Matching and mobility in the market for Australian rules football coaches.
The relative explanatory power of different theories of worker mobility is an empirical question. Two main approaches to testing for the determinants of worker mobility have been adopted in empirical studies. One focuses on the mobility behavior of large samples of individuals (for example, Topel and Ward 1992; Farber 1992); the other closely analyzes the mobility patterns in an individual labor market.
In this study we concern ourselves with matching theories of worker mobility, and we take the approach of examining a single, narrowly delimited labor market. Specifically, to try to establish the role of matching effects as a determinant of mobility patterns, we examine the labor market for Australian Rules football coaches. Australian Rules football is the main spectator sport in Australia; in 1994, approximately 4.7 million spectators attended regular season games.
Restricting attention to mobility behavior in the labor market for football coaches has several advantages. Perhaps most important for our purposes, it facilitates a focus on the effects of labor market matching, because most other theories of mobility are unlikely to be relevant for this labor market. For example, because a coach's activities and performance are observable to his own team and to other teams, theories that attribute worker mobility to information asymmetries in the labor market are unlikely to have much explanatory power for the mobility of football coaches (for example, Gibbons and Katz 1991). Two other advantages of studying the mobility of football coaches are that it provides a sample of employment spells from a reasonably lengthy time period; and precise measures of team performance are available.
It must be noted, however, that in some respects the market for football coaches differs from the stylized market in theoretical matching models. First, whereas studies of matching generally assume that workers control separation decisions, the market for football coaches is a "thin" market in which firms initiate separations. Second, the interdependence between tire performance of teams in a sporting league may introduce external effects into the matching process that are not present in existing matching models. Similar interdependencies may, however, also exist in the labor market for top management employees of firms that operate in an oligopolistic market.
Most theories of worker mobility adopt the standard neoclassical postulate that a worker has equal productivity at any firm. Several groups of such theories are identifiable. Sectoral-shift theories suggest that worker mobility occurs in response to changes in labor demand due to intertemporal variation in the composition of product demand or in production technology across industries (for example, Lilien 1982). In theories of internal labor markets, mobility occurs through up-or-out employment rules designed to improve worker incentives (for example, Huberman and Kahn 1988). Theories of career mobility posit that workers shift between jobs that provide different learning opportunities (for example, Rosen 1972). And in theories of asymmetric information, where a worker's current employer acquires private information on the worker's productivity, the market infers that retained workers at a firm are high-productivity and bids up wages of those workers so that it is unprofitable for a firm to retain low-productivity workers (for example, Gibbons and Katz 1991).
Models of labor market matching are distinguished from other theories of worker mobility by two assumptions: a worker's productivity differs between firms, and all workers and firms initially share symmetric imperfect information on a worker's productivity at a firm. Information on the value of a worker-firm match, which is initially uncertain, is obtained through either of two means. First, it may be obtained through worker search (Jovanovic 1979a). In that case, the value of a worker's current match is assumed to be known but the value of available matches with other firms is unknown and must be discovered through search activity by the worker. Second, it may be obtained through the experience of a worker in a match (Jovanovic 1979b; Harris and Weiss 1984), in which case the value of the worker's current match is initially uncertain and is learned over time but the value of any alternative match has constant expected value. Some models - such as Jovanovic (1984), Topel (1986), and McLaughlin (1991) - allow for learning about both the value of the current match and available alternative matches.(1)
Common to both types of matching models is the assumption that the values of worker-firm matches for a worker differ between firms, and the associated prediction that mobility of workers across jobs will occur in response to these differences. Hence a direct test of matching models is to examine whether an individual's productivity differs between matches. This empirical approach has previously been applied to test for matching effects in the markets for U.S. and Japanese baseball managers (Chapman and Southwick 1991; Ohtake and Ohkusa 1994).
Matching models can also be applied to predict the relation between the probability of worker-firm separations and the value of a worker-firm match, and between the probability of worker-firm separations and years of worker tenure and experience. In theoretical matching models positing that the value of a worker-firm match is obtained through a worker's on-the-job experience, it is assumed that each period's output provides a signal of future output in that match. Output from a match in each period depends on a match-specific parameter and a random component. In any period, expected match output equals the expected value of the match-specific parameter. Ex-ante, the expected value of the match-specific parameter is equal in each alternative match. Once a worker moves to a new match, each period's output provides a signal of the value of the match-specific parameter and hence of future expected match output. The information generation process provides a signal of expected future output that has increasing precision as a worker's tenure with a firm increases.
