Life-Cycle Demand for Major League Baseball

Seung C. Ahn (480) 965-6574

miniahn@asu.eduArizona State University

Tempe, AZ 85827 U.S.A.

and

Young H. Lee (822) 760-4066

yhnlee@hansung.ac.krHansung University

Seoul, 136-792 Korea

June 20, 2003

Draft prepared for the Western Economics Association Conference, July 11-15, 2003.

1

Life-Cycle Demand for Major League Baseball

1. Introduction

Professional sports teams are known to be business-oriented and to have a degree

of monopoly power. If owners focus on profit-maximization, the optimal ticketing

pricing should guarantee the condition that marginal revenue (MR) is equal to marginal

cost (MC). Therefore, optimal pricing is to set price in the elastic range of attendance

demand. However, a puzzle is that a recurrent empirical finding for nearly thirty years

has not supported this basic microeconomic theory. Empirical work on sports

attendance demand has almost uniformly found that teams do not set ticket prices in the

elastic portion of gate demand. A long list of price elasticity estimates turns out to be

insignificant and/or to be less than unitary.

Some researchers offer their explanations for non-elastic pricing. Noll (1974)

and Fort (2003a) suggests that admission is only part of the cost of attendance, so that

price elasticity estimates are under-stated. By lowering ticket price, a team has some

revenue gain in parking, concessions and merchandise sales which might be larger than

the revenue loss in ticket sales. Fort (2003b) also uses a general specification of the

team revenue in Fort and Quirk (1995, henceforth, FQ)) to explain previous empirical

results. The FQ model considers local TV revenue in addition to attendance revenue

2

and its first order condition shows how inelastic pricing of winning can result from a

particular relationship between a team’s marginal local TV revenue, the marginal cost

of talent, and the average marginal local TV revenue for the rest of the teams in the

league. Fort (2003b) shows that this particular relationship holds for Major League

Baseball (MLB) using data from the period 1975-1988.

However, the inclusion of travel and/or other opportunity costs of attendance in

the regression models has not changed the empirical result of inelastic pricing [Bird

(1982), Fort and Roseman (1999a, 1999b)]. The proposition derived by Fort (2003b)

does not explain solely the puzzle since we still find inelastic pricing in the sports where

local TV revenue is negligible such as National Football League (NFL). Therefore, the

puzzle is still unsolved.

Consumption of sports games might be strongly habit-forming. Season ticket

holders in current season might have a tendency to buy a season ticket next season.

Given habitual consumption of sports games, we develop a dynamic life-cycle profit

model. The first order condition shows that inelastic pricing is consistent with life-

cycle profit maximizing team behavior. The rationale is that owners lower their ticket

prices to increase current attendance which will derive more future attendance because

of habitual behavior, and then they can raise their life-cycle profits.

3

Next, we apply time nonseparable life-cycle consumption models [Becker,

Grossman, and Murphy (1994, henceforth, BGM), Dynam (2000)] to MLB in order to

find whether there is a habit-formation in consumption of MLB games. According to

our empirical results, there is strong habit-formation in the consumption of MLB games

which explains why short-run price elasticity estimates are under-stated. The MLB

attendance in the current period does not respond to the ticket price in the same period

enough to be elastic because of habitual consumption.

This paper proceeds as follows. First, past empirical works on attendance

demand are briefly reviewed. Second, the theory of inelastic pricing is presented

based on life-cycle profit maximization. Third, the life-cycle consumption model is

introduced. Fourth, empirical results of MLB game attendance are presented.

Conclusions round out the paper.

2. Review

An extensive empirical literature reveals that sports teams set their price

attendance in inelastic demand range. One exception is the result of Demmert (1973)

which estimate a price elasticity of -0.93 for MLB. Most of other estimates of price

elasticity are insignificant and/or clearly less than unitary, and some of them have a

4

positive sign. We can find broad reviews in Cairns, Jennett, and Sloane (1985), Coffin

(1996), and Fort (2003).

Noll (1974) reports price estimate in the inelastic region of attendance for MLB,

with a point elasticity estimate of -0.14 and it is insignificant at the 95% level of

confidence. Noll also includes a price variable in attendance equation for NBA and

NFL, but both estimates of price are also insignificant. Siegfried and Eisenberg (1980)

find a point price elasticity estimate of -0.25 for minor league professional baseball but

it is not significant either. Jennett (1984) finds attendance unresponsive to price

changes for Scottish football. Scully (1989) estimates a price elasticity of demand of -

0.61. The estimation results of Welki and Zlatoper (1994) also follow the same line of

previous works. The estimate of price for NHL turned out to be significant, but the

price elasticity of -0.27 was far below unitary. Coffin (1996) estimates an annual

attendance model for MLB, separately for the 1962-1975 seasons and the 1976-1992

seasons. The price estimate is significant in the second period, but insignificant in the

first period and the price elasticity of demand is -0.67 in 1976-1992, as opposed to -0.11

in 1962-1975. Fort and Rosenman (1999a, 1999b) estimate a game-by-game

attendance equation for the 1989 and 1990 seasons. The price variables turn out to be

significant for both of NL and AL. According to the calculation by Fort (2003b), the

5

average values of price elasticity estimates are -0.36 and -0.14 for the AL and NL,

respectively, over the two years considered. Lee (2003) also estimates the attendance

equation of Korean professional baseball and finds attendance of low price elasticity.

