Non-linear Methods in Econometrics

Abstract

Economic theory guides empirical research primarily by suggesting which variables ought to enter a relationship. But as to the functional form that this relationship ought to take, it only gives general information such as stating that certain first and second partial derivatives of a relationship must be positive or such as ruling out certain functional forms. In some applications, notably consumer demand systems, the theory rules out models that are linear in the parameters such as $y=\sum {x}_i{\beta}_i+e$ and thus provides a natural impetus to the development of statistical methods for models that are non-linear in the parameters such as

$$ y=\left(\sum {x}_i{\beta}_i\right)/\left(\sum {x}_i{\gamma}_i-1\right)+e. $$

A more subtle but more profound influence in the same direction is exerted by the converse aspect of suggesting what variables ought to enter a relationship, that is variables not suggested ought not be present. Thus, when searching for a model that explains data better than an existing model, one will prefer a more complicated model involving only the suggested variables to a model of equal complexity in additional variables. One will inevitably fit models to data that are nonlinear in the parameters during the search.