In a competitive labor market, the wage payment to a worker for each period of employment will equal expected output in that period. At any point in time, the worker chooses between staying in an existing match or moving to an alternative match. For a risk-neutral worker, the decision whether to stay in an existing match or move to an alternative match will depend on a comparison of the wage in the existing match - which is determined by accumulated information on expected future out-put in that match - with the expected wage in any alternative match minus the cost of mobility between matches.
A number of predictions concerning patterns of labor mobility can be derived from this model:(2)
(a) The probability of worker-firm separation is decreasing with the value of a worker-firm match. Where there is a higher level of expected future output in an existing match, it is less likely that output in the existing match will be exceeded by the fixed threshold of expected output in alternative matches.
(b) The probability of a worker-firm separation initially increases, and then decreases, with years of worker tenure. The relation between years of worker tenure and probability of separation is determined by interaction between mobility costs and the increasing precision of the signal of future output in the worker's current match. In the initial period of a worker's employment, new information on the value of a match will not change expected match output enough to outweigh mobility costs, and hence there will be a low rate of separation. As time proceeds, however, expected output in a match may decrease to a level at which the expected gain in wage payments from shifting to an alternative match will be sufficient to outweigh mobility costs, and a higher rate of separation will occur. Eventually, as years of worker tenure increase, the increasing precision of the measure of expected output means that if there has not already been a decrease in expected future output sufficient to induce separation, it is unlikely that future output information will cause a revision in expected output sufficient to induce separation, and hence the rate of separation will decline.
(c) The probability of a worker-firm separation initially increases, and then decreases, with years of worker experience. For the initial years of worker experience, the rate of separation will follow the same pattern as for worker tenure. Later, when some threshold of years of experience has been attained, the probability of separation will decline with further years of experience, for two reasons: the probability of separation is decreasing with years of tenure, and workers will only shift from matches with lower to higher expected output - an opportunity that occurs with less frequency as years of experience increase.
Construction of the Data Set
In this study we apply two main tests of the role of matching effects in the market for Australian Rules football coaches: a direct test of coach-team match effects on team performance; and a test of the predicted relation - derived from matching theories of worker mobility - between the probability of coach-team separation and a coach's years of tenure and years of experience. In order for us to undertake the analysis of matching effects; data are therefore required on two variables: the length of employment spells of coaches, and the value of a match between a coach and team.
The sample of employment spells of coaches in this study includes all spells in the Victorian Football League (VFL), and subsequently the Australian Football League (AFL), that commenced in or after the 1931 season. Information on coaching spells is from AFL (1994). In calculating the length of employment spells, we have made a number of adjustments to the data. In one case, a team (Geelong) did not compete in 1943-44 (during World War II), but had the same coach prior to and immediately following the break. We treated this case as a continuous spell of employment. Similarly, in a case in which a coach was granted a year's leave of absence because of ill health, we imputed a continuous spell for that coach, as well as for the coach who took over for that year and then later resumed coaching at the same club. Twelve coaching spells concluded in mid-season. In those cases, we assumed spell length for the old coach to be equal to years of tenure at the end of that season, and the spell of the new coach to commence at the start of that season. However, in calculating career performance measures for each coach, we weighted the season according to the proportion of the season over which the coach participated (in four cases in which it was not possible to determine the exact timing of the change of coach, separation was assumed to occur at the midpoint of the season). Unlike in studies of CEO turnover, where censoring of employment spells is a significant problem (see, for example, Welsbach 1988), censoring of spells due to mandatory retirement does not seem to be an issue for the employment of football coaches.
A shortcoming of the data on employment spells is that it has not been possible to distinguish for all spells between separations that occur as quits and those that occur as dismissals. One cause of this problem is the absence of historical records for some teams, but another is that it is often difficult to know how to correctly classify a separation. Fortunately, being able to distinguish quits from dismissals is not necessary in empirical analysis of the type of symmetric information matching model examined in this study. In a matching model in which workers and firms have symmetric information and all separations are efficient, the label of a quit or dismissal is inconsequential. That is, for any separation that occurs, both worker and firm would have an incentive to initiate the separation (see Hall and Lazear 1984; McLaughlin 1991). Hence, predictions from the matching model relate to the pattern of separations rather than to the pattern of quits and dismissals.
The basic measure of the value of a coach-team match in this study is derived from the team's winning percentage during the regular season. Winning percentage is the percentage of games won by a team in a season. Information on team performance is from Rodgers (1991) and AFL (1994). To measure the value of a coach-team match, it is necessary to adjust data on team winning percentage to take account of differences in the set of teams played by each team during a season, and of the effects of coach and player quality on team performance.
