ch02 - The Simple Regression Model y = 0 1 x u Economics 20...

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Economics 20 - Prof. Anderson 1 The Simple Regression Model y = β 0 + β 1 x + u
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Economics 20 - Prof. Anderson 2 Some Terminology In the simple linear regression model, where y = β 0 + β 1 x + u , we typically refer to y as the Dependent Variable, or Left-Hand Side Variable, or Explained Variable, or Regressand
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Economics 20 - Prof. Anderson 3 Some Terminology, cont. In the simple linear regression of y on x, we typically refer to x as the Independent Variable, or Right-Hand Side Variable, or Explanatory Variable, or Regressor, or Covariate, or Control Variables
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Economics 20 - Prof. Anderson 4 A Simple Assumption The average value of u , the error term, in the population is 0. That is, E( u ) = 0 This is not a restrictive assumption, since we can always use β 0 to normalize E( u ) to 0
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Economics 20 - Prof. Anderson 5 Zero Conditional Mean We need to make a crucial assumption about how u and x are related We want it to be the case that knowing something about x does not give us any information about u, so that they are completely unrelated. That is, that E( u | x ) = E( u ) = 0, which implies E( y | x ) = β 0 + β 1 x
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Economics 20 - Prof. Anderson 6 . . x 1 x 2 E( y|x ) as a linear function of x , where for any x the distribution of y is centered about E( y|x ) E( y | x ) = β 0 + β 1 x y f( y )
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Economics 20 - Prof. Anderson 7 Ordinary Least Squares Basic idea of regression is to estimate the population parameters from a sample Let {( x i ,y i ): i =1, …, n } denote a random sample of size n from the population For each observation in this sample, it will be the case that y i = β 0 + β 1 x i + u i
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Economics 20 - Prof. Anderson 8 . . . . y 4 y 1 y 2 y 3 x 1 x 2 x 3 x 4 } } { { u 1 u 2 u 3 u 4 x y Population regression line, sample data points and the associated error terms E( y|x ) = β 0 + β 1 x
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Economics 20 - Prof. Anderson 9 Deriving OLS Estimates To derive the OLS estimates we need to realize that our main assumption of E( u | x ) = E( u ) = 0 also implies that Cov( x,u ) = E( xu ) = 0 Why? Remember from basic probability that Cov(X,Y) = E(XY) – E(X)E(Y)
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Economics 20 - Prof. Anderson 10 Deriving OLS continued We can write our 2 restrictions just in terms of x , y , β 0 and β 1 , since u = y – β 0 β 1 x E( y – β 0 β 1 x ) = 0 E[ x ( y – β 0 β 1 x )] = 0 These are called moment restrictions
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Economics 20 - Prof. Anderson 11 Deriving OLS using M.O.M. The method of moments approach to estimation implies imposing the population moment restrictions on the sample moments What does this mean? Recall that for E(X), the mean of a population distribution, a sample estimator of E(X) is simply the arithmetic mean of the sample
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Economics 20 - Prof. Anderson 12 More Derivation of OLS
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