Baseball: An Economics Story

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Baseball An Economics Story
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Discrimination

• During the early stages of the MLB association,
  like many other social activities, there was
  discrimination in baseball
• The purpose of this presentation is to look at
  the cost that is associated with having a
  discriminatory employment policy verses one
  that integrates all able body individuals
• Discrimination is hard to measure, this is why we
  have chosen baseball, where discrimination was
  evident in the prohibition of black players in
  the MLB
• Therefore, the period of the mid-1940s, where
  MLB teams started to employ black players from
  the negro league is the perfect starting point in
  which to study there contribution (MPL) to there
  employers
• Simpson's clip (sorry, file was too big!)

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Overview of Article

• Major League Baseball had policy of
  discrimination, until the colour barrier was
  broken by Jackie Robinson in 1942.
• Once colour line was broken ceteris paribus firms
  had an incentive to substitute less expensive and
  in some cases more productive black players for
  higher-priced white players.
• The five teams that employed the most black
  player years through 1956 comprised five of the
  top six teams from 1952-56.
• Contrary to what many owners thought, attendance
  improved greatly for teams who employed more
  black player years.
• In 1955 blacks comprised 7.7 of all Major League
  players, yet they made up 18 of the all-star
  team.

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Overview of Marginal cost and Marginal Benefit

• Example we have a production line, if we were to
  add addition machines to this line the output
  would increase by our benefit column

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Marginal Cost and Benefit
Marginal benefit is defined as the value added
from the addition of one more machine/employee in
a production process. Marginal benefit
diminishes with the addition of more employees.
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Important Economic Intuition

• Low discriminators obtained a competitive
  advantage in Major League Baseball relative to
  other teams ? Productivity gains, low wages, more
  productive inputs.
• However, this advantage has its limits due to
  economic law of diminishing marginal returns.

                                    
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Variables

• Black47 Cumulative number of black player
  years, 1947-1956
• Black52 Cumulative number of black player
  years, 1952-1956
• Won Percentage of games won, 1952-1956
  (dependent)
• Rank Ranking (won-lost record), 1952-1956
• Prob Probit for percentage games won, 1952-1956
• f Ordinate of the standard normal curve for
  percentage of games won, 1952-1956

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Basic Question ? Question 1

• Won a bXi ei
• Xi black47
• b 0.0043187 My Pictures\graphfor1.gif
• H0 b 0, HA b not equal 0 _at_ 5 level
• t-stat 2.330, tcrit-14 d.f. 2.145
• t-stat gt t-crit, therefore reject H0, Beta is
  significant at 5 level
• Note Constant is highly significant using same
  method as above

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Question 1d)

• Important to test beta at 5 level of
  significance, to test whether our model has any
  significance, and whether we should continue with
  our analysis
• Our Beta is consistent with the economic
  intuition presented in the paper that black
  players have positive impact on winning
  percentages.
• Our Beta value from equation 1 is the marginal
  product of an additional black player on a team
  in terms of an increase in the percentage of
  games won.

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Question 2

• a) USING 95 AND 90 CONFIDENCE INTERVALS
• CONFIDENCE INTERVALS BASED ON NORMAL
  DISTRIBUTION WITH CRITICAL VALUES 1.960 AND
  1.645
• NAME LOWER 2.5 LOWER 5 COEFFICIENT
  UPPER 5 UPPER 2.5 STD. ERROR
• BLACK47 0.6862E-03 0.1270E-02
  0.43187E-02 0.7367E-02 0.7951E-02
  0.002
• b) Our true value of Beta is in our significant
  range
• c) In economic terms, we can state with 90
  confidence that the addition of black players has
  a significant marginal benefit at the extreme
  bounds of the interval.

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Question 3

• a) Xi black52
• b) b 0.0053592, t-stat 1.857, t-crit 2.145
• -Beta is insignificant at 5 level of
  significance
• c) The impact that black players had on winning
  percentage has declined over the period of
  1952-1956. The cause of this is two-fold.
  However, this is a strong assumption that
  requires further analysis.

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Question 4

• Note!
• Mean of Won 0.55567
• Std. Error 0.069661
• t-crit 2.015
• 90 Confidence interval
• 0.41530 gt b gt 0.69604
• What does this mean??
• We can state with 90 confidence that teams
  with
• at least 10 black player years are expected
  to have a winning percentage between 41.5 and
  69.6.

