320_LectureNotes-Nonlinear Models

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

This preview shows pages 1–3. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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 .
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern