041)
		[0.672]	[0.582]	[0.187]	[0.633]	[0.000]	[0.011]
	Pub	0.2804	0.4658 1.0211 0.6232 -0.1300	0.5118
		(0.076)	(0.208)	(0.202)	(0.333)	(0.102)	(0.104)
		[0.000]	[0.025]	[0.000]	[0.061]	[0.205]	[0.000]
	Woman	0.3794	0.3271 0.1028 0.1323 0.2581	0.5490
		(0.034)	(0.097)	(0.125)	(0.114)	(0.038)	(0.037)
		[0.000]	[0.001]	[0.410]	[0.244]	[0.000]	[0.000]
	Age	0.0279	0.0394 0.0260 0.0132 0.0185	0.0412
		(0.002)	(0.007)	(0.009)	(0.010)	(0.003)	(0.002)
		[0.000]	[0.000]	[0.004]	[0.178]	[0.000]	[0.000]
	Income	-0.0004 -0.0007 -0.0042 -0.0037 0.0017 -0.0012
		(0.000)	(0.001)	(0.002)	(0.002)	(0.000)	(0.000)
		[0.315]	[0.594]	[0.026]	[0.044]	[0.000]	[0.009]
	Education	0.0175	0.0200 -0.0660 -0.0643 0.1115	0.0078
		(0.007)	(0.020)	(0.022)	(0.024)	(0.009)	(0.008)
		[0.012]	[0.318]	[0.003]	[0.006]	[0.000]	[0.305]
	alpha	9.8375	0.4879 0.4579 0.4146 2.8892 12.7021
		(0.224)	(0.590)	(0.173)	(1.151)	(0.078)	(0.313)
		[0.000]	[0.408]	[0.008]	[0.719]	[0.000]	[0.000]
	gam1/Const2 1.0000 -1.5483	na 0.2949 -0.0948	2.3131
		(0.096)	(0.732)		(0.323)	(0.055)	(0.131)
		[0.000]	[0.034]		[0.361]	[0.084]	[0.000]
	gam2/alpha2	2.5659 17.5843	na	na 0.0030 56.6040
		(0.166)	(0.546)			(0.036)	(0.400)
		[0.000]	[0.000]			[0.934]	[0.000]
	mix	0.2278	0.5515	na	na	na	0.6577
		(0.461)	(1.354)				(0.933)
		[0.621]	[0.684]				[0.

Among the numerous examples of application of the NB model to health care demand are Cameron et al. (1988), Pohlmeier and Ulrich (1995), Geil et al. (1997) and Dismuke and Guimares (2002).

The 1999 paper is based upon a 1996 working paper, which explains the dates of these references.

The consistency proofs of[START_REF] Gallant | Seminonparametric maximum likelihood estimation[END_REF] and[START_REF] Gurmu | Semiparametric estimation of count regression models[END_REF] are for continuous random variables. While it seems reasonable to expect that the proofs could be adapted to discrete random variables, this has not yet been done, to our knowledge.

The programs used to process the raw data, as well as the resulting data files are available upon request from the authors.

Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript

¢¡ ¤£ ¦¥ ¨ § ¤© £ © ¤© ¨ § ! "research was supported by grants SEJ 2006-00712/ECON and CONSOLIDER- INGENIO 2010 (CSD2006-00016