Where differences exist in each team's regular season schedule, the ranking of teams on the basis of winning percentage may be sensitive to that schedule. For the greatest part of the sample period in this study, this consideration is a relevant one. Only between 1970 and 1987 did the regular season consist of a set of "home and away" games with every team playing every other team twice during the season. Hence, to account for the influence of the relative strength of its opponents on a team's winning percentage, we derived an adjusted performance measure using an iterative procedure based on the method of paired comparisons (Cowden 1974). A detailed description of this procedure is provided in Appendix A.(3)
To obtain a measure of the value of a coach-team match, it is also necessary to adjust the measure of team performance for the effects of player quality and coach quality. To the extent that player quality is the main determinant of a team's winning percentage, team performance is less meaningful as a measure of the coach's contribution to team output. In their studies of matching in the markets for baseball managers in the United States and Japan, Chapman and Southwick (1991) and Ohtake and Ohkusa (1994) adjusted team performance for the influence of pitching, hitting, and fielding performance variables.
The measures we use to adjust for player quality in this study are the number of players in a team receiving Brownlow medal votes in each season, and the total number of Brownlow medal votes received by players in a team in each season. The Brownlow medal is an award similar to the MVP in American football. Brownlow medal votes are assigned by the umpires in each game to the three best players in that game out of the eighteen players from each team involved in the game. Over the period of the study, the number of players in a team receiving Brownlow medal votes in a season has varied from a minimum of 7 to a maximum of 20; the mean number of players receiving votes is 11.4. Aggregate Brownlow medal votes, and the number of players receiving Brownlow votes, can therefore be regarded as providing a measure of the quality of the group of best-performing players in a team.
Two aspects of the player quality variables - which have different implications for estimates of coach-team match effects - should be noted. First, where the Brownlow vote variables do not completely control for intertemporal variation between teams in differences in player quality, it is possible that the match dummy variables will also proxy for those differences. In this case, estimates of the effect of a coach-team match on team performance will be biased upward. Second, to the extent that coach quality and coach-team match quality affect measures of player performance, some of the effect of a coach-team match will be reflected in the estimated effect of player quality on team performance. Hence, the estimated match effect may represent a lower bound on the true effect.(4)
To control for fixed effects of coach quality on team performance, we included a dummy variable for each coach.(5) In addition, years of experience of a coach can be included as a proxy for skill or information acquisition. We assume that the information on performance in the VFL/AFL is the only available measure of the ability of a coach. In reality, of course, many coaches enter or re-enter the VFL/AFL after a spell in a minor league. Hence, teams may have information on coaching performance beyond the measures considered in this paper.(6)
Table 1. Sample Descriptive Information. All Spells Spell 1 Spells 2+ Number of Spells 231 172 59 Number of Incomplete Spells 12 5 7 Coaches Coaches Total with with Coaches Spell = 1 Spells [greater than] 1 Coaches 172 131 41 Spell Adjusted Winning Percentage: mean 36.6 34.6 42.6 S.D. 23.0 23.6 20.1 min. 0.0 0.0 7.7 max. 89.2 89.2 81.4 Premierships per Season: mean 0.079 0.052 0.102 S.D. 0.269 0.222 0.303 Source: Authors' calculations from Rodgers (1991) and AFL (1994).
A final point regarding the sample period for this study is that there is some reason to be concerned that the role of the coach within the team may have altered between 1931 and 1994. In the 1930s, almost one-half of the teams in the league had a playing coach, but by the 1980s this practice had become virtually nonexistent. To attempt to capture the possible influence of a change in production technology, we examined sub-samples of spells for non-playing coaches, as well as time periods before and after 1950, but we discerned little difference in the pattern or determinants of mobility between these samples and the sample of all coaches.
Table 2. Average Annual Separations per Team. Years Separations 1931-34 0.32 1940-44 0.33 1950-54 0.20 1960-64 0.20 1970-74 0.27 1980-84 0.38 1990-94 0.22 1935-39 0.32 1945-49 0.23 1955-59 0.23 1965-69 0.18 1975-79 0.40 1985-89 0.29 Note: Average annual separations per team are calculated as (Total number of separations) / (Number of years x Number of teams in league). Source: Authors' calculations from AFL (1994).
Tables 1 and 2 present descriptive information on employment spells of Australian Rules football coaches examined in this study. The sample of coaching spells is distributed over 172 coaches, of whom 131 have had a single spell of employment and 41 have had multiple (up to four) spells. The total number of spells in the sample is 231; 12 spells in the sample are incomplete (that is, unfinished as of the end of the period of study).
