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lecture-11 - ECON 103 Lecture 11 Functional Form and Dummy...

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ECON 103, Lecture 11: Functional Form and Dummy Variables Maria Casanova February 18th (version 0) Maria Casanova Lecture 11
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Requirements for this lecture: Chapter 8 of Stock and Watson Maria Casanova Lecture 11
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1. Functional Form of Regression Models In this course we cover linear regressions . By linear we mean that they are linear in parameters They may or may not be linear in the X ’s The Y variable does not need to be linear either. Maria Casanova Lecture 11
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1.1 Linear Model Ex1: Linear Model It is linear in variables Y = β 0 + β 1 X + ε β 1 gives the increase in Y given a unit increase in X Maria Casanova Lecture 11
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1.2 Log-Linear Model Ex2: Log-Linear Model It is not linear in X Y = AX β 1 However, we can linearize it by taking the natural ln on both sides: ln Y = ln ( AX β 1 ) = ln A + β 1 ln X ln Y = β 0 + β 1 ln X If the least squares assumptions are satisfied, we can estimate β 0 and β 1 by OLS from a regression like: ln Y = β 0 + β 1 ln X + ε Maria Casanova Lecture 11
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1.2 Log-Linear Model In the log-linear model β 1 measures the percentage change in Y for a given percentage change in X (that is, the the elasticity of Y with respect to X ). For example, in the following model β 1 is the price-elasticity of demand : ln Y = β 0 + β 1 ln X + ε, where: Y = quantity demanded X = price β 1 = d ln Y d ln X = dY / Y dX / X = ξ p Δ Y Y Δ X X = % increase in Y % increase in X ! In this model, when we conduct hypothesis tests about β 1 we are testing hypothesis about the price-elasticity of demand. Maria Casanova Lecture 11
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1.3 Semilog Model Ex3: Semilog Model Only one variable appears in log form. ln Y = β 0 + β 1 X + ε In this model β 1 measures the percentage change in Y given a unit change in X : β 1 = d ln Y dX = dY / Y dX Δ Y Y Δ X = % increase in Y Increase in X ! When are we interested in computing the percentage change in one variable (e.g. Y ) per unit increase of other variable (e.g. X )? Maria Casanova Lecture 11
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1.3 Semilog Model 1 Example: wage equation ln wage = β 0 + β 1 educ + ε, where educ is measured in years. In this case, β 1 measures the percentage change in income for a one year increase in education. Maria Casanova Lecture 11
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1.3 Semilog Model 2 Example: demand equation ln Q = β 0 + β 1 P + ε We know that β 1 = d ln Q dP = dQ / Q dP Remember that the price elasticity of demand was: ξ p = dQ / Q dP / P In this case the elasticity of demand is: ξ p = β 1 P that is, the elasticity varies for different price levels.
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