Hayden Economics

Intrinsically Linear And Intrinsically Nonlinear Regression Models

When we started our discussion of linear regression models in Chapter 2, we stated that our concern in this book is basically with models that are linear in the parameters; they may or may not be linear in the variables. If you refer to Table 2.3, you will see that a model that is linear in the parameters as well as the variables is a linear regression model and so is a model that is linear in the parameters but nonlinear in the variables. On the other hand, if a model is nonlinear in the parameters it is a nonlinear (in-the-parameter) regression model whether the variables of such a model are linear or not.

1We noted in Chap. 4 that under the assumption of normally distributed error term, the OLS estimators are not only BLUE but are BUE (best unbiased estimator) in the entire class of estimators, linear or not. But if we drop the assumption of normality, as Davidson and MacKinnon note, it is possible to obtain nonlinear and/or biased estimators that may perform better than the OLS estimators. See Russell Davidson and James G. MacKinnon, Estimation and Inference in Econometrics, Oxford University Press, New York, 1993, p. 161.

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However, one has to be careful here, for some models may look nonlinear in the parameters but are inherently or intrinsically linear because with suitable transformation they can be made linear-in-the-parameter regression models. But if such models cannot be linearized in the parameters, they are called intrinsically nonlinear regression models. From now on when we talk about a nonlinear regression model, we mean that it is intrinsically nonlinear. For brevity, we will call them NLRM.

To drive home the distinction between the two, let us revisit exercises 2.6 and 2.7. In exercise 2.6, Models a, b, c, and e are linear regression models because they are all linear in the parameters. Model d is a mixed bag, for fa2 is linear but not ln p1. But if we let a = lnfa1, then this model is linear in a and fa.

In exercise 2.7, Models d and e are intrinsically nonlinear because there is no simple way to linearize them. Model c is obviously a linear regression model. What about Models a and b? Taking the logarithms on both sides of a, we obtain ln Yi = fa + faXi + ui, which is linear in the parameters. Hence Model a is intrinsically a linear regression model. Model b is an example of the logistic (probability) distribution function, and we will study this in Chapter 15. On the surface, it seems that this is a nonlinear regression model. But a simple mathematical trick will render it a linear regression model, namely,

Therefore, Model b is intrinsically linear. We will see the utility of models like (14.1.1) in the next chapter.

Consider now the famous Cobb-Douglas (C-D) production function.

Letting Y = output, X2 = labor input, and X3 = capital input, we will write this function in three different ways:

where a = ln fa1. Thus in this format the C-D function is intrinsically linear. Now consider this version of the C-D function:

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