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Lecture28-2010

# Lecture28-2010 - Simple Linear Regression Lecture XXVIII...

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Simple Linear Regression: Lecture XXVIII Charles B. Moss November 30, 2010 I. Overview A. 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 y i = α + βx i + i (1) where y i is a random variable and x i is a variable hypothesized to affect or drive y i . a) The coeﬃcients α and β are the intercept and slope pa- rameters, 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 E [ i ] = 0. e) Thus, the expected value of y i given x i then becomes E [ y i ] = α + βx i (2) 2. The goal of regression analysis is to estimate α and β 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. 1

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AEB 6571 Econometric Methods I Professor Charles B. Moss Lecture XXVIII Fall 2010 4. This relationship is a linear regression in that the relationship is linear in the parameters α and β . Abstracting for a mo- ment, the traditional Cobb-Douglas production function can be written as y i = αx β i (3) taking the natural logarithm of both sides yields ln ( y i ) = ln ( α ) + β ln ( x i ) (4) Noting that ln ( α ) = α , 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 ( x 1 , y 1 ) , · · · ( x n , y n ) . 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 ¯ x = 1 n n i =1 x i , ¯ y = 1 n n i =1 y i . (5) 2. The sums of squares S xx = n i =1 ( x i ¯ x ) 2 , S yy = n i =1 ( y i ¯ y ) 2 s xy = n i =1 ( x i ¯ x ) ( y i ¯ y ) (6) 3. The most common estimators given this formulation are then given by b = S xy S xx , α = ¯ y b ¯ x (7) 2
AEB 6571 Econometric Methods I Professor Charles B. Moss Lecture XXVIII Fall 2010 C. Least Squares: A Mathematical Solution 1. Following on our theme in the discussion of linear projections.

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Lecture28-2010 - Simple Linear Regression Lecture XXVIII...

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