1. LINEARITY AND NONLINEARITY 1 This sequence introduces the topic of fitting nonlinear regression models. First we need a definition of linearity. Linear in variables and parameters:

2. The model shown above is linear in two senses. The right side is linear in variables because the variables are included exactly as defined, rather than as functions. 2 LINEARITY AND NONLINEARITY Linear in variables and parameters:

3. It is also linear in parameters since a different parameter appears as a multiplicative factor in each term. 3 LINEARITY AND NONLINEARITY Linear in variables and parameters:

4. The second model above is linear in parameters, but nonlinear in variables. 4 LINEARITY AND NONLINEARITY Linear in parameters, nonlinear in variables: Linear in variables and parameters:

5. Such models present no problem at all. Define new variables as shown. 5 LINEARITY AND NONLINEARITY Linear in parameters, nonlinear in variables: Linear in variables and parameters:

6. With these cosmetic transformations, we have made the model linear in both variables and parameters. 6 LINEARITY AND NONLINEARITY Linear in parameters, nonlinear in variables: Linear in variables and parameters:

7. Nonlinear in parameters: This model is nonlinear in parameters since the coefficient of X 4 is the product of the coefficients of X 2 and X 3. As we will see, some models which are nonlinear in parameters can be linearized by appropriate transformations, but this is not one of those. 7 LINEARITY AND NONLINEARITY Linear in parameters, nonlinear in variables: Linear in variables and parameters:

8. We will begin with an example of a simple model that can be linearized by a cosmetic transformation. The table reproduces the data in Exercise 1.4 on average annual rates of growth of employment and GDP for 25 OECD countries. 8 LINEARITY AND NONLINEARITY Average annual percentage growth rates Employment GDP Employment GDP Australia1.683.04Korea2.577.73 Austria0.652.55Luxembourg3.025.64 Belgium0.342.16Netherlands1.882.86 Canada1.172.03New Zealand0.912.01 Denmark0.022.02Norway0.362.98 Finland–1.061.78Portugal0.332.79 France0.282.08Spain0.892.60 Germany0.082.71Sweden–0.941.17 Greece0.872.08Switzerland0.791.15 Iceland–0.131.54Turkey2.024.18 Ireland2.166.40United Kingdom0.661.97 Italy–0.301.68United States1.532.46 Japan1.062.81

9. A plot of the data reveals that the relationship is clearly nonlinear. We will consider various nonlinear specifications for the relationship in the course of this chapter, starting with the hyperbolic model shown. 9 LINEARITY AND NONLINEARITY

10. This is nonlinear in g, but if we define z = 1/g, we can rewrite the model so that it is linear in variables as well as parameters. 10 LINEARITY AND NONLINEARITY

11. Here is the data table a second time, showing the values of z computed from those of g. There is no need in practice to perform the calculations oneself. Regression applications always have a facility for generating new variables as functions of existing ones. 11 LINEARITY AND NONLINEARITY Average annual percentage growth rates e g z e g z Australia1.683.040.3289Korea2.577.730.1294 Austria0.652.550.3922Luxembourg3.025.640.1773 Belgium0.342.160.4630Netherlands1.882.860.3497 Canada1.172.030.4926New Zealand0.912.010.4975 Denmark0.022.020.4950Norway0.362.980.3356 Finland–1.061.780.5618Portugal0.332.790.3584 France0.282.080.4808Spain0.892.600.3846 Germany0.082.710.3690Sweden–0.941.170.8547 Greece0.872.080.4808Switzerland0.791.150.8696 Iceland–0.131.540.6494Turkey2.024.180.2392 Ireland2.166.400.1563United Kingdom0.661.970.5076 Italy–0.301.680.5952United States1.532.460.4065 Japan1.062.810.3559

12. Here is the output for a regression of e on z. 12 LINEARITY AND NONLINEARITY. gen z = 1/g. reg e z Source | SS df MS Number of obs = 25 -------------+------------------------------ F( 1, 23) = 26.06 Model | 13.1203665 1 13.1203665 Prob > F = 0.0000 Residual | 11.5816089 23.503548214 R-squared = 0.5311 -------------+------------------------------ Adj R-squared = 0.5108 Total | 24.7019754 24 1.02924898 Root MSE =.70961 ------------------------------------------------------------------------------ e | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- z | -4.050817.793579 -5.10 0.000 -5.69246 -2.409174 _cons | 2.604753.3748822 6.95 0.000 1.82925 3.380256 ------------------------------------------------------------------------------

13. The figure shows the transformed data and the regression line for the regression of e on z. 13 LINEARITY AND NONLINEARITY ------------------------ e | Coef. -----------+------------ z | -4.050817 _cons | 2.604753 ------------------------

14. Substituting 1/g for z, we obtain the nonlinear relationship between e and g. The figure shows this relationship plotted in the original diagram. The linear regression of e on g reported in Exercise 1.4 is also shown, for comparison. 14 LINEARITY AND NONLINEARITY

15. 15 LINEARITY AND NONLINEARITY In this case, it was easy to see that the relationship between e and g was nonlinear. In the case of multiple regression analysis, nonlinearity might be detected using the graphical technique described in a previous slideshow.

16. Copyright Christopher Dougherty 2012. These slideshows may be downloaded by anyone, anywhere for personal use. Subject to respect for copyright and, where appropriate, attribution, they may be used as a resource for teaching an econometrics course. There is no need to refer to the author. The content of this slideshow comes from Section 4.1 of C. Dougherty, Introduction to Econometrics, fourth edition 2011, Oxford University Press. Additional (free) resources for both students and instructors may be downloaded from the OUP Online Resource Centre http://www.oup.com/uk/orc/bin/9780199567089/http://www.oup.com/uk/orc/bin/9780199567089/. Individuals studying econometrics on their own who feel that they might benefit from participation in a formal course should consider the London School of Economics summer school course EC212 Introduction to Econometrics http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx or the University of London International Programmes distance learning course EC2020 Elements of Econometrics www.londoninternational.ac.uk/lsewww.londoninternational.ac.uk/lse. 2012.11.03