Selecting Non-Linear Models
Selecting Non-Linear Models
The selection of a non-linear model often begins from a previous linear model and adds non-linear terms. Such an approach is specific-to-general in two respects. First, between studies, advances are bound to be generalizations as new knowledge accumulates, which is in part why scientific progress is so difficult. Second, however, one should not just extend the best earlier model, which was implicitly selected to accommodate all omitted effects, but commence with an identified and congruent general non-linear approximation which enters all the linear terms unrestrictedly and includes a complete set of impulse indicators so that non-linearities do not mis-represent breaks or outliers. As a prior step, a test for non-linearity can check whether any extension is needed. We then use squares, cubics, and exponentials of the principal components of the variables to approximate a range of possible non-linearities. Once a selection has been made therefrom, if non-linear terms remain, any proposed theory-based functions (such as logistic or squashing) can be entered to check if they further simplify the approximation. Such an approach avoids the issue of lack of identification under the null and directly tests that the postulated functions are valid reductions.
Keywords: Non-linearities, principal components, approximating functions, encompassing
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