The average adjusted winning percentage per spell for all coaches is 36.6%. The average winning percentage in a spell is below 50% due to a large number of short spells in which the coach's winning percentage was below 50%. For individual spells, the average adjusted winning percentage varies between zero and 89.2%, although the distribution of coaches' average winning percentages in individual spells is largely concentrated in the range from 40% to 60%. Coaches with more than one spell of employment have a higher average spell-adjusted winning percentage, and also a higher rate of winning premierships (league championships), than coaches with only one spell, but in neither case is the difference statistically significant (at the 5% level).
Table 2 presents information on intertemporal variation in coach-team separations. For example, in 1931-34, the probability of a coach-team separation occurring at a team in any year was 0.32. It is evident from Table 2 that some time-series variation exists in the pattern of separations. In particular, mobility appears to have been higher in 1931-45 and 1975-85 than during other periods.
Test of Match-Specific Effects on Team Performance
If we interpret a team's season winning percentage as a measure of team output, and coach quality, player quality, and the coach-team match as potential inputs, a possible test for the effect of the coach-team match on team output is through a comparison of the regression equations:
(1) [P.sub.jt] = [summation over c [element of] C] [[Beta].sub.c] [COACH.sub.cj] + [Delta]P[P.sub.jt] + [Phi][EXP.sub.cjt] + [Eta] [([EXP.sub.cjt]).sup.2] + [[Epsilon].sub.jt]
(2) [P.sub.jt] = [summation over c [element of] C][[Beta].sub.c][COACH.sub.cj] + [summation over m [element of] M] [[Gamma].sub.m][MATCH.sub.mcj] + [Delta]P[P.sub.jt] + [Phi][EXP.sub.cjt] + [Eta][([EXP.sub.cjt]).sup.2] + [[Epsilon].sub.jt],
where [P.sub.jt] equals team performance in year t of spell j (j = 1, ..., 231), [COACH.sub.cj] is a dummy variable for the identity of the coach in spell j (c = 1, 2, ..., 172), [MATCH.sub.mcj] is a dummy variable for the identity of the coach-team match in spell j, P[P.sub.jt] is the vector of player performance measures in year t of spell j (total Brownlow medal votes awarded to players in the team, and number of players from the team awarded Brownlow medal votes), and [EXP.sub.cjt] is years of experience of coach c in year t of spell j. In equation (2) the set of match dummy variables is restricted to the set of repeat coaching spells (minus one spell) to avoid collinearity with the set of team dummy variables. A comparison of the relative explanatory power of equation (1) and equation (2) provides a direct test of matching effects on team output. If a coach's value differs between matches, the coefficients on the match dummy variables in equation (2) should be statistically significant.
The results from estimation of the regression equations are reported in Table 3. We estimated all equations using OLS with White's heteroskedastic-consistent covariance standard errors. Further details of this procedure are described in Appendix B.(7) Column (i) of Table 3 shows results from equation (1), which includes a full set of coach dummies, and column (ii) shows results from equation (2) with the coach dummies and a restricted set of match dummies. A Wald test rejects at the 1% level of significance the hypothesis that the match dummies do not explain variation in winning percentage. Note that this approach underestimates the role of match-specific effects in explaining team performance, since it is not possible to identify a match-specific effect for coaches with only a single spell. Of the other explanatory variables in equations (1) and (2), the proxy for player quality, total number of Brownlow medal votes awarded, has a significant positive effect on team performance in both regressions. Variables for a coach's years of experience and the number of players awarded Brownlow medal votes are not significant.
Since performance may also vary systematically across individual teams, it seems important to establish that the coach-team match dummies are not proxying for team-specific factors. Unfortunately, the match and coach dummies are collinear with the team dummies, so it is not possible to include the complete set of dummy variables in the same regression. Nevertheless, a regression equation that includes a full set [TABULAR DATA FOR TABLE 3 OMITTED] of coach dummy variables and a restricted set of team dummy variables (excluding one team dummy variable, to prevent collinearity between the sets of coach and team dummy variables) provides a limited test of the role of team-specific effects:
(3) [P.sub.jt] = [summation over c [element of] C] [[Beta].sub.c] [COACH.sub.cj] + [summation over z [element of] Z] [[Theta].sub.z] [TEAM.sub.zj] + [Delta]P[P.sub.jt] + [Phi][EXP.sub.cjt] + [Eta] [([EXP.sub.cjt]).sup.2] + [[Epsilon].sub.jt],
where [TEAM.sub.zj] is dummy variable for the identity of team in spell j (excluding one team).
