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Unformatted text preview: 5/18/2010 1 Logarithmic Regression Models • Next we will look at three specific nonlinear models that are used in practice quite often. All involve taking (natural) log transformations of the dependent variable and/or the regressor variable. ( 29 ( 29 ( 29 ( 29 1 1 1 1) (loglog model) : ln = ln 2) (linearlog model): = ln 3) (loglinear model): ln = i i i i i i i i i Y X u Y X u Y X u β β β β β β + + + + + + Logarithmic Regression Models  I • These three models represent different nonlinear relationships between X and Y . These can be either concave or convex depending on the specification used and the value of β 1 . • Instead of looking at this, let’s focus on exactly what the slope coefficients β 1 measure in each of the 3 models. These slope coefficients have specific interpretations that are important to know in order to correctly interpret results of these regressions. 5/18/2010 2 Logarithmic Regression Models  II • First recall that in our linear model: we can think of the slope coefficient β 1 satisfies the equation: • In words, this says that if β 1 = 5, a 1 unit change in X i (i.e. ∆ X i = 1) will result in a 5 unit change in Y i (i.e. ∆ Y i = 5). A 3 unit change in X i (i.e. ∆ X i = 3) will result in a 15 unit change in Y i (i.e. ∆ Y i = 15) 1 = i i i Y X u β β + + 1 = i i Y X β ∆ ∆ Logarithmic Regression Models  III • In our logarithmic regression models, the slope coefficients measure 3 different things: • 1) loglog: so the slope coefficient satifies ∆ ln( Y i )= β 1 ∆ ln( X i ) • 2) linearlog: so the slope coefficient satisfies ∆ Y i = β 1 ∆ ln( X i ) • 2) loglinear: so the slope coefficient satisfies ∆ ln( Y i )= β 1 ∆ X i ( 29 ( 29 1 ln = ln i i i Y X u β β + + ( 29 1 ln = i i i Y X u β β + + ( 29 1 = ln i i i Y X u β β + + 5/18/2010 3 Logarithmic Regression Models  IV • How can we interpret these relationships, i.e. what does it mean that in these three models, ∆ ln( Y i )= β 1 ∆ ln( X i ), ∆ Y i = β 1 ∆ ln( X i ), or ∆ ln( Y i )= β 1 ∆ X i ? • The ln( ) function is special because: “For small changes, the change in the natural log of a variable is approximately equal to the percentage change of that variable”, i.e. • Notation: “% ∆ in Z i ” = 0.30 means a 30% increase in Z i . ( 29 ln % in i i i i Z Z Z Z ∆ ∆ ≈ = ∆ Logarithmic Regression Models  V • Some examples to confirm this: ln(51) – ln(50) = 3.9318 – 3.9120 = 0.0198 u2248 2% change ln(102) – ln(100) = 4.6250 – 4.6052 = 0.0202 u2248 2% change ln(1010)ln(1000)=6.91771– 6.90776=0.00995 u2248 1% change ln(1.1) – ln(1) = 0.09531 – 0 = 0.0953 u2248 10% change 5/18/2010 4 Logarithmic Regression Models  VI • Given this result, consider our loglog model, i.e....
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This note was uploaded on 06/17/2010 for the course ECON 103 taught by Professor Sandrablack during the Spring '07 term at UCLA.
 Spring '07
 SandraBlack
 Econometrics

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