9252012 p kolm 7 the two variable linear regression

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Unformatted text preview: 7 The Two-Variable Linear Regression Model: Assumptions Model: y = b0 + b1x + u Important assumptions: 1. Zero mean of u: The average value of u in the population is 0. That is, E (u ) = 0 o This is not a restrictive assumption since we can always use b0 to normalize E (u ) to 0 2. Zero conditional mean of u: o We need to make a crucial assumption about how u and x are related o 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 ) = 0 , which implies E (y | x ) = b0 + b1x VER. 9/25/2012. © P. KOLM 8 The Two-Variable Linear Regression Model: Figure E (y | x ) = b0 + b1x (linear function of x ), where for any x the distribution of y is centered about the regression line represented by E (y | x ) VER. 9/25/2012. © P. KOLM 9 Example (1/2) A simple savings equation: savings = b0 + b1income + u Remarks: b0 represents the savings when income = 0 b1 represents the change in wage for a unit change in income when everything else (u) is held constant How can we think about u? o u is the unexplained part o Includes other variables that explain savings such as age, wealth, savings ethic, etc. VER. 9/25/2012. © P. KOLM 10 Example (2/2) VER. 9/25/2012. © P. KOLM 11 How Do We Fit This Line? (1/2) Basic idea of linear regression is to estimate the population parameters from a sample using ordinary least squares Let {(x i , yi ) : 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 yi = b0 + b1x i + ui We call this the data generation process (DGP) VER. 9/25/2012. © P. KOLM 12 How Do We Fit This Line? (2/2) VER. 9/25/2012. © P. KOLM 13 Deriving OLS Estimates Recall: y = b0 + b1x + u with the restrictions: 1. E (u ) = 0 2. E (u | x ) = 0 o This implies that u and x are independent, and therefore Cov(x , u ) = E (xu ) = 0 Writing there restrictions in terms of x , y, b0 and b1 , we have 1. E [y - b0 - b1x ] = 0 2. E éêëx (y - b0...
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