A Wald test that compares equation (3) with equation (1) rejects at the 1% level of significance the hypothesis that the team dummies do not explain variation in winning percentage. This result may seem to suggest that it is difficult to separate the influence of match and team dummies. It is, however, possible to exploit the relation existing between match and team dummies to provide further information on the relative explanatory power of these factors. Since each match dummy is associated with some individual team, a team dummy is equal to the sum of the match dummies specific to that team. That is,
(4) [summation over m [element of] z] [MATCH.sub.mcj] = [TEAM.sub.zj],
where m [element of] z denotes the set of match dummies specific to team z. Hence equation (3) can be regarded as a version of equation (2) in which the coefficients on each match dummy for the same team are restricted to be equal. A Wald test of this restriction finds that it is possible to reject at the 1% level of significance the hypothesis that the coefficients on each match dummy variable for the same team are equal ([[Chi].sup.2](45) value of 789.91 against a critical value at the 1% level of significance of 50.89). Hence, the estimated match effects are not simply proxying for fixed team effects.
Can the results from estimation of equations (1)-(3) be interpreted as evidence that variation in team performance is not explained wholly by differences in coach quality or by team-specific effects, but in addition will depend on the coach-team match? Before accepting that conclusion, we must consider the possibility that the regression analysis of team performance is biased toward rejecting the hypothesis of no matching effects.
One potential source of bias is from correlation between the set of coach-team match dummy variables and the error term. Identification of match effects is from repeat spells of coaches with multiple spells. If the sample of coaches with multiple spells is non-random due to selection effects, the expected value of the sum of the error terms may be non-zero over the duration of the initial spells of those coaches. Hence, the error term will constitute a source of systematic variation in the team performance variable for a coach's initial spell. Since the coefficient on the match dummy variable will depend on the difference in the unexplained variation in team performance across the different spells of the coach, a systematic effect of the error term on unexplained variation in team performance in the coach's initial spell may cause the coefficient on the match dummy variable to be biased away from zero.
How can the impact of this potential source of bias on estimates of the matching effect be assessed? One possible approach derives from the relation between spell length and the expected value of the sum of the error terms. Since the probability that the sum of the error terms in the coach's initial spell is non-zero will decline with the length of that spell, the problem of systematic variation in unexplained team performance in a coach's initial spell caused by a non-zero error term should be less severe as the length of the spell increases. Hence, a possible method for assessing the effects of bias from selection effects in the sample of coaches with multiple spells is to estimate equations (1)-(3) on subsamples of coaches with multiple spells who had successively longer initial spells of employment.(8)
Table 4a reports the results from estimating equations (1)-(3) on samples of coaches with multiple spells, and initial spells of at least two years, three years, and four years. These results are similar to those obtained from estimating the equations for the complete sample of coaches. A series of Wald tests shows that the match effects are statistically significant at the 1% level in each regression. In addition, Table 4b shows that for each regression a Wald test rejects at the 1% level of significance the hypothesis that the match effects are equivalent to team effects. Hence, the results obtained do appear to provide direct evidence for effects of coach-team matches on team performance, even controlling for the possibility of non-random selection of coaches with multiple spells.(9)
Matching theories of worker mobility also yield predictions regarding the relation between the probability of worker-firm separation and years of worker tenure or experience, and between the probability of separation and the value of the coach-team match. Figures 1, 2, and 3 present non-parametric Kaplan-Meier hazard rates by years of tenure for all coaching spells, disaggregated between initial spells and repeat spells for coaches with multiple spells, and disaggregated between initial spells of coaches who had only one spell and initial [TABULAR DATA FOR TABLE 4A OMITTED] spells of those who had multiple spells.(10) Figure 4 presents hazard rates by years of experience for all coaches. In Figures 1 through 4, hazard rates are displayed only for years of tenure or experience for which at least seven observations are available.
From Figure 1 and Figure 4 it is evident that both for years of tenure and for years of experience, the probability of separation initially increases and then decreases. For the hazard function by years of tenure it is possible to reject at the 1% level of significance the hypothesis that the hazard rate is constant ([[Chi].sup.2](7) of 22.5 against a critical value of 18.5); and for the hazard function by years of experience the same hypothesis is rejected at the 5% level of significance ([[Chi].sup.2](7) of 14.5 against a critical value of 14.1). These findings are consistent with the predictions of a matching theory in which information on match productivity is generated through on-the-job experience.
