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lecture6 - ECON 103 Lecture 6 The linear regression...

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ECON 103, Lecture 6: The linear regression model (contd) Maria Casanova January 28th (version 0) Maria Casanova Lecture 6
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Requirements for this lecture: Chapter 4 of Stock and Watson Maria Casanova Lecture 6
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1. Introduction Suppose we are interested in estimating β 0 and β 1 in the following model: Y i = β 0 + β 1 X i + ε i We may estimate the unknown β 0 and β 1 by OLS: ˆ β 0 = ¯ Y - ˆ β 1 ¯ X ˆ β 1 = X i ( X i - ¯ X )( Y i - ¯ Y ) X i ( X i - ¯ X ) 2 Next we review the assumptions on the linear regression model and the sampling scheme under which OLS provides an appropriate estimator of β 0 and β 1 . Maria Casanova Lecture 6
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2. The least squares assumptions Ass1: The conditional distribution of ε i given X i has a mean of zero. E ( ε i | X i ) = 0 This is a statement about the underlying model. This assumption refers to the ”other factors” affecting Y i which are captured by ε i . It says that these other factors are unrelated to X i in the sense that, given a value of X i , the mean of their distribution is zero. Maria Casanova Lecture 6
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2. The least squares assumptions Figure: Conditional mean wage given fitness 0 1 2 3 4 5 6 7 8 9 10 0 500 1000 1500 Fitness Wage Population regression function1 β 0 + β 1 X 1 X 1 = (lowest) (highest) Maria Casanova Lecture 6
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2. The least squares assumptions Figure: Conditional mean wage given age 20 25 30 35 40 45 50 0 500 1000 1500 age wage population regression function X 2 = δ 0 + δ 1 X 2 δ 1 = 250 δ 0 = 500 Maria Casanova Lecture 6
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2. The least squares assumptions Figure: Assumption 1 holds for this linear model 0 1 2 3 4 5 6 7 8 9 10 0 500 1000 1500 fitness wage (highest) (lowest) X 1 = β 0 = 1000 β 1 = 0 Maria Casanova Lecture 6
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2. The least squares assumptions Figure: Assumption 1 holds for this linear model 0 1 2 3 4 5 6 7 8 9
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