10 Functional Form Specification

# 10 Functional Form Specification - Economics 140A...

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Functional Form Specification regressors, the functional form, and the stochastic error. Today we focus on is Y t related to the regressors? Do we expect a graph of the two to be linear or curved? Does the impact of a regressor peak at one value and then decline? As in selecting regressors, we look to economic theory to guide our choice of functional form. The baseline regression model is linear in both the coe¢ cients and the regres- sors Y t = 1 + 2 X t; 2 + + K X t;K + U t : Each coe¢ cient is given by k = @Y t @X t;k ; that is, the change in Y t brought about by a small change in X t;k holding all other regressors constant. If k = 4 , then an increase of 1 unit in X t;k leads to a 4 unit increase in Y t . For such a model, the partial derivative (or slope) is constant. A scale invariant measure is obtained directly from the slope coe¢ cient by standardizing all variables, that is Y t ± Y S Y = 1 + 2 X t; 2 ± X 2 S X 2 + + K X t;K ± X K S X K + U t : Logarithmic Forms Often economic theory tells us that the elasticity of Y t with respect to X t;k is constant. Because this elasticity is @Y t Y t @X t;k X t;k = k X t;k Y t ; the elasticity depends on the values of the variables in the baseline model. To obtain a constant elasticity, we work with the logarithms of the variables

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10 Functional Form Specification - Economics 140A...

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