The point estimate of price elasticity is -0.48.

3. The Life-cycle Profit Maximization Model

We assume that sports fans have habitual behavior to attend games. Since the

past attendance influences on the current attendance, sports team ticket pricing must

consider dynamic effects of the current price on the future attendance. Therefore, the

current attendance is a function of the past attendance as well as the current price.

ATTt = ATTt (Pt, ATTt-1) (1)

The habitual behavior affects optimal monopoly pricing. Then, the owner’s problem is

Max E [∑ βt-1 πt(ATTt, ATTt-1)] (2)

where β is a time discount factor and πt = ATTt (Pt, ATTt-1) Pt – c ATTt(Pt, ATTt-1).

Here, we assume a constant marginal cost of c for simplicity, but it does not affect the

main results. The associated first-order condition is

0][)( 1 =∂∂

+∂∂ +

t

t

t

t

PE

Pπ

βπ (3)

6

Under the assumption that people can perfectly forecast one-period-ahead ticket price,

the equation (3) is identical to

))(()11( 11

t

tt

pt ATT

ATTcPcP

∂∂

−−=−− ++β

ε (3’)

where εp is price elasticity.

The terms in the left-hand side of the equation (3’) are marginal revenue and

marginal cost, while the right-hand side is the discounted marginal effect on next

period’s profit of today’s attendance. When there is not habitual behavior

)0)(( 1 =∂∂ + tt ATTATT or when future price follows competitive pricing (Pt+1 = c), the

equation (3’) reduces to the profit maximization condition of MR=MC in the usual

static model. Then, the price elasticity is either unitary or greater than one.

Since an increase in current attendance causes future attendance to increase,

)( 1 tt ATTATT ∂∂ + is positive and the right-hand side of the equation (3’) has a negative

sign if the firm is a monopolist. This implies that marginal revenue is always less than

marginal cost in the life-cycle profit maximization. In case of MC=0, a monopolist set

its price low enough to keep in inelastic region because 0)11( <− ptP ε in the first-

order condition. In case of MC>0, the elasticity depends on the sign

of )])(([ 11 ttt ATTATTcPc ∂∂−− ++β . If it is negative, then the attendance demand is

inelastic. The smaller marginal cost and/or the stronger habit-formation, the less

7

elastic demand is. Therefore, inelastic pricing can be consistent with profit

maximizing behavior.

The implications of the above simple monopoly pricing model are intuitive. Since

greater current attendance increase future attendance, future profits are higher when

current price is lower and then current consumption is larger. Therefore, a sports team

maintains its price low enough to get more current attendance which will derive more

future attendance. With strong habitual behavior, which brings a sufficiently large

positive effect on future attendance of a lower current ticket price, a sports team might

set its price in the region of inelastic demand. One implication we get from this result

is that a sports team may offer lower ticket price to potential consumers who currently

do not buy tickets, if it can engage in price discrimination or variable (dynamic) pricing.

3. The Model

We will apply life-cycle consumption models to MLB to test whether there is

habit-formation of MLB consumption. In the past two decades, many studies on

consumption examine behavior when preferences are assumed to be time separable.

More recently, there has been growing interest in the implications of preferences that

are not time separable. In consumption studies, Becker, Grossman, and Murphy

8

(1994, henceforth, BGM) and Dynam (2000) present empirical results using time

nonseparable life-cycle consumption models.

The BGM model is developed in which utility maximizing consumers may

become “addicted” to the consumption of a product. It examines whether lower past

and future prices for cigarettes raise current cigarette consumption. Their empirical

results support the implication of addictive behavior that long-run responses exceed

short-run responses.

The Dynam model focuses on a specific class of time nonseparable preferences.

With habit formation, current utility depends not only on current consumption, but also

on a habit stock formed by lagged consumptions. A habit formation model implies a

condition relating the strength of habits to the evolution of consumption over time.

Her empirical results using consumption data from the Panel Study on Income

Dynamics yield no evidence of habit formation.

Our model is based on the Dynam model, since it imposes less strict assumptions

than the BGM model does. The BGM model assumes perfect foresight. With habit

formation, current utility depends not only on current consumption, but also on a habit

stock formed by lagged consumption. Habit formation causes consumers to adjust

9

slowly to shocks to permanent income. This model allows utility in each period to

depend on consumption services in that period;

U(Yt, C*t, εt)

(4)

Here C*t is consumption service in period t where C*

t = Ct – γ1Ct-1. Consumption

services in period t are positively related to current consumption and negatively related

to lagged consumption. The parameter γ1 measures the strength of habit formation.

The assumptions are basically the same as the Dynam model;

(i) The utility function is separable; that is, the utilities from the consumption of

different goods and services are separable. Assume that the utility from the

consumption of MLB is the form of Relative Risk Aversion:

11 1( )( )

1t t

tC Cx

ργαρ

−−−′

− ,

(5)

Here, xt is a vector of observable variables that can influence life-cycle variables on

utility.

(ii) Unlike the BGM model, this model does not assume perfect foresight for future.