There are, however, a number of effects that may be confounded in the aggregate hazard function by years of tenure displayed in Figure 1:
(a) Heterogeneity in coach quality. The shape of the hazard function can be affected if the function differs between coaches with different levels of ability and the distribution of coaches by ability varies within or between spells. To understand the effect of heterogeneity on the shape of an aggregate hazard function derived from the same spell of a set of coaches, suppose that there are two groups of coaches with different hazard functions. At each point in time a coach in group 1 has a higher probability of separating from employment than a coach in group 2. Then even if the hazard rate is constant for each group, the aggregate hazard rate will decrease within a given spell. The reason is that as time elapses, coaches in group 1 will complete spells at a faster rate than coaches in group 2, and therefore the proportion of coaches in group 2 increases. Since coaches in group 2 have a lower probability of separating from employment, the aggregate hazard rate will decrease.
Heterogeneity may also affect the shape of an aggregate hazard function derived for a set of coaches some of whom have single spells of employment and others [TABULAR DATA FOR TABLE 4B OMITTED] multiple spells. This can occur if part of the transition process of coaches from initial to repeat spells involves sorting by ability. In this case, the length of initial and repeat spells will be generated by hazard functions with different functional forms. Both types of hazard functions will be confounded in the aggregate hazard function (Kiefer 1988:672).
(b) Coach's years of experience. Since all spells of employment are included in Table 1, for a coach with multiple spells, differences in the hazard function between spells may exist due to the effects of experience (such as skill or information acquisition) on the probability of separation.
To provide information on the role of ability and experience effects in determining the form of the hazard function, in Figure 2 and Figure 3 we graph disaggregated hazard rates for different groups of coaches. Figure 2 presents hazard rates in initial and repeat spells for the subsample of coaches with multiple spells. This comparison holds constant coach ability, but allows years of experience to vary between spells. Figure 3 displays hazard rates in the initial spell of employment for coaches who had only one spell and for those who had more than one spell. This comparison holds constant years of experience, but may allow coach ability to vary between the groups.
An examination of hazard functions for initial and repeat spells of coaches with multiple spells in Figure 2 reveals that the hypothesis of no significant difference between the functions cannot be rejected at the 10% level ([[Chi].sup.2](6) value of 4.90 against a critical value of 10.65). Hence, there is not strong evidence of an effect of years of experience on the hazard function. Is there still evidence of matching effects? For the hazard function for the repeat spells of coaches it is possible to reject the hypothesis of a constant hazard rate at the 10% level ([[Chi].sup.2](5) value of 7.89 against a critical value of 7.82); on the other hand, for the hazard function for initial spells the hypothesis of a constant hazard rate cannot be rejected at the 10% level ([[Chi].sup.2](5) value of 4.75). Therefore, the findings on matching effects for the disaggregated sample of spells are mixed. It is important to note, however, that the sample of initial spells is small, which may explain the failure to reject the hypothesis of a constant hazard rate.
An examination of hazard functions for the initial spell of coaches with a single career spell and with multiple career spells in Figure 3 establishes that it is possible to reject at the 1% level ([[Chi].sup.2](6) value of 33.74 against a critical value of 16.81) the hypothesis that there is no significant difference between the hazard functions of those groups of coaches. Over the initial years of tenure, coaches with only a single spell of employment have a significantly higher rate of separation than coaches who have multiple spells. This difference in the form of the hazard function might therefore be interpreted as evidence of heterogeneity in ability of the coaches, with sorting by ability between coaches with a single spell and with multiple spells of employment. For the initial spells of coaches with only one spell of employment, it is possible to reject at the 1% level the hypothesis of a constant hazard rate level ([[Chi].sup.2](5) value of 16.53 against a critical value of 15.09). Figure 3 shows that the hazard rate for this group is increasing to four years of tenure, and then decreases. Therefore, for the group of [TABULAR DATA FOR TABLE 5 OMITTED] [TABULAR DATA FOR TABLE 6 OMITTED] coaches with a single spell, there is still evidence of matching effects in the form of the hazard function.
Tables 5 and 6 provide information on the relation between the probability of a coach-team separation in a season and a team's adjusted winning percentage in that season, and the relations between the probability of coach-team separation and a coach's average adjusted winning percentage during the current spell. For either measure of team performance, the probability of separation declines with increases in winning percentage. In fact, perhaps surprisingly, it is difficult to distinguish between the two measures as predictors of separation. This may indicate that teams are primarily concerned with the recent performance of a coach. It also appears that a career or season winning percentage below 50% is more likely to be associated with a separation in a coach's initial spell than in repeat spells.
