320_LectureNotes-Nonlinear Models

320_LectureNotes-Nonlinear Models - University of Oregon...

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University of Oregon Rosie Mueller Department of Economics Fall 2016 Lecture: Nonlinear Models and Transformations of Variables (Chapter 4 in Dougherty) EC 320: Econometrics I. Linear Regression Models To this point we have focused on linear regression models . Consider the model: Y i = β 0 + β 1 X 1 i + β 2 X 2 i + β 3 X 3 i + u i This model is linear because every term is a parameter multiplied by an independent variable. One implication is that the effect of a one-unit change in X ki , holding everything else constant, is given by β k , which is a constant. In other words, the effect of a one-unit change in X ki does not depend on the value of X ki . In some situations, this might be an unsatisfactory assumption. Suppose we are modeling the effect of income on the demand for a particular good or service. Such relationships are called Engel Curves . We would not expect the effect of a one dollar increase in income to be the same at high income levels than at low income levels. Similarly, the effect of an extra year of schooling on earnings likely has a larger effect going from 15 to 16 years (finishing a college degree) than going from 8 to 9 years. In order to model such phenomena, we work with nonlinear regression models . Again, consider the model: Y i = β 0 + β 1 X 1 i + β 2 X 2 i + β 3 X 3 i + u i This model is linear in variables . This means that the explanatory variables enter as a weighted sum, where the weights are given by the parameters. This model is linear in parameters . This means that the parameters enter as a weighted sum, where the weights are given by the explanatory variables. Nonlinear models can depart from the linear regression model in both of these dimensions. We will first consider models that are nonlinear in variables . For example consider a quadratic regression model: Y i = β 0 + β 1 X i + β 2 X 2 i + u i This model implies the effects of changes in X i are larger (or smaller) at higher values of X i . 1
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Nonlinearity in variables is easy to handle using the techniques we have already developed. We can define: Z 1 i = X i and Z 2 i = X 2 i , and write the same model as Y i = β 0 + β 1 Z 1 i + β 2 Z 2 i + u i This model is linear in both parameters and variables and can be estimated using OLS. Interpretation of Quadratic Models In our standard linear model: Y i = β 0 + β 1 X 1 i + β 2 X 2 i + u i we interpret β 1 , as a one-unit increase in X 1 i results in a β 1 unit increase on Y i .
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