That is, the marginal utility of wealth is allowed to vary over time. However, it is

10

assumed that people can perfectly forecast one-period-ahead ticket prices and teams’

performances

Dynam considers the case of composite consumption so that price variable is not

included in the model. However, we have to take into account a price factor since we

consider consumption of a specific good, a MLB game. An approximation of the first-

order conditions of the life-cycle utility maximization given the information available at

time t leads to the log-difference form as follows:

1 1 2 1 3ln ln ln ,t o t t t tC C P x 1γ γ γ γ+ +′∆ = + ∆ + ∆ + ∆ + ε +

(6)

where the parameters are the functions of the parameters in the first-order conditions.

xt is a vector of observable variables that can influence life-cycle variables on utility and

εt+1 is an error term related with forecast error.

This model also indicates that the consumption level at time t should not depend

on income at time t and the past if people make their current consumption levels

rationally. Thus, a way to test people’s rationality is to estimate the model including

current and past income levels and test whether the income variables are statistically

significant or not.

11

If the Ct were observed without measurement errors, the above equation can be

estimated by OLS. However, Dynam shows that the error term εt+1 follows MA(2), if

the Ct are observed with measurement errors. Then, the regressor, ∆lnCt is

endogenous. Our attendance data are the number of attendees. This means one

attendee at the third class seat and one at the first class seat are counted with the same

weight. Therefore, there should be some measurement error at the annual attendance

data, and then, we need to estimate the above equation using instrumental variables.

4. Data and Empirical Results

(1) Data

The data consist of a time series of MLB team cross sections covering the period

from 1969 through 2000. Table 1 contains means and standard deviations of primary

variables in the data set. MLB ticket price data from Doug Pappas are obtained at

Rodney Fort’s Sports Business Data Pages. We use Per Capita Personal Income in the

metropolitan statistical area in which the team is located. All prices and income

measures are real terms. The win/loss records of MLB teams are taken from

Baseballstat.net.

12

There are 30 MLB teams but 4 teams (Arizona Diamondbacks, Tampa Bay Devil

Rays, Montreal Expos, and Toronto Blue Jays) are dropped from the sample. We delete

Arizona and Tampa Bay since the two expansion teams do not provide enough

information and two Canadian teams in order for the fluctuation of exchange rate not to

influence on our empirical results. The time series begins at 1969 because of the

limitation of city-specific data and the period of 1986-1990 is dropped because ticket

price data are missing. Since the sample includes some expansion teams such as

Colorado Rockies, it is an unbalanced panel data set.

(2) Empirical Results

For MLB, the regression equation (6) becomes as below;

∆lnATTt+1 = γ0 + γ1∆lnATTt + γ2∆lnPt+1 + γ3∆WPCTt+1 + γ4∆GBt+1

+ γ5∆DUMNEWt+1 + εt+1, (7)

We include into xt, WPCT (winning percentage), GB (games back), and one dummy

variable. DUMNEW is a variable that allows a new or a renovated stadium to have a

linearly decreasing impact on attendance. DUMNEW is set to be 4-year trend since new

stadium or renovated stadium. DUMNEW is equal to 4 in the first year of a new

stadium, 3 in the second year, 2 in the third year, and 1 in the fourth year. We do not

include strike dummies since time effects will be considered.

13

We add individual effects as well as time effects to the demand function (7) and

we assume these effects are fixed since our sample are close to population in aspect of

teams. The equality of individual effects can not be rejected since the chi-squared

statistic is 12.05 and its p-value is 0.99. Therefore, the equality of individual effects is

not rejected at 1% significance level. The estimates of time and individual effects are

presented in Appendix Table A1, A2 and A3.

We test the quality of the instruments as shown in Table 2. The overall

significant test statistics are 919.34 in the estimation time effects only and 968.23 in the

estimation with time effects and individual effects, and both p-values are near zero.

According to the EHS exogeniety test developed by Eichenbaum, Hansen and Singleton

test (1981), ∆lnATTt turns out to be endogenous since the p-value is near zero as shown

Table 3. This is consistent with the model specification.

We estimate the demand function (7) by the Generalized Method of Moments

(GMM) controlling autocorrelations by the Newey-West method (setting bandwidth=3).

The choice of the bandwidth is reasonable if the et are not autocorrelated..

Table 3 and 4 present the GMM results with time effects and individual effects and

with time effects only, respectively, when the dependant variable is growth of

attendance. The first four columns of Table 3 contain GMM estimates by the two-step

14

estimation, while the last two columns contain an OLS estimate. The first four

columns of Table 4 contain GMM estimates, while the last column contains an OLS

estimate. The first two columns show the results of the two-step GMM estimation and

the next two columns show those of the continuous-updating GMM estimation.

The point estimates of γ1 are more than 0.5 and are significant at 1%. This

implies that there is strong habit-formation in the demand for attending MLB games.

The estimates of ∆INCt+1 are not significant at all in GMM estimation. Therefore, the

result is consistent with the notion that the MLB demand is habitual but rational. The

estimated price effects are negative, but not significant. Our explanation is that current

attendance may not responsive to current price because of strong habit-formation.

