Section 6

# Section 6 - Section 6 Econ 140 GSI Edson Severnini 1...

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Section 6 - Econ 140 GSI: Edson Severnini 1 Nonlinear regression Y i = f ( β 0 1 X 1 i 2 X 2 i ,...,β k X ki ) + ε i , i = 1 ,...,N Eﬀect on Y i of a change in X 1 i : Δ b Y i = b f ( X 1 i + Δ X 1 i ,X 2 i ,...,X ki ) - b f ( X 1 i ,X 2 i ,...,X ki ) 1.1 Polynomials Y i = β 0 + β 1 X i + β 2 X 2 i + ... + β r X r i + ε i Test that population regression is linear H 0 : β 2 = β 3 = ... = β r = 0 vs H 1 : at least one β j 6 = 0 , j = 2 ,...,r Test F with r - 1 restrictions Test for the degree of the polynomial: sequential test t 1.2 Logarithms Logarithms can be used to transform the dependent variable Y , an independent variable X , or both (but they must be positive). Below, we are going to discuss the interpretation of the regression coeﬃcient β 1 for these three possible cases. In each case, β 1 can be estimated by applying OLS after taking the logarithm of the dependent and/or independent variable. 1. Y i = β 0 + β 1 ln ( X i ) + ε i 0 . 01 β 1 = 1 100 β 1 = ∂Y ∂lnX = ∂Y ∂X ∂X ∂lnX = ∂Y ∂X 1 ∂lnX ∂X = ∂Y ∂X 1 1 X = ∂Y ∂X X = Δ Y Δ X X = Δ Y Δ X X 100 1% change in X change in Y of 0 . 01 β 1 2. ln ( Y i ) = β 0 + β 1 X i + ε i 100 β 1 = ∂lnY ∂X = ∂lnY ∂Y ∂Y ∂X = 1 Y ∂Y ∂X = 1 Y Δ Y Δ X = Δ Y Y 100 Δ X change in X by 1 unit 100 β 1 % change in Y 3. ln ( Y i ) = β 0 + β 1 ln ( X i ) + ε i β 1 = ∂lnY ∂lnX = ∂lnY ∂Y ∂Y ∂X ∂X ∂lnX = 1 Y ∂Y ∂X 1 ∂lnX ∂X = 1 Y ∂Y ∂X 1 1 X = X Y ∂Y ∂X = X Y Δ Y Δ X = Δ Y Y 100 Δ X X 100 1% change in X β 1 % change in Y ( β 1 : elasticity of Y with respect to X ) 1

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1.3 Interactions between independent variables o Between dummies: - Yr : 0o + hDu * 7zDzt -t ls(Du. x D2;) * ei
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## This note was uploaded on 08/06/2009 for the course ECON 140 taught by Professor Duncan during the Summer '08 term at Berkeley.

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Section 6 - Section 6 Econ 140 GSI Edson Severnini 1...

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