Y [team finish] = ? ?X [spending]

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• Example
• Y team finish ? ?X spending
• Expressing the model in words, values of the Y
  variable (team finish) are a function of some
  constant, plus some amount of the X variable.
• The question we are interested in is how much
  change in the Y variable (team finish) is
  associated with a one-unit change in the X
  variable (spending). The answer lies in ß (beta),
  this is know as the regression coefficient. In
  terms of the baseball example, it would be the
  amount of improvement in team finish (1first, 2
  second, . and 7 last) associated with an
  additional 1 million in spending on players
  salaries.
• Q Would we expect the relationship to be
  positive or negative?

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• Todd Donovan using 1999 season data and a
  bivariate regression found
• Team finish 4.4 0.029 x spending (in
  millions)
• This means that the slope of the relationship
  between spending and team finish was 0.029. Or,
  for each million dollars that a team spends,
  there is only a 3 percent change in division
  position. This result is significant at the .01
  level. These results show that a team spending
  70 million on players will finish close to
  second place. We can also show that any given
  team would have to spend 35 million more to
  improve its team finish by one position (-0.029 x
  35million 1.105).
• Pearsons r for this example was -0.39 which
  means that spending explains only 15 percent of
  variation in the teams finish (r2 .15 -0.39
  x 0.39).

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• Coefficient of determination r-squared

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• Coefficient of nondetermination
• 1- r2

5

• Sum of Squares for X
• Some of Squares for Y
• Sum of produces

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Multiple Regression

• I. Multiple Regression
• Multiple regression contains a single dependent
  variable and two or more independent variables.
  Multiple regression is particularly appropriate
  when the causes (independent variables) are
  inter-correlated, which again is usually the
  case.
• ßi is called the called a partial slope
  coefficient as it is what mathematicians call the
  slope of the relationship between the independent
  variable Xi and the dependent variable Y holding
  all other independent variables Constant.

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• II. Assumptions
• Normality of the Dependent Variable Inference
  and hypothesis testing require that the
  distribution of e is normally distributed.
• Interval Level Measures The dependent variable
  is measured at the interval level
• The effects of the independent variables on Y are
  additive For each independent variable Xi, the
  amount of change in E(Y) associated with a unit
  increase in Xi (holding all other ind. Variables
  constant) is the same.
• The regression model is properly specified. This
  means there is no specification bias or error in
  the model, the functional form, i.e., linear,
  non-linear, is correct, and our assumptions of
  the variable are correct.

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• Why do we need to hold the independent variables
  constant?
• Figures 1 and 2 may help clarify. Each circle may
  be thought of as representing the variance of the
  variable. The overlap in the two circles
  indicates the proportion of variance in each
  variable that is shared with the other. Since the
  proportion of variance in a variable that is
  shared or explained by another is defined as the
  coefficient of determination, or r2.

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Y
X2
X1
11
Y
X2
X1
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• In figure 1 the fact that X1 and X2 do not
  overlap means that they are not correlated, but
  each is correlated with Y. In figure 2 X1 and
  X2 are correlated. The area c is created by the
  correlation between X1 and X2 c represents the
  proportion of the variance in Y that is shared
  jointly with X1 and X2. To which variable should
  we assign this jointly shared variance? If we
  computed two separate bivariate regression
  equations, we would assign this variance to both
  independent variables, and it would clearly be
  incorrect to count it twice. Therefore bivariate
  regression would be wrong.