According to our brief application of the BGM model, we also get the empirical

result of habit formation. The estimated effect of past attendance on current attendance

is significantly positive as shown in Table 7 & 8. The estimate of 0.410-0.451 is more

or less equal to the coefficient (0.373-0.481) of past cigarette consumption estimated by

BGM. It implies the strength of habit in the consumption of MLB games may be

similar to those in the consumption of cigarette.

6. Conclusion

15

It has been a puzzle that previous literature analyzing the attendance demand

empirically finds inelastic ticket pricing consistently. This contradicts microeconomic

theory of profit maximization. In the inelastic range of demand, a monopoly firm can

raise its profit simply by reducing its output since marginal revenue is negative.

In this paper, we develop a theoretical model in which inelastic pricing can happen

as a result of profit maximization. We also broaden the discussions of attendance

demand to dynamic consumption models by taking account into the habitual aspects of

consumption. We apply the rational expectation life-cycle models to the demand for

attending MLB games.

Because of a habitual behavior, owners keep their ticket prices in the inelastic

portion of demand in order to increase current attendance. There might be some loss

of revenue in current period, but the increased attendance will encourage greater future

attendance enough to raise their life-cycle profits.

According to our empirical results, there is significant addiction or habit-

formation in the consumption of MLB games. This strong habit may cause demand

for attending MLB games to be insensitive to current ticket price change. Therefore,

short-run price elasticity is estimated to be less than unitary and/or to be insignificant

and sometimes to be positive in the previous empirical studies.

16

References

BaseballStats.net, 2003. [online], available: http://www16.brinkster.com/bbstats. Becker, G., M. Grossman, and K. Murphy, 1994. “An Empirical Analysis of Cigarette

Addiction,” The American Economic Review 84(June):396-418. Cairns, J., N. Jennet, and P.J. Sloane, 1985. “The Economics of Professional Team

Sports: A Survey of Theory and Evidence,” Journal of Economic Studies 13(3):179-186.

Coffin, D.A. 1996. “If You Build It, Will They Come? Attendance and New Stadium

Construction,” In E. Gustafson and L. Hadley (eds.) Baseball Economics, (Westport, CT: Praeger).

Dynam, K.E. 2000. “Habit Formation in Consumer Preerences: Evidence from Panel

Data,” The American Economic Review 90(June):391-406. Demmert, H.G. 1973. The economics of Professional Team Sports, (Lexington, Mass:

Lexington Books). Eichenbaum, M.S, L.P. Hansen, and K.J. Singleton, 1981. “A Time Series Analysis of

Representative Agent Models of consumption and Leisure Choice under Uncertainty,” The Quarterly Journal of Economics 103(1):51-78.

Fort, R. 2003a. Sports Economics (Upper Saddle River, NJ: Prentice Hall). Fort, R. 2003b. “Inelastic Sports Pricing,” Managerial and Decision Economics. Fort, R., and Quirk, J. 1995. “Cross-Subsidization, Incentives, and Outcomes in

Professional Team Sports Leagues.” Journal of Economic Literature XXXIII(September):1265-1299.

Fort, R. and R. Rosenman, 1999a. “Streak Management,” In J. Fizel, E. Gustafson, and

L. Hadley (eds.) Sports Economics: Curent Research (Westport, CT: Praeger Publishers).

17

Fort, R. and R. Rosenman, 1999b. “Winning and Managing for Streaks,” Proceedings of the joint Statistical Meetings of 1998, Section on Sports Statistics (Alexandrea, VA: American Statistical Association).

Hansen, L.P. 1982. “Large Sample Properties of Generalized Methods of Moments

Estimators,” Econometrica 50:1029-1055. Jennett, N. 1984. “Attendances, Uncertainty of Outcome and Policy in Scottish League

Football,” Scottish Journal of Political Economy 31(June):176-198. Lee, Y.H. 2002. “Decline of Attendance in Korean Baseball League: Economic Crisis

or Competitive Imbalance,” Unpublished Manuscript, Department of Economics, Hansung University, Seoul, Korea.

Newey, W. and K. West, 1987. “A Simple Positive Semi-Definite, Heteroskedasticity

and Autocorrelation Consistent Covariance Matrix,” Econometrica 55:703-708. Noll, R.G.. 1974. “Attendance and Price-Setting,” In R. Noll (eds.), Government and

the Sports Business, (Washington, D.C.: The Brookings Institution). Quirk, J., and Fort, R.D. 1992. Pay Dirt: The Business of Professional Team Sports

(Princeton, NJ: Princeton University Press). Scully, G.W. 1989. The Business of Major League Baseball (Chicago, IL: University

of Chicago Press). Siegfried, J.J., and J.D. Eisenberg, 1980. “The Demand for Minor League Baseball,”

Atlantic Economic Journal 8: 56-69. Welki, A.M. and T. J. Zlatoper, 1994. “US Professional Football: the Demand for

Game-Day Attendance in 1991,” Managerial and Decision Economics 15(5):489-495.

