Abstract
The Fourier Flexible form provides a global approximation to an unknown data generating process. In terms of limiting function specification error, this form is preferable to functional forms based on second-order Taylor series expansions. The Fourier Flexible form is a truncated Fourier series expansion appended to a second-order expansion in logarithms. By replacing the logarithmic expansion with a Box-Cox transformation, we show that the Fourier Flexible form can reduce approximation error by 25% on average in the tails of the data distribution. The new functional form allows for nested testing of a larger set of commonly implemented functional forms.
Highlights
Functional form selection can be difficult in applied work, especially in cases where economic theory is not a useful guide
With a second-order approximation, the TL is rejected in favor of the Box-Cox Fourier (BCF) at the 1% level of significance for all data generating processes, while the TLF is rejected in favor of the BCF at the 5% level for one data generating process and at the 1% level for the remaining two
We have demonstrated that the Fourier flexible form (TLF) of Gallant (1981, 1982) suffers from serious approximation bias at the boundaries of the data
Summary
Functional form selection can be difficult in applied work, especially in cases where economic theory is not a useful guide. From the early 1970’s, Diewert-flexible forms such as the generalized Leontief, quadratic (normalized, square-rooted and symmetric) and translog have dominated applied parametric analysis These functional forms have many desirable properties, placing no restrictions on derived measures that are functions of their first and second derivatives (Creel 1997).. Unless the true function subject to approximation happens to be in the same family as the approximating function, least squares will not consistently estimate the true value of the function in a global sense (White 1980). This limitation was addressed by Gallant (1981, 1982) Fourier flexible form. Based on the composition of a truncated Fourier series expansion of orthogonal polynomials and a second-order
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