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Online Econometrics Textbook - Regression Extensions - Assumption Violations of Linear Regression - Nonlinearities in Linear Regression

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III.I.3 Nonlinearities in Linear Regression

If the linearity assumption of OLS (II.I.3-1) or (II.II.1-2) is violated it is sometimes possible to solve this problem by using transformed series in stead of the original series. The simplest transformation is the natural logarithm which can be applied if we have to estimate for instance COBB-DOUGLAS-like equations

Online Econometrics Textbook - Regression Extensions - Assumption Violations of Linear Regression - Nonlinearities in Linear Regression


Transforming (III.I.3-1) yields thus


which is now a (log)linear relationship. Therefore we can, by first transforming each series, apply OLS to (III.I.3-2) without any problems.

Another nice result of the loglinear specification is that the sensitivity parameters alpha and beta can now be interpreted as elasticities. To illustrate this, assume an equation


which can be rewritten as


The elasticity of changes in X affecting Y is defined as


We find the sensitivity parameter of (III.I.3-4) as


and now the remaining question is


which can be obviously shown to be true since (III.I.3-7) is equivalent with


Another possible solution to the nonlinearity problem is substitution of the variables. This method can only be applied if there is a nonlinearity in the variables but not in the parameters.

A simple example will illustrate this substitution method.

Assume an equation



which solves the problem.

Finally consider an equation which cannot be transformed and which is nonlinear in the parameters


which is quite hard to solve with a least squares formula, let alone impossible to solve with OLS.

In general there exists no formula which can be applied to likewise equations. In stead, another method should be used: the nonlinear least squares (mostly by means of an iterative MLE algorithm).

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