18

TABLE 1 Descriptive Statistics for Sample Data

Variable Mean Standard

DeviationMaximum Minimum

ATT: Attendance(mils.) 1.722 0.764 4.483 0.307

P: Ticket Price 5.023 3.205 20.618 1.457

INC: Income(thousands) 11.820 6.987 42.250 2.969

WPCT: Winning Percentage 0.502 0.069 0.704 0.321

GB: Games Back 14.132 11.548 52.000 0.000

19

TABLE 2

Testing Quality of Instrument Variables by First Stage OLS (∆lnATTt-1)

Dependent Variable Model 1 Model 2 Variable estimate t-value estimate t-value constant -0.184 -0.28

Pt-2 -0.180 -1.24 -0.196 -1.40 Pt-1 -0.096 -0.62 -0.096 -0.63 Pt 0.171 1.85 0.182 1.96

WPCTt-2 -1.059 -4.19 -1.138 -4.56 WPCTt-1 1.684 5.75 1.648 5.82 WPCTt -0.105 -0.42 -0.141 -0.62 GBt-2 0.002 1.54 0.002 1.45 GBt-1 -0.002 -1.32 -0.002 -1.35 GBt 0.000 -0.15 0.000 -0.17

DUMNEW t-2 -0.035 -2.45 -0.035 -2.59 DUMNEW t-1 0.050 2.32 0.049 2.23 DUMNEW t 0.015 0.96 0.011 0.72

INCt-2 -0.542 -1.43 0.413 1.18 INCt-1 0.518 1.45 -0.403 -1.09

1)ˆ,( yyCorr 0.648 0.638

Significant Test 968.23 [0.00] 919.34 [0.00]

The numbers in [] are p-values. 1 is a correlation coefficient. Model 1 is with time effects and individual effects. Model 2 is with time effects only.

20

TABLE 3

GMM Estimation with Time Effects and Individual Effects

GMM(i) GMM(ii) OLS Variable

estimate t-value estimate t-value estimate t-value ∆lnATTt-1 0.508 7.84 0.509 7.78 0.037 0.83

∆lnPt -0.070 -0.60 -0.031 -0.24 0.151 1.33

∆WPCTt 1.781 6.66 1.795 6.58 1.482 6.09

∆GBt -0.002 -1.60 -0.002 -1.44 -0.001 -1.02

∆DUMNEWt 0.028 1.87 0.026 1.69 0.040 3.27

∆lnINCt -1.019 -1.00 1)ˆ,( yyCorr 0.522 0.513 0.618

Hansen Test 12.85 [0.17] 11.40 [0.18]

The numbers in [] are p-values. 1 is a correlation coefficient. GMM(i) & (ii) are the two-step GMM.

21

TABLE 4 GMM Estimation with Time Effects Only

GMM OLS Independent

variable (i) (ii) (iii) (iv) (v)

constant 0.061 (2.00)

0.112(1.74)

0.058(1.80)

0.127 (1.86)

0.035(1.13)

∆lnATTt-1 0.508 (7.72)

0.505(7.57)

0.558(8.17)

0.542 (7.86)

0.050(1.17)

∆lnPt -0.049 (-0.43)

-0.017(-0.14)

-0.063(-0.55)

-0.020 (-0.16)

0.154(1.39)

∆WPCTt 1.788 (6.61)

1.799(6.55)

1.877(6.84)

1.876 (6.73)

1.481(6.12)

∆GBt -0.002 (-1.61)

-0.002(-1.47)

-0.002(-1.36)

-0.002 (-1.23)

-0.001(-1.08)

∆DUMNEWt 0.031 (2.11)

0.030(2.02)

0.025(1.71)

0.024 (1.61)

0.039(3.19)

∆lnINCt -0.812

(-0.89) -1.101 (-1.13)

1)ˆ,( yyCorr 0.517 0.511 0.502 0.498 0.611Hansen Test. 11.72

[0.23] 10.69[022]

10.77[0.29]

9.77 [0.28]

The numbers in () are t-values. The numbers in [] are p-values. 1 is a correlation coefficient. (i) & (ii) is the two-step GMM. (iii) & (iv) is the continuous-updating GMM.

22

TABLE 5 Testing Quality of Instrument Variables by First Stage OLS in the BGM Model

with Time effects and Individual Effects

Dependent Variable ATTt-1 ATTt+1

Variable estimate t-value estimate t-value

Pt-1 0.088 1.94 -0.019 -0.41 Pt 0.045 1.18 0.040 0.93

Pt+1 0.006 0.21 0.059 1.91 WPCTt-1 2.844 5.07 0.207 0.42 WPCTt -0.257 -0.53 1.354 3.14

WPCTt+1 -0.368 -0.73 2.255 4.66 GBt-1 -0.007 -1.96 -0.003 -1.14 GBt -0.003 -1.15 -0.005 -1.77

GBt+1 -0.004 -1.34 -0.006 -2.06 DUMNEW t-1 0.080 2.97 0.066 2.41 DUMNEW t -0.001 -0.04 0.001 0.05

DUMNEW t+1 -0.001 -0.04 0.094 4.41 INCt-1 -0.033 -0.45 -0.042 -0.56 INCt 0.150 1.50 0.004 0.05

INCt+1 -0.112 -1.85 0.016 0.30 1)ˆ,( yyCorr 0.791 0.830

Overall Significant Test 1136.44 [0.00] 2577.13 [0.00]

The numbers in [] are p-values. 1 is a correlation coefficient.

