Estimation of Montana winter wheat response to nitrogen and precipitation : an application of the Fourier flexible functional form
The principal focus of this study was to use a Fourier function, a specific flexible functional form, to test and reexamine two-dimension yield-input relationships which have been explored previously by agricultural economists. Based on previous literature, three debated crop response functions, i.e., von Liebig, quadratic and Mitscherlich-Baule, their estimation, and their pros and cons were reviewed regarding the matter of whether plant nutrients (including water) are substitutable factors in crop production response functions. The fundamental issue is the nature of the isoquant pattern exhibited. A flexible functional form was used in this research. Global flexibility, Sobolev-flexibility, appears to be a more attractive criterion than local flexibility, Diewert-flexibility. A Fourier-flexible form is the only form that has been shown to be the Sobolev-flexible which can endow a parametric statistical methodology with the semi-non-parametric property. In the estimation process of the Fourier functional form, establishing an acceptable relationship for yield response of winter wheat to fertilizers and various soil and climatic factors failed because of the high noise level in the 900-observation data set. The attempt to delete ill-conditioned plots did not improve the performance of the model. Therefore, the last step in estimation was based on a "stylized" data set containing 388 arbitrarily-trimmed observations. Since the Fourier function is very sensitive to the data, one should use this highly flexible form only when a data set with normal statistical properties is available. Once armed with a good data set, the Fourier function should perform a fairly good approximation to a differentiable function. Finally, the results using the stylized data set seem to support the hypothesis of the Mitscherlich-Baule for nitrogen, i.e., a smooth upward curve followed by a plateau as suggested by Frank, Beattie and Embleton (1989). However, given that the data were stylized, nothing definitive can be said in this regard.