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Unformatted text preview: 1 Simple Linear Regression Lecture XXVIII I. Overview A. 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. 554577. B. The purpose of regression analysis is to explore the relationship between two variables. 1. In this course, the relationship that we will be interested in can be expressed as: i i i x y where i y is a random variable and i x is a variable hypothesized to affect or drive i y . a) The coefficients and are the intercept and slope parameters, respectively. b) These parameters are assumed to be fixed, but unknown. c) The residual i is assumed to be an unobserved, random error. d) Under typical assumptions i E . e) Thus, the expected value of i y given i x then becomes: i i x y E 2. The goal of regression analysis is to estimate and and to say something about the significance of the relationship. 3. 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. 4. This relationship is a linear regression in that the relationship is linear in the parameters and . Abstracting for a moment, the traditional CobbDouglas production function can be written as: i i x y taking the natural log of both sides yields: i i x y ln ln ln AEB 6933 Mathematical Statistics for Food and Resource Economics Lecture XXVIII Professor Charles Moss Fall 2007 2 Noting that * ln , this relationship is linear in the estimated parameters and, thus, can be estimated using a simple linear regression. II. Simple Linear Regression A. The setup for simple linear regression is that we have a sample of n pairs of variables 1 1 , , , n n x y x y . Further, we want to summarize this relationship using by fitting a line through the data. B. Based on the sample data, we first describe the data as follows: 1. The sample means n i i n i i y n y x n x 1 1 1 1 2. The sums of squares: n i i i xy n i i yy n i i xx y y x x...
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 Fall '09
 CARRIKER

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