23

TABLE 6 Testing Quality of Instrument Variables by First Stage OLS in the BGM Model

with Time effects only

Dependent Variable ATTt-1 ATTt+1

Variable estimate t-value estimate t-value

Pt-1 -0.011 -0.17 -0.120 -2.09 Pt 0.080 1.54 0.073 1.36

Pt+1 0.034 0.87 0.100 2.40 WPCTt-1 2.774 4.25 -0.131 -0.21 WPCTt -0.395 -0.70 1.046 1.96

WPCTt+1 0.070 0.12 2.528 4.41 GBt-1 -0.008 -1.78 -0.005 -1.24 GBt -0.004 -1.25 -0.006 -1.85

GBt+1 -0.004 -1.05 -0.007 -1.85 DUMNEW t-1 0.094 2.30 0.068 1.78 DUMNEW t -0.001 -0.02 -0.002 -0.07

DUMNEW t+1 -0.007 -0.12 0.071 1.87 INCt-1 0.045 0.47 0.073 0.77 INCt 0.009 0.07 -0.134 -1.22

INCt+1 -0.088 -1.27 0.020 0.31 1)ˆ,( yyCorr 0.596 0.656

Overall Significant Test 736.55 [0.00] 783.72 [0.00]

The numbers in [] are p-values. 1 is a correlation coefficient.

24

TABLE 7

GMM Estimation of the BGM Model With Time Effects and Individual Effects

GMM OLS

Variable estimate t-value estimate t-value

ATTt-1 0.451 8.48 0.439 26.61 ATTt+1 0.371 1.94 0.484 20.84

Pt 0.018 1.07 0.007 0.86 WPCTt 1.741 4.68 1.467 4.88

WPCTt+1 -0.882 -1.63 -1.142 -3.77 GBt -0.005 -2.42 -0.005 -2.78

GBt+1 0.002 0.92 0.003 1.57 DUMNEWt 0.044 2.37 0.054 2.96

DUMNEWt+1 -0.002 -0.06 -0.025 -1.39 INCt 0.020 0.62 0.028 1.09

INCt+1 -0.025 -0.97 -0.029 -1.37 1)ˆ,( yyCorr 0.938 0.942

Hansen Test 4.73 [0.32]

The numbers in [] are p-values. 1 is a correlation coefficient. GMM is the two-step GMM.

25

TABLE 8

GMM Estimation of the BGM Model with Time Effects Only

GMM(i) GMM(ii) OLS

Variable estimate t-value estimate t-value estimate t-value

constant -0.067 -0.46 -0.081 -0.56 -0.050 -0.34 ATTt-1 0.410 10.59 0.411 10.75 0.442 29.50 ATTt+1 0.539 5.38 0.544 5.71 0.521 29.62

Pt 0.006 0.60 0.005 0.52 0.003 0.47 WPCTt 1.545 5.58 1.550 5.64 1.385 4.75

WPCTt+1 -1.297 -3.40 -1.296 -3.44 -1.201 -4.06 GBt -0.004 -2.24 -0.004 -2.18 -0.005 -2.67

GBt+1 0.003 1.50 0.003 1.59 0.003 1.89 DUMNEWt 0.036 2.23 0.031 1.98 0.051 2.89

DUMNEWt+1 -0.018 -0.96 -0.016 -0.85 -0.030 -1.69 INCt 0.033 1.32 0.034 1.26 0.029 1.22

INCt+1 -0.035 -1.67 -0.036 -1.55 -0.030 -1.45 1)ˆ,( yyCorr 0.941 0.941 0.941

Hansen Test 5.26 [0.26] 4.96 [0.29]

The numbers in [] are p-values. 1 is a correlation coefficient. GMM(i) is the two-step GMM. GMM(ii) is the continuous-updating GMM.

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TABLE A1 Time Effects Estimation of the Model with Time Effects and Individual Effects

GMM(i) GMM(ii) OLS

Year estimate t-value estimate t-value estimate t-value

71 -0.068 -1.24 -0.082 -1.48 -0.036 -0.74 72 -0.170 -3.79 -0.154 -3.20 -0.126 -2.87 73 0.114 2.22 0.125 2.32 0.099 2.10 74 -0.125 -2.13 -0.102 -1.61 -0.034 -0.59 75 -0.058 -1.08 -0.064 -1.17 -0.046 -1.00 76 -0.007 -0.12 0.033 0.45 0.000 -0.01 77 0.026 0.52 0.065 1.00 0.057 1.24 78 -0.070 -1.29 -0.022 -0.29 0.002 0.05 79 -0.011 -0.21 0.027 0.40 0.021 0.42 80 -0.085 -1.43 -0.072 -1.18 -0.026 -0.46 81 -0.572 -11.85 -0.562 -11.28 -0.557 -9.95 82 0.763 11.86 0.743 10.95 0.541 10.23 83 -0.327 -6.57 -0.342 -6.59 -0.067 -1.18 84 -0.078 -1.31 -0.042 -0.60 -0.041 -0.82 93 0.093 2.14 0.052 0.87 0.085 1.88 94 -0.514 -11.13 -0.545 -9.93 -0.420 -9.13 95 0.132 2.32 0.108 1.73 -0.030 -0.65 96 0.146 3.46 0.131 2.93 0.170 3.97 97 -0.096 -2.35 -0.109 -2.59 0.009 0.21 98 -0.032 -0.69 -0.031 -0.68 -0.007 -0.17 99 -0.012 -0.29 -0.022 -0.52 -0.004 -0.10

2000 -0.061 -1.35 -0.051 -1.07 -0.038 -0.87 (i) is the two-step GMM without ∆lnINCt.

