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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. 554-577. 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 Cobb-Douglas 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