‎Why Linear (and Piecewise Linear) Models Often Successfully Describe Complex Non-Linear Economic‎ ‎and Financial Phenomena‎: ‎A~Fuzzy-Based Explanation

Abstract

Economic and financial phenomena are highly complex and‎ ‎non-linear‎. ‎However‎, ‎surprisingly‎, ‎in many cases‎, ‎these phenomena‎ ‎are accurately described by linear models‎ -- ‎or‎, ‎sometimes‎, ‎by‎ ‎piecewise linear ones‎. ‎In this paper‎, ‎we show that fuzzy‎ ‎techniques can explain the unexpected efficiency of linear and‎ ‎piecewise linear models‎: ‎namely‎, ‎we show that a natural‎ ‎fuzzy-based precisiation of imprecise (``fuzzy'') expert knowledge‎ ‎often leads to linear and piecewise linear models‎.‎We show this by applying invariance ideas to analyze which membership functions‎, ‎which fuzzy ``and''-operations (t-norms)‎, ‎and which‎ ‎fuzzy implication operations are most appropriate for applications to economics and finance‎. ‎We also discuss which expert-motivated nonlinear models should be‎ ‎used to get a more accurate description of economic and financial‎ ‎phenomena‎: ‎specifically‎, ‎we show that a natural next step is to add cubic‎ ‎terms to the linear (and piece-wise linear) expressions‎, ‎and‎, ‎in general‎, ‎to consider polynomial (and piece-wise polynomial) dependencies‎.

Keywords

• linear models‎
• ‎piece-wise linear models‎
• ‎fuzzy logic‎
• ‎economics and finance‎