(ii) is the two-step GMM with ∆lnINCt.

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TABLE A2 Time Effects Estimation of the Model with Time Effects Only

GMM OLS

Year (i) (ii) (iii) (iv) (v)

71 -0.068(-1.22) -0.077(-1.38) -0.057(-1.00) -0.065(-1.15) -0.037(-0.76) 72 -0.173(-3.84) -0.161(-3.43) -0.169 (-3.55) -0.154(-3.16) -0.126(-2.89) 73 0.110(2.15) 0.117(2.22) 0.116(2.15) 0.126(2.30) 0.100(2.15) 74 -0.127(-2.15) -0.108(-1.72) -0.127(-2.10) -0.103(-1.61) -0.036(-0.62) 75 -0.062(-1.16) -0.068(-1.23) -0.059(-1.06) -0.065(-1.13) -0.045(-1.01) 76 -0.011(-0.18) 0.020(0.28) -0.018(-0.29) 0.028(0.38) 0.000(-0.01) 77 0.024(0.47) 0.055(0.90) 0.034(0.65) 0.076(1.19) 0.057(1.25) 78 -0.077(-1.44) -0.041(-0.61) -0.079(-1.44) -0.027(-0.39) 0.000(0.01) 79 -0.017(-0.32) 0.012(0.19) -0.013(-0.24) 0.027(0.41) 0.020(0.41) 80 -0.078(-1.32) -0.068(-1.13) -0.077(-1.27) -0.062(-1.00) -0.027(-0.48) 81 -0.575(-11.61) -0.568(-11.29) -0.575 (-11.38) -0.565(11.01) -0.558(-10.04) 82 0.758(11.71) 0.738(10.70) 0.783(11.73) 0.753(10.60) 0.547(10.55) 83 -0.332(-6.58) -0.344(-6.56) -0.349(-6.78) -0.362(-6.75) -0.075(-1.35) 84 -0.081(-1.35) -0.053(-0.78) -0.076(1.23) -0.039(-0.55) -0.042(-0.82) 93 0.095(2.17) 0.062(1.07) 0.103(2.29) 0.058(0.97) 0.086(1.90) 94 -0.517(-11.09) -0.542(-10.02) -0.523(-10.97) -0.556(-10.00) -0.423(-9.27) 95 0.120(2.07) 0.099(1.56) 0.155(2.62) 0.119(1.84) -0.035(-0.73) 96 0.136(3.19) 0.124(2.75) 0.138(3.15) 0.122(2.61) 0.161(3.76) 97 -0.107(-2.66) -0.117(-2.81) -0.112(-2.68) -0.124(-2.89) -0.003(-0.08) 98 -0.044(-0.95) -0.045(-0.98) -0.042(-0.90) -0.044(-0.95) -0.017(-0.34) 99 -0.020(-0.48) -0.027(-0.65) -0.018(-0.44) -0.027(-0.63) -0.014(-0.34)

2000 -0.069(-1.50) -0.062(-1.31) -0.059(-1.24) -0.053(-1.07) -0.048(-1.08) The numbers in () are t-values. (i) & (ii) is the two-step GMM. (iii) & (iv) is the continuous-updating GMM.

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TABLE A3

Individual Effects Estimation of the Model with Time Effects and Individual Effects

GMM(i) GMM(ii) OLS

Team estimate t-value estimate t-value estimate t-value

ANA 0.054 1.54 0.110 1.69 0.042 1.05 BAL 0.076 2.28 0.139 2.01 0.057 1.61 BOS 0.055 1.66 0.127 1.64 0.011 0.32

CHI(W) 0.063 1.38 0.119 1.66 0.036 0.74 CLE 0.061 1.68 0.116 1.77 0.048 1.15 DET 0.063 1.91 0.125 1.79 0.036 0.98 KC 0.061 1.79 0.126 1.75 0.040 1.05 MIL 0.047 1.23 0.107 1.49 0.033 0.82 MIN 0.040 1.05 0.107 1.39 0.004 0.09

NY(Y) 0.071 2.10 0.132 1.96 0.052 1.45 OAK 0.053 1.14 0.122 1.49 0.028 0.45 SEA 0.072 1.81 0.135 1.80 0.031 0.72 TEX 0.072 2.16 0.140 1.88 0.039 1.09 ATL 0.051 1.37 0.121 1.57 0.025 0.59

CHI(C) 0.073 2.16 0.132 2.00 0.043 0.99 CIN 0.055 1.46 0.118 1.66 0.026 0.62 COL -0.001 -0.01 0.072 0.82 -0.052 -1.11 FLO -0.045 -0.77 0.000 0.00 -0.115 -1.76 HOU 0.070 1.88 0.147 1.80 0.040 0.89 LA 0.053 1.58 0.110 1.67 0.032 0.89

NY(M) 0.049 1.26 0.108 1.52 0.013 0.25 PHI 0.050 1.39 0.111 1.60 0.043 0.96 PIT 0.063 1.39 0.121 1.70 0.015 0.39 STL 0.052 1.52 0.113 1.64 0.029 0.76 SD 0.064 1.59 0.133 1.70 0.071 1.57 SF 0.070 1.66 0.148 1.70 0.037 0.76

(i) is the two-step GMM without ∆lnINCt.

