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Lecture 28-2007 - Lecture XXVIII Most of the material for...

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Lecture XXVIII
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Most of the material for this lecture is from George Casella and Roger L. Berger Statistical Inference (Belmont, California: Duxbury Press, 1990) Chapter 12, pp. 554-577.
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The purpose of regression analysis is to explore the relationship between two variables. In this course, the relationship that we will be interested in can be expressed as: where y i is a random variable and x i is a variable hypothesized to affect or drive y i . The coefficients α and β are the intercept and slope parameters, respectively. i i i x y ε β α + + =
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These parameters are assumed to be fixed, but unknown. The residual ε i is assumed to be an unobserved, random error. Under typical assumptions E[ ε i ]=0. Thus, the expected value of y i given x i then becomes: [ ] i i x y E β α + =
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The goal of regression analysis is to estimate α and β and to say something about the significance of the relationship. From a terminology standpoint, y is typically referred to as the dependent variable and x is referred to as the independent variable. Cassella and Berger prefer the terminology of y as the response variable and x as the predictor variable.
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This relationship is a linear regression in that the relationship is linear in the parameters α and β . Abstracting for a moment, the traditional Cobb-Douglas production function can be written as: taking the natural log of both sides yields: β α i i x y = ( 29 ( 29 ( 29 i i x y ln ln ln β α + =
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The setup for simple linear regression is that we have a sample of n pairs of variables ( x i , y i ),…( x n , y n ). Further, we want to summarize this relationship using by fitting a line through the data.
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