As a further test of the effect of match value on the probability of separation, we estimated a regression equation that examines the relation between the coach-team separation variable and the coefficient estimates for the match dummies from equation (2).(11) This exercise can be undertaken for repeat spells of coaches with more than one spell; since match dummy variables can only be estimated for repeat spells of a coach, it is not possible to identify the match-specific performance for the initial spell of any coach. Table 7 presents information on the results from the OLS regression equation. Consistent with the hypothesis of matching theory, the results show that the value of a match is inversely related to the probability of separation. However, the effect is not statistically significant.
The empirical findings of this study provide support for the hypothesis that mobility in the market for Australian Rules football coaches involves a process of matching of coaches across teams. We have applied two main tests for matching effects on mobility, using data on employment spells of Australian Rules football coaches between 1931 and 1994. First, regression analysis of the determinants of team performance provides direct support for the existence of coach-team match-specific effects on team output. Second, consistent with the implications of matching theory, we find that the probability of a coach-team separation decreased with the coach-team match effect on team performance and with years of coach tenure and coach experience. There is also some evidence that sorting of coaches by ability occurred between groups of coaches with only one spell and those with multiple spells.
Table 7. Regression Analysis of Relation Between Coach-Team Separation and Match-Specific Performance. (Dependent Variable: Coach-Team Separation Dummy Variable) Standard Variable Coefficient Error Constant 0.222 0.027 Match-Specific Effect on Team Performance -0.099 0.190 Number of Observations = 237 Note: The reported results are from an OLS regression equation. The sample is all observations from repeat spells of coaches with more than one spell. Sources: Coach-team separation is from Rodgers (1991) and AFL (1994). Match-specific effect on team performance is from regression equation - reported in column (ii) of Table 3 - with adjusted team performance as the dependent variable and player performance variables, experience, experience squar ed, and coach and m atch dummies as explanatory variables.
Although each of the matching models considered in this study generates predictions that can be used to test for the existence of matching effects, less seems to be known about how to differentiate between the theories of matching. For example, to generate observed patterns of mobility, is it necessary to have heterogeneity in worker ability? Is it necessary for there to be learning about the value of both current and alternative matches? Can it be assumed that the productivity of a match remains fixed for the duration of a match - or that alternative employers do not learn about potential match quality from observing output from a worker's current match? In a similar vein, Topel and Ward (1992:441) observed, regarding models of labor turnover, that "except under extreme circumstances, they are observationally equivalent in data on job durations alone. This suggests that additional information on the productivities of particular job pairings may provide important identifying leverage for distinguishing the importance of competing theories." Evaluating competing labor market matching models therefore is an important task for future research.
CALCULATING AN ADJUSTED PERFORMANCE MEASURE
The most common method of ranking teams competing in sporting events is to compare the win-loss records of the teams. This approach has the shortcoming that in competitions wherein each team does not play all other teams the same number of times, it does not take into account the relative strength of the set of opponents of each team. One method for calculating an adjusted performance measure with a correction for opponents' strength is developed in Cowden (1974). This approach is based on an iterative method whereby, at each iteration, an estimate of the relative performance of each team is obtained as follows. Let k denote the iteration number, [a.sub.ij] denote the number of wins of team i over team j, and [b.sub.ij] denote the number of losses of team i to team j. If [Mathematical Expression Omitted] equals the "win" scores of team i at iteration k and [Mathematical Expression Omitted] equals the "loss" scores of team i at iteration k, then
[Mathematical Expression Omitted], [Mathematical Expression Omitted]
[Mathematical Expression Omitted], [Mathematical Expression Omitted]
Starting values for the win score and loss score are chosen as [Mathematical Expression Omitted]. This implies that at the first iteration [Mathematical Expression Omitted] will equal the proportion of matches that team i wins. At subsequent iterations, [Mathematical Expression Omitted] represents an updated estimate of the relative ability of the different teams, where team i's wins are weighted according to [Mathematical Expression Omitted] and team i's losses are weighted according to [Mathematical Expression Omitted]. At successive iterations there may be changes in both win and loss scores, and in rankings between teams. The performance measure is assumed to have converged when [Mathematical Expression Omitted].
HETEROSKEDASTIC-CONSISTENT T-STATISTICS AND TESTS OF HYPOTHESES
If errors in equations (1)-(3), [[Epsilon].sub.jt], corresponding to year t of spell j, are heteroskedastic, standard estimated variances of the OLS estimates will be biased and inconsistent and hence statistical inferences will no longer be valid. White (1980) showed that it is possible to obtain an estimator of the covariance matrix of OLS estimates that is asymptotically valid when heteroskedasticity is present. Such an estimator is called a heteroskedasticity-consistent covariance matrix estimator.
