Slides28-2010 - Simple Linear Regression: Lecture XXVIII...

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Simple Linear Regression: Lecture XXVIII Charles B. Moss November 14, 2010 Charles B. Moss () Simple Linear Regression November 14, 2010 1 / 28
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1 Overview 2 Simple Linear Regression A Mathematical Solution Working’s Law 3 Best Linear Unbiased Estimators Charles B. Moss () Simple Linear Regression November 14, 2010 2 / 28
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Overview The purpose of regression analysis is to explore the relationship between two variables. I In this course, the relationship that we will be interested in can be expressed as y i = α + β x i + ± i (1) Charles B. Moss () Simple Linear Regression November 14, 2010 3 / 28
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Continued I Where y i is a random variable and x i is a variable hypothesized to affect or drive y i . F The coefficients α and β are the intercept and slope parameters, respectively. F These parameters are assumed to be fixed, but unknown. F The residual ± i is assumed to be an unobserved, random error. F Under typical assumptions E [ ± i ] = 0. F Thus, the expected value of y i given x i then becomes E [ y i ] = α + β x i (2) Charles B. Moss () Simple Linear Regression November 14, 2010 4 / 28
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Continued I The goal of regression analysis is to estimate α and β and to say something about the significance of the relationship. I 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. I 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 y i = α x β i (3) taking the natural logarithm of both sides yields ln ( y i ) = ln ( α ) + β ln ( x i ) (4) Charles B. Moss () Simple Linear Regression November 14, 2010 5 / 28
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Continued I Noting that ln ( α ) = α * , this relationship is linear in the estimated parameters and, thus, can be estimated using a simple linear regression. Charles B. Moss () Simple Linear Regression November 14, 2010 6 / 28
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Simple Linear Regression The setup for simple linear regression is that we have a sample of n pairs of variables ( x 1 , y 1 ) , ··· ( x n , y n ) . Further, we want to summarize this relationship using by fitting a line through the data. Based on the sample data, we first describe the data as follows: I The sample means ¯ x = 1 n n X i =1 x i , ¯ y = 1 n n X i =1 y i . (5) I The sums of squares S xx = n X i =1 ( x i - ¯ x ) 2 , S yy = n X i =1 ( y i - ¯ y ) 2 s xy = n X i =1 ( x i - ¯ x ) ( y i - ¯ y ) (6) Charles B. Moss () Simple Linear Regression November 14, 2010 7 / 28
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Continued 3. The most common estimators given this formulation are then given by b = S
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Slides28-2010 - Simple Linear Regression: Lecture XXVIII...

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