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Slides_Econ140A_Regression_Continued

# Slides_Econ140A_Regression_Continued - Regression continued...

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Regression continued… Copyright Dick Startz 1

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Goals Prove OLS is unbiased Derive variance Hypothesis tests t- test versus z More on reading what a regression really tells us Copyright Dick Startz 2
Regression as a theoretical model Assumptions 1. ? ? = ?? ? + 𝜀 ? , true model of DGP 2. 𝑋 nonrandom (or exogenous) 3. ? 𝜀 ? = 0 ∀? 4. ? 𝜀 ? 𝜀 ? = 𝜎 2 , ? = ? 0, ? ≠ ? 5. (sometimes) 𝜀~????𝑎? Copyright Dick Startz 3

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Substitute in the DGP ? = ∑?? ∑? 2 = ?? + 𝜀 ? ∑? 2 = ∑?? 2 + ∑𝜀? ∑? 2 ? = ? + ∑𝜀? ∑? 2 ? = ? + 𝜀 1 ? 1 ∑? 2 + 𝜀 2 ? 2 ∑? 2 + ⋯ + 𝜀 𝑛 ? 𝑛 ∑? 2 Copyright Dick Startz 4
Prove OLS is unbiased ? = ? + 𝜀 1 ? 1 ∑? 2 + 𝜀 2 ? 2 ∑? 2 + ⋯ + 𝜀 𝑛 ? 𝑛 ∑? 2 ? ? = ? + ? 𝜀 1 ? 1 ∑? 2 + ? 𝜀 2 ? 2 ∑? 2 + ⋯ + ? 𝜀 𝑛 ? 𝑛 ∑? 2 ? ? = ? + 0 + 0 … + 0 = ? Copyright Dick Startz 5

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var ? for ? = 2 ? = ? + 𝜀 1 ? 1 ? 1 2 + ? 2 2 + 𝜀 2 ? 2 ? 1 2 + ? 2 2 var ? = ? ? − ? ? 2 = ? 𝜀 1 ? 1 ? 1 2 + ? 2 2 + 𝜀 2 ? 2 ? 1 2 + ? 2 2 2 = ? 𝜀 1 2 ? 1 2 ? 1 2 + ? 2 2 2 + 𝜀 2 2 ? 2 2 ? 1 2 + ? 2 2 2 + 2𝜀 1 𝜀 2 ? 1 ? 2 ? 1 2 + ? 2 2 2 Copyright Dick Startz 6
var ? for ? = 2 continued ? 𝜀 1 2 ?

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