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Question 4 Contd

• c) Note!
• We now use our coefficients from our
  regression in question 3 and set Xi (black52)
  equal to 10.
• Won 0.45897 0.0053592 (black52)
• Now we can derive a 90 prediction interval
• 0.512562 /- 1.761 Se
• ?0.50748gtbgt0.51764

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Question 5

• Glejser test for Heteroskedasticity by
  construction.
• What is heteroskedasticity by construction?
• -Gujarati in his text states that you should run
  an OLS on the absolute value of your residuals
  against your explanatory variable(s).
• -If your beta is significant, it is an
  indication that your explanatory variables have a
  relationship with your absolute residual,
  indicating your model possesses
  heteroskedasticity.

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Question 5 Contd

• -If your explanatory variable(s) is used to
  explain your dependent variable, your Beta should
  be insignificant, because if your explanatory
  variables explain your dependent variable well,
  there should be no relationship between your
  residual error and explanatory variable.
• Economic significance for our case ? In our
  regression if by construction we have
  heteroskedasticity, black player years are not
  necessarily the only factor adding to an increase
  in winning percentage.

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Question 5 Contd

• Gujarati equation ? ei bXi ui
• constant in this case is irrelevant to our
  calculation- J.D. Han
• T-stat from regression equals, 3.541 and
  t-crit 1.753.
• Therefore, our Beta is significant at the 10
  level of significance, indicating
  heteroskedasticity.

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Question 5 contd

• c) Note GLS minimizes the weight sum of squared
  residuals. In the case of heteroskedasiticity
  observations expected to have large error terms
  with large variances are given less weight then
  in an observations with small variances.
• This can be corrected, (as quoted by Professor
  Ibbott), by the hetcov command in shazam on our
  initial regression equation with Xi black47.
• The hetcov command uses the heteroskedastic
  consistent-covariance matrix, which standardizes
  our errors.

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Question 6

• Linear probability is flawed, the model assumes
  constant returns to scale, which is not
  consistent with the economic theory of our model.
• We now proceed to use the logit model
• log (pi/1-pi) a bXi e

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Question 6 Contd

• c) ..\..\..\SHAZAM\question6output.txt
• At 10 level of significance tcrit 1.761,
  therefore our beta is significant.
• Note!
• Our constant has been proven to be insignificant
  in our model.
• d) As indicated by the significance of our Beta
  coefficient, black players do in fact appear to
  have a positive impact on winning percentage.
  However, their contribution declines as more
  teams begin to desegregate and employ more black
  players, thus saturating the league.

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Question 6 contd

• To help you visualize these contributions, we
  have a graph of the function.
• ..\..\..\SHAZAM\Q6graph.gif
• To further emphasize this theory, we have the set
  of Beta values.
• e) The logit model, as depicted in the graph
  above, satisfies the expected theory of
  diminishing marginal returns experienced when
  major league baseball teams began employing black
  players at an increasing rate.

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Question 7

• In our new equation we now are given a function
  for our variance of the error, to be added into
  our logit equation. The only purpose of this we
  found was that it standardizes the error into a
  specific functional form to reduce random error
  fluctuations. Hence reducing the probability of
  heteroskedasticity
• ei 1 / Ni Pi(1 Pi)
• -Ni number of total games played
• -Pi expected probability of winning for team i

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Question 7 Contd

• Note!
• We are using expected probability of winning
  instead of actual probabilities. This is the
  formula used for expected probabilities
• logit function 0.5
• - logit function is our marginal product of
  winning, our 0.5 is the probability of winning
  holding all contributing factors to winning
  constant. ? ..\..\..\SHAZAM\teams expected to
  win2.txt

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Question 7 Contd

• log(pi/1-pi) a bXi ei
• Xi black47
• we run the OLS using the hetcov command
• Here is our output ? ..\..\..\SHAZAM\question7outp
  ut.txt

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Question 7 Contd

• b) Test for 10 level of significance with
• t-crit 1.761 ? since our t-stat of our beta is
  3.539, we reject the null hypothesis. Our
  constant is insignificant along with our
  variance.
• A regression exists due to the significance of
  our beta, even though our constant is
  insignificant in this case ? the importance lies
  in the beta being significant, due its important
  property.
• Note!
• We could also use the F-test to test for a
  regression.
• We found the F-stat to be 5.39, with our F-crit
  equal to 3.07. Since
• F-stat gt F-crit we have a regression at the 10
  level of significance.
• Ignore low value of R2, most likely random
  walk produced that result.

