simplereg - Economics 20 - Prof. Anderson 1 The Simple...

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Unformatted text preview: Economics 20 - Prof. Anderson 1 The Simple Regression Model y = + 1 x + u Economics 20 - Prof. Anderson 2 Some Terminology In the simple linear regression model, where y = + 1 x + u , we typically refer to y as the Dependent Variable, or Left-Hand Side Variable, or Explained Variable, or Regressand 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 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 to normalize E( u ) to 0 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 ) = + 1 x 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 ) = + 1 x y f( y ) 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 = + 1 x i + u i 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 ) = 0 + 1 x 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) Economics 20 - Prof. Anderson 10 Deriving OLS continued We can write our 2 restrictions just in terms of x , y , and 1 , since u = y 1 x E( y 1 x ) = 0 E[ x ( y 1 x )] = 0 These are called moment restrictions Economics 20 - Prof. Anderson 11 Deriving OLS using M.O.M....
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simplereg - Economics 20 - Prof. Anderson 1 The Simple...

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