week3 - Econ 508 Correlation and Regression Juan Fung MSPE...

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Econ 508: Correlation and Regression Juan Fung MSPE February 11, 2011 Juan Fung (MSPE) Econ 508: Correlation and Regression February 11, 2011 1 / 17
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Introduction: the purpose of correlation and regression analysis Suppose you observe n realizations of two random variables, X and Y . Example For example, let X = parents’ income (in $1000), Y = student GPA. X Y X Y 21 4.0 12 3.0 15 3.0 18 3.5 15 3.5 6 2.5 9 2.0 12 2.5 Using this data, what questions might you ask regarding X and Y ? 1. Is there a relationship between X and Y ? 2. If so, what is the form of the relationship? For example, do you expect that an increase in parent income would be associated with a proportional increase in student GPA? Juan Fung (MSPE) Econ 508: Correlation and Regression February 11, 2011 2 / 17
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Correlation Correlation analysis Recall the correlation coefficient, ρ = Cov [ X , Y ] p Var [ X ] Var [ Y ] σ XY σ X σ Y If X and Y are independent, then ρ = 0 (since Cov [ X , Y ] σ XY = 0). Equivalently, if ρ 6 = 0 , then X and Y are not independent . Even more, they share a linear relationship . The direction and strength of this relationship is determined by the sign and magnitude of ρ . The goal of correlation analysis is to identify whether a linear relationship exists, and if it does, its direction and strength. Juan Fung (MSPE) Econ 508: Correlation and Regression February 11, 2011 3 / 17
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Correlation Recall example of GPA and parent income. An excel scatter plot suggests positive correlation: But we can’t say anything precise about their relationship. 1. First, ρ is unobservable: we need population moments. 2. Even if we knew they shared a strong positive relationship, ρ tells us nothing about how one variable (say X ) influences another (say Y ). Juan Fung (MSPE) Econ 508: Correlation and Regression February 11, 2011 4 / 17
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Correlation ρ is unobservable Luckily, the MLE for ρ (try deriving) has an intuitive formulation, r = S XY S XX S YY , where S XX = X i ( X i - X ) 2 = ( n - 1) d Var ( X ) S YY = X i ( Y i - Y ) 2 = ( n - 1) d Var ( Y ) S XY = X i ( X i - X )( Y i - Y ) = ( n - 1) d Cov ( X , Y ) .
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