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Question 7 Contd

• c) Coefficients have not changed, but our
  significance of our t-test has actually
  increased, putting greater significance on the
  positive contribution of black players to their
  teams.

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Question 8

• Note!
• probit model uses standard normal distribution
  that corresponds to observed relative frequency
  eg. pi that equals 0.03 in probit zi would be
  -1.88.
• Distribution Tables

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Question 8 Contd

• Probit will be our dependent variable, with
  black52 being our explanatory variable.
• Our regression results ? ..\..\..\SHAZAM\question8
  output.txt
• We test our null hypothesis at the 10 level of
  significance. With b 0, and t-crit 1.761.
  Our t-stat gt t-crit, hence our beta is
  significant.

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Question 8 Contd

• Heres a graph to better understand the probit
  function ? My Pictures\question8graph.gif
• Here is also an output displaying our varying
  values of beta (since obviously beta isnt
  constant, you just look at it and you know T.
  Osborne) ?
• A regression exists, since our most important
  variable is significant
• Note!
• We could also use the F-test to test for
  regression, we found our F-stat to be 3.42, and
  our F-crit was found to be 3.07, hence if F-stat
  gt F-crit we have a regression at the 10 level of
  significance

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Question 9

• Group data probit model suffers from
  heteroskedasticity from construction ? Glejser
  Test.
• ? Our auxilliary regression with absolute
  residual as the dependent and black52 as the
  independent ..\..\..\SHAZAM\question9output.txt
• t-crit 1.753 ? t-stat gt t-crit, therefore we
  have heteroskedasticity consistent with our
  Glejser.

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Question 9 Contd

• Note!
• The Glejser Test, although it calculated the
  presence of heteroskedasticity, Gujarati states
  its an unreliable test for a small sample size,
  because it computes absolute residual values,
  instead of the more reliable squared residual.

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Question 9 Contd

• Solution ? Use the White Procedure! (NOT the
  White TEST)
• Method ? e2 a b1Xi b2(Xi)2
• ? From this we take our R2 and multiply it by n,
  to get LM.

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Question 9 Contd

• -Our White Procedure Output ? ..\..\..\SHAZAM\ques
  tion9whiteprocedure.txt
• -LM 0.1020 15
• 1.53
• -Chi Square Stat at 10 level of significance
• Chi square stat 6.251 with 3 d.f.
• H0? No Hetero, HA? Hetero
• Since chi square gt LM we do not reject H0.

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Question 10

• -Similar to question 7, we are given an equation
  for our variance of the error term, to use in our
  probit model ? ei Pi(1 Pi) / Ni fi2
• Note!
• We once again use expected probability of winning
  instead of the actual values. Ni is still equal
  to 154. f is the ordinate of the standard normal
  curve for percentage games won.
• - The only purpose of our function of our
  variance of error, may be to standardize the
  error into a functional form, to reduce the
  opportunity for heteroskedasticity with highly
  fluctuating error values.

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Question 10 Contd

• We estimate our model, with Probit as the
  dependent variable with black52, and our error
  equation as the explanatory variables. We also
  use the GLS method, by using the hetcov command
  on shazam
• Our Results ? ..\..\..\SHAZAM\question10output.txt
  
• t-crit at 10 level of significance with 14 d.f.
  1.761.
• t-stat gt t-crit, therefore our b1 is significant
• We do have a regression, at the 10 level of
  significance, since our most important
  explanatory variable remains significant in this
  case.
• Note!
• We will once again use the F-test to verify the
  above result. Our F-stat
• was found to be 3.42. Since our F-crit is equal
  to 3.07, we do have a
• regression at the 10 level of significance.

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Question 10 Contd

• Comparison of heteroskedastic corrected
  regression, and standard regression with black
  52. ?..\..\..\SHAZAM\question10Coutput.txt

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Question 11

• We will now test our original probit model for
  heteroskedasticity using the White test at the
  10 level of significance
• White Test stat 1.632, with critical value of
  4.605 with 2 d.f.
• ? Since our t-stat lt t-crit, there is not
  heteroskedasticity in our probit model, according
  to the white test.