1. Nonlinear Models

2. Agenda Omitted Variables Dummy Variables Nonlinear Models Nonlinear in variables Polynomial Regressions Log Transformed Regressions Interactions in Regressions Nonlinear in parameters Estimation Issues

3. Omitted Variable Bias What is it? Leaving out (omitting) a variable that should have been in the model Particularly problematic if the variable left out is correlated with what was put in Why? Solutions: Don’t omit relevant variables!! Controlling for unobservable variables Experiments Instrumental Variable Regression EXCEL Example demonstrating OVB

4. Dummy Variables Why Dummy Variables Natural for Nominal Data Useful in describing nonlinearities Construction of Dummy Variables Use Logic Equations (similar to if-else) Interpret Dummy Variables as difference in means Let’s do this in EXCEL

5. Nonlinear Models What is a nonlinear model? Two types: Nonlinear in variables Where either the X, Y (or both) are transformed but the regression is still linear in the  ’s. Nonlinear in parameters Where the model is such that regression is no longer a linear function of the  ’s. Why does this matter?

6. Polynomial Regression This model allows the X’s to have higher order effects. Note: This is essentially a multivariate regression! Predicted Values: Effect of Changes in X

7. Log Transformed Models In these models the X’s, Ys or both are log (ln) transformed Three Cases These models are referred to as (a) Log-linear (b) Linear-log and (c) Log-Log models Note Again: These are essentially multivariate regressions!

8. Log Transformed Models The opposite of the ln function is the exp function, further The original models therefore are

9. Log Models: Prediction Note that if u is normal with mean 0 and variance  2 then Since the models which have ln(Y) as the dependent variable are going to predict exp(Y) we need to make some corrections

10. Log Models: Prediction Log linear model: Log-Log Model

11. Interpretation of Log Models Log-Linear Model: if X changes by 1 unit it changes Y by 100  % Linear-Log Model: if X changes by 1% then it changes Y by 0.01  units. Log-Log Model: if X changes by 1%then it changes Y by  % In other words  is the elasticity! Note that these differences make comparing the  s across models impossible but elasticities can be compared as can marginal effects ( ) of a change in X

12. Variable Interactions Interactions: Using the product of two (or more) independent variables in the regression Types: Dummy Interactions Dummy-Continuous Interactions Continuous Interactions Interpretation Examples… in EXCEL

13. Nonlinear in Parameters Beyond the scope of this course… but… Sometimes we cannot use simple regression software to estimate the  For example: In such cases we can use solver or a more sophisticated statistical software There are a number of such models for particular kinds of data (e.g. shares)