Reducing Approximation Error in the Fourier Flexible Functional Form

Abstract

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.

Farm‐level determinants of product conversion: Organic milk production

“…To increase the robustness of our first-stage estimation to problems associated with functional form misspecification, we use the BCF functional form of Skolrud (2017). In stochastic frontier analysis, the original Fourier-flexible form as proposed by Gallant (1981) has often been used as a more robust alternative to the popular translog functional form (Berger & DeYoung, 1997;Berger & Mester, 1997;Fenn, Vencappa, Diacon, Klumpes, & O'Brien, 2008).…”

“…In stochastic frontier analysis, the original Fourier-flexible form as proposed by Gallant (1981) has often been used as a more robust alternative to the popular translog functional form (Berger & DeYoung, 1997;Berger & Mester, 1997;Fenn, Vencappa, Diacon, Klumpes, & O'Brien, 2008). However, this form-which we refer to as the translog Fourier-flexible form (TLF) due to the nesting of the translog-has significant approximation problems at the lower and upper tails of the data (Skolrud, 2017). To circumvent this problem, we employ a version of the BCF functional form that has been shown to ensure the robustness of the Fourier-flexible form with improved approximation properties at the boundaries of the data (Skolrud, 2017).…”

“…However, this form-which we refer to as the translog Fourier-flexible form (TLF) due to the nesting of the translog-has significant approximation problems at the lower and upper tails of the data (Skolrud, 2017). To circumvent this problem, we employ a version of the BCF functional form that has been shown to ensure the robustness of the Fourier-flexible form with improved approximation properties at the boundaries of the data (Skolrud, 2017). The BCF functional form is composed of a truncated Fourier-series appended to a second-order expansion in Box-Cox polynomials:…”

“…Second, there is evidence that the relationship between TE and size is similar between Canada and the United States, that is, that TE is highest for the largest dairy farms (Slade & Hailu, 2016), suggesting that smaller, less efficient farms in both countries may show a higher propensity to convert. 2 This article is also the first to employ the new Box-Cox Fourier (BCF) functional form of Skolrud (2017) for a multi-output input distance function. Previous researchers seeking a more robust functional form than the translog for stochastic frontier analysis used the Fourier-flexible form of Gallant (1981Gallant ( , 1982, which allows for a global approximation to an unknown data generating process.…”

“…However, due to approximation problems at the tails of the data, authors resorted to data truncation, introducing a source of bias into their results (Berger & Mester, 1997). The BCF functional form retains the robustness of the original Fourier-flexible form without the associated boundary approximation errors (Skolrud, 2017).…”