Lecture 8

Lecture 8 - LECTURE 8: FUNCTION FORM AND DUMMY VARIABLES...

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LECTURE 8: FUNCTION FORM AND DUMMY VARIABLES FUNCTIONAL FORM OF REGRESSION MODELS Y’s, X’s don’t necessarily have to be linear to be a linear regression. Linear regression is linear in the parameters . 1. Linear Model Linear in Variables Y i = β 0 + β 1 X i + ε i β 1 gives the increase in Y for an increase in X 2. Log Linear Model Y i = A 1 β i X NOT linear in X. BUT, can transform by taking the ln of both sides: lnY i = lnA + β 1 lnX i β 0 = lnA lnY i = β 0 + β 1 lnX i We call this log-linear because it is linear in the log of the variables. IF the assumptions of the Classical Linear Regression Model are satisfied for the transformed model, we can use OLS. β 1 tell us the elasticity of Y with respect to X; i.e., the percentage change in Y for a given percentage change in X.
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Example lnY i = β 0 + β 1 lnX i Y i = quantity demanded X i = price β 1 = price elasticity of demand NOTE : Since β 1 is constant (does not vary with X), this is a constant elasticity model 100 * X X Δ 100 * Y Y Δ X ln d Y ln d β 1 Conduct hypothesis tests the same way (on transformed model) 3. Semilog Model In this case, only one variable appears in log form Example : How do we measure a growth rate (r = compound rate of growth, t = time trend)? Y t = Y 0 (1 + r) t lnY t = t ln(1 + r) + lnY 0 β 0 = lnY 0 β 1 = ln(1 + r) lnY t = β 0 + β 1 t Example : Wage Equations lnY i = β 0 + β 1 X i + ε i
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Y = income X = education β 1 = X Δ Y Y Δ dX Y ln d β 1 = percentage change in income for a one year increase in education. 4. Polynomial Regression Model If you don’t think the relationship between X and Y is linear, can add non-linear terms Example Y i = earnings X i = experience To date, we have assumed: Y i = β 0 + β 1 X i + ε i BUT, what if Y i = β 0 + β 1 X i + β 2 X i 2 + ε i In this case, there is NOT A CONSTANT SLOPE (slope depends on X): i 2 1 i i X β 2 β dX dY
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How do we decide which functional form to use? 1. Economic Theory If Economic theory doesn’t tell you, you can: 2. Plot the data (to see if the relationship looks linear, etc). This can be difficult in the context of multiple regression 3. Choose the model with the higher Adjusted R 2 (IF the models have the same dependent variable) 4. Intuition
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Dummy Variables = Indicator Variables = Binary Variables Indicate presence, or absence, of a “quality” or attribute Often used for qualitative variables To quantify: Create artificial variables that take on a value of 0 or 1 Use D to denote dummy variables: Y i = β 0 + β 1 D i + ε i Where Y = starting salary D i = { otherwise 0 graduate college a is i individual if 1 What is mean starting salary of non-college graduates? E(Y
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This note was uploaded on 03/20/2009 for the course ECON 103 taught by Professor Sandrablack during the Winter '07 term at UCLA.

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Lecture 8 - LECTURE 8: FUNCTION FORM AND DUMMY VARIABLES...

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