linearity and nonlinearity 1 this sequence introduces the topic of fitting nonlinear regression...
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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
Linear in variables and parameters:
It is also linear in parameters since a different parameter appears as a multiplicative factor in each term.
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uXXXY 4433221
Linear in variables and parameters:
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:
Linear in variables and parameters:
Such models present no problem at all. Define new variables as shown.
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uXXXY 4433221
uXXXY 44332221 log
4433222 log,, XZXZXZ
Linear in parameters, nonlinear in variables:
Linear in variables and parameters:
With these cosmetic transformations, we have made the model linear in both variables and parameters.
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uXXXY 4433221
uXXXY 44332221 log
4433222 log,, XZXZXZ
uZZZY 4433221
Linear in parameters, nonlinear in variables:
Linear in variables and parameters:
Nonlinear in parameters:
uXXXY 4433221
uXXXY 44332221 log
4433222 log,, XZXZXZ
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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Linear in parameters, nonlinear in variables:
Linear in variables and parameters:
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 ------------------------
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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.
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2012.11.03