Consider the regression equation
y = X[Beta] + u,
where y is a Nx1 vector of observations on the dependent variable, X is a NxK matrix of observations of the explanatory variables, [Beta] is a Kx1 vector of coefficients, and u is a Nx1 vector of random disturbances. The asymptotic covariance matrix of OLS assuming heteroskedasticity of the form [Omega] is given by
[Mathematical Expression Omitted].
One possible heteroskedastic-consistent covariance matrix estimator is
[Mathematical Expression Omitted],
[Mathematical Expression Omitted]
and [Mathematical Expression Omitted] are the OLS residuals.
To perform a test of q independent linear restrictions of the form
[H.sub.0]: R[Beta] = r,
a Wald test can be applied. This test is given by the equation
[Mathematical Expression Omitted],
which is asymptotically distributed as [[Chi].sup.2] with q degrees of freedom.
The data set used in this study is available on request from the authors.
1 See Mortensen (1988) for a survey of the matching literature.
2 Similar predictions are obtained from a matching model in which information generation occurs through on-the-job search by workers. Since the set of potential employers of football coaches is small, it does seem reasonable in the market for Australian Rules football coaches that experience should constitute a more important mechanism than search activity for learning match productivity.
3 Tamura and Neumann (1992) applied an alternative version of the method of paired comparisons to calculate adjusted performance measures in the National Football League in the United States. Whether a measure of adjusted or unadjusted winning percentage is more appropriate for the purposes of this study depends on which measure is chosen by football teams as an indicator of the value of the coach-team match. In this paper, the results we report apply the adjusted performance measure; however, results we obtained for an unadjusted performance were similar.
4 For example, Kahn (1993) found that coach quality can affect player performance. However, excluding controls for player performance has no qualitative effect on the estimates of coach-team match effects, which are reported in sections below.
5 Porter and Scully (1982) also derived a measure of match-invariant coach quality from a coach-specific dummy variable in a regression model for team performance; alternative methods for measuring coach quality apply information on coach salary, experience, and team performance (Kahn 993), or historical data on team performance or coach experience (Pfeffer and Davis-Blake 1986).
6 Mobility of coaches between minor leagues and the VFL/AFL might represent a progression between jobs with different opportunities for learning or human capital investment potential (see, for example, Rosen 1972).
7 Chapman and Southwick (1991) also proposed a GLS estimator to take account of the problem that since winning percentages within a given season must average 0.5, contemporaneous errors will be correlated.
8 This approach might not be valid if coaches with longer initial spell lengths had, on average, shorter repeat spells than the full sample of coaches with multiple spells. For example, suppose that all coaches with an initial spell of at least four years experienced a large negative draw of the error term in the first year of their second spells and for this reason had a completed duration in that spell of one year. In this case, the source of bias in estimated matching effects for coaches with an initial spell of at least four years would derive from a systematic effect of the error term on the performance measure in the repeat spell. However, this does not seem to be a problem. The average duration of initial spells for all coaches with multiple spells is 4.5 years, and average duration of completed repeat spells for those coaches is 3.9 years. For the restricted samples of coaches with spell lengths of at least two years, at least three years, and at least four years, the average durations of initial spells and completed repeat spells are, respectively, 4.97 years and 3.87 years, 5.61 years and 3.98 years, and 6.47 years and 4.09 years. Hence, spell length in a repeat spell is weakly positively correlated with initial spell length.
9 The results from estimation of equations (1)-(3) are robust to the omission of interrupted spells (from WW2 or from illness) and to omission of the player quality and coach experience variables.
10 See Kiefer (1988) for an introduction to the derivation of hazard functions.
11 The coefficient estimates on the match dummy variable constitute a generated regressor. Pagan (1984, Theorem 3 (iii), p. 226) demonstrated that with a single generated explanatory variable, where the true coefficient is zero, tests of whether the coefficient estimate is significantly different from zero are not biased. If the true coefficient is non-zero, the standard error from the OLS procedure is an underestimate of the true standard error.
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Jeff Borland is Associate Professor in the Department of Economics at the University of Melbourne, and Visiting Fellow at the Centre for Economic Policy Research at the Australian National University. Jenny Lye is Senior Lecturer in the Department of Economics at the University of Melbourne. For helpful comments, the authors thank John Kennan, Mico Loretan, Robert Tamura, Jim Walker, and seminar participants at the University of Chicago. Barrie Gillespie, John Knox, and Richard Home provided excellent research assistance.
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|Author:||Borland, Jeff; Lye, Jenny|
|Date:||Oct 1, 1996|
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