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TOPIC 4: NONLINEAR REGRESSION FUNCTIONS
Until now, the population regression was assumed to be linear. In other words, the slope of the population regression function was
constant, so the effect on Y of a unit change of X does not itself depend on the value of X. Therefore, in a nonlinear regression, the
slope of the population regression function depends on the value of one or more independent variables.
The effect on Y of a change in the independent variable(s) can be computed by evaluating the regression function at two values of the independent variables(s). PROCEDURE: INTERPRETATION OF THE COEFFICIENTS: In the multiple regression model, the regression coefficients had a natural interpretation. But this is not the case in a nonlinear model. To do the best interpretation we will need to graph and calculate the predicted effect on Y of changing one or more of the independent variables.
Laura Aparicio 52 ECONOMETRICS I There are two methods for modelling a nonlinear regression function: ① POLYNOMIALS A polynomial regression includes powers of X as regressors: A quadratic regression includes X and X2.
A cubic regression includes X, X2 and X3.
Which degree polynomial should I use? The answer balances a trade-off between flexibility and statistical precision. Increasing the degree r introduces more flexibility into the regression function and allows it to match more shapes.
Laura Aparicio 53 ECONOMETRICS I ② LOGARITHMS Small changes in logarithms can be interpreted as proportional or percentage changes in a variable. Regression involving logarithms are used to estimate proportional changes and elasticities.
USEFUL PROPERTIES OF LOGARITHMS Finally, the product of two variable is called and INTERACTION TERM. When interaction terms are included as regressors, they allow the regression slope of one variable to depend on the value of another variable.
1) Interactions between two binary variables. First compute the expected values of Y for each possible case described by the set of binary variables. Next compare these expected values. Each coefficient can then be expressed either as an expected value or as the difference between two or more expected values.
Laura Aparicio 54 ECONOMETRICS I 2) Interactions between a continuous and a binary variable 3) Interactions between two continuous variables Laura Aparicio 55 ...