(ii) is the two-step GMM with ∆lnINCt.

29

TABLE A4 Time Effects Estimation of the BGM Model

With Time Effects and Individual Effects

GMM OLS Year

estimate t-value estimate t-value70 -0.024 -0.20 0.041 0.59 71 -0.027 -0.19 0.057 0.92 72 -0.178 -1.44 -0.113 -1.85 73 -0.010 -0.08 0.046 0.79 74 -0.063 -0.59 -0.013 -0.21 75 -0.049 -0.49 0.000 0.00 76 -0.046 -0.55 -0.014 -0.23 77 0.051 0.70 0.075 1.30 78 0.013 0.20 0.026 0.46 79 0.088 1.52 0.099 2.02 80 0.277 1.65 0.370 6.07 81 -0.647 -10.26 -0.638 -10.74 82 0.340 5.58 0.334 6.53 83 0.072 1.05 0.081 1.27 92 -0.046 -0.40 -0.090 -1.44 93 0.555 5.41 0.580 7.71 94 -0.363 -3.65 -0.303 -4.58 95 -0.088 -1.20 -0.133 -2.08 96 0.211 2.55 0.187 3.08 97 0.119 1.37 0.092 1.31 98 0.147 1.49 0.101 1.42 99 0.144 1.61 0.120 1.63

GMM is the two-step GMM.

30

TABLE A5 Time Effects Estimation of the BGM Model

With Time Effects only

GMM(i) GMM(ii) OLS Year

estimate t-value estimate t-value estimate t-value 70 0.049 0.53 0.049 0.53 0.059 0.95 71 0.077 0.76 0.082 0.82 0.082 1.46 72 -0.095 -1.03 -0.091 -1.01 -0.091 -1.62 73 0.066 0.75 0.070 0.80 0.068 1.24 74 0.007 0.09 0.010 0.12 0.004 0.07 75 0.009 0.11 0.010 0.13 0.014 0.24 76 -0.011 -0.15 -0.011 -0.16 -0.005 -0.08 77 0.078 1.18 0.078 1.20 0.083 1.46 78 0.033 0.53 0.033 0.54 0.031 0.55 79 0.105 1.98 0.105 1.98 0.103 2.10 80 0.410 3.86 0.413 4.01 0.400 6.58 81 -0.630 -10.38 -0.631 -10.36 -0.631 -10.53 82 0.326 5.83 0.326 5.85 0.339 6.62 83 0.095 1.44 0.095 1.44 0.086 1.33 92 -0.118 -1.36 -0.121 -1.40 -0.107 -1.81 93 0.618 8.39 0.624 8.50 0.595 8.31 94 -0.285 -4.15 -0.297 -4.31 -0.285 -4.61 95 -0.104 -1.61 -0.100 -1.56 -0.127 -2.26 96 0.180 2.61 0.179 2.62 0.190 3.52 97 0.090 1.23 0.088 1.20 0.092 1.55 98 0.108 1.28 0.105 1.27 0.099 1.72 99 0.122 1.56 0.121 1.54 0.123 1.96

GMM(i) is the two-step GMM. GMM(ii) is the continuous-updating GMM.

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TABLE A6 Individual Effects Estimation of the BGM Model

With Time Effects and Individual Effects GMM OLS

Team estimate t-value estimate t-value

ANA -0.076 -0.39 -0.018 -0.10 BAL -0.153 -0.67 -0.062 -0.33 BOS -0.140 -0.66 -0.044 -0.25

CHI(W) -0.151 -0.69 -0.055 -0.31 CLE -0.160 -0.75 -0.061 -0.34 DET -0.112 -0.54 -0.030 -0.17 KC -0.173 -0.80 -0.069 -0.40 MIL -0.136 -0.63 -0.042 -0.23 MIN -0.201 -0.87 -0.074 -0.42

NY(Y) -0.171 -0.79 -0.072 -0.40 OAK -0.226 -0.96 -0.087 -0.51 SEA -0.135 -0.65 -0.046 -0.26 TEX -0.171 -0.81 -0.068 -0.39 ATL -0.147 -0.70 -0.039 -0.23

CHI(C) -0.106 -0.53 -0.042 -0.25 CIN -0.160 -0.75 -0.071 -0.40 COL 0.060 0.24 0.031 0.17 FLO -0.122 -0.57 -0.015 -0.08 HOU -0.161 -0.74 -0.062 -0.35 LA 0.025 0.11 0.014 0.08

NY(M) -0.094 -0.45 -0.024 -0.13 PHI -0.066 -0.33 -0.017 -0.09 PIT -0.258 -1.08 -0.118 -0.69 STL -0.079 -0.40 -0.027 -0.16 SD -0.091 -0.47 -0.022 -0.13 SF -0.173 -0.74 -0.064 -0.35

GMM is the two-step GMM.

32