linearity and nonlinearity 1 this sequence introduces the topic of fitting nonlinear regression...

LINEARITY AND NONLINEARITY

1

This sequence introduces the topic of fitting nonlinear regression models. First we need a definition of linearity.

uXXXY 4433221

Linear in variables and parameters:

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.

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uXXXY 4433221


It is also linear in parameters since a different parameter appears as a multiplicative factor in each term.

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uXXXY 4433221


The second model above is linear in parameters, but nonlinear in variables.

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uXXXY 4433221

uXXXY 44332221 log

Linear in parameters, nonlinear in variables:


Such models present no problem at all. Define new variables as shown.

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uXXXY 4433221

uXXXY 44332221 log

4433222 log,, XZXZXZ



With these cosmetic transformations, we have made the model linear in both variables and parameters.

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uXXXY 4433221

uXXXY 44332221 log


uZZZY 4433221



Nonlinear in parameters:

uXXXY 4433221

uXXXY 44332221 log


uZZZY 4433221

uXXXY 43233221

This model is nonlinear in parameters since the coefficient of X4 is the product of the coefficients of X2 and X3. As we will see, some models which are nonlinear in parameters can be linearized by appropriate transformations, but this is not one of those.

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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.

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Average annual percentage growth rates

Employment GDP Employment GDP

Australia 1.68 3.04 Korea 2.57 7.73Austria 0.65 2.55 Luxembourg 3.02 5.64Belgium 0.34 2.16 Netherlands 1.88 2.86Canada 1.17 2.03 New Zealand 0.91 2.01Denmark 0.02 2.02 Norway 0.36 2.98Finland –1.06 1.78 Portugal 0.33 2.79France 0.28 2.08 Spain 0.89 2.60Germany 0.08 2.71 Sweden –0.94 1.17Greece 0.87 2.08 Switzerland 0.79 1.15Iceland –0.13 1.54 Turkey 2.02 4.18Ireland 2.16 6.40 United Kingdom 0.66 1.97Italy –0.30 1.68 United States 1.53 2.46Japan 1.06 2.81

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.

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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.

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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.

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Average annual percentage growth rates

e g z e g z

Australia 1.68 3.04 0.3289 Korea 2.57 7.73 0.1294Austria 0.65 2.55 0.3922 Luxembourg 3.02 5.64 0.1773Belgium 0.34 2.16 0.4630 Netherlands 1.88 2.86 0.3497Canada 1.17 2.03 0.4926 New Zealand 0.91 2.01 0.4975Denmark 0.02 2.02 0.4950 Norway 0.36 2.98 0.3356Finland –1.06 1.78 0.5618 Portugal 0.33 2.79 0.3584France 0.28 2.08 0.4808 Spain 0.89 2.60 0.3846Germany 0.08 2.71 0.3690 Sweden –0.94 1.17 0.8547Greece 0.87 2.08 0.4808 Switzerland 0.79 1.15 0.8696Iceland –0.13 1.54 0.6494 Turkey 2.02 4.18 0.2392Ireland 2.16 6.40 0.1563 United Kingdom0.66 1.97 0.5076Italy –0.30 1.68 0.5952 United States 1.53 2.46 0.4065Japan 1.06 2.81 0.3559

Here is the output for a regression of e on z.

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. 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------------------------------------------------------------------------------

The figure shows the transformed data and the regression line for the regression of e on z.

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------------------------ e | Coef. -----------+------------ z | -4.050817 _cons | 2.604753 ------------------------

ze 05.460.2ˆ

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.

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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.

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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/.

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

or the University of London International Programmes distance learning course

EC2020 Elements of Econometrics

www.londoninternational.ac.uk/lse.

2012.11.03

linearity and nonlinearity 1 this sequence introduces the topic of fitting nonlinear regression...

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