Lecture 19-2005 - Limitations, Aggregation, and Constraints...

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Limitations, Aggregation, and Constraints Lecture X I. Limitations to Flexible Functional Forms A. The limitations of Flexible Functional Forms, particularly with respect to the limitations imposed by the Taylor series expansion varieties can be demonstrated in several ways. 1. Chambers demonstrates the limitations of the functional forms based on limitations in imposing separability. 2. These arguments are similar to arguments related to imposing separability on various demand systems (i.e. the AIDS models). B. I prefer to demonstrate the limitations to Flexible Function Forms by resorting to the basic notions behind the Taylor Series expansion on which it is based. Specifically, as presented last time: () ( ) ( ) () ( ) 0 0 0 2 2 00 2 0 1 1 2 1 ! xx i i i i fx fx fx ix = = = = ∂∂ =+ + + =− L 0 1. Thus, for an infinite expansion, or for a true n th order polynomial, this expression is exact. Otherwise, we know that: ( ) ( ) ( ) [] 0 23 3 2 * 0 * 0 11 26 for some , f xf x x x x x x x x x = == + + 2. Focusing on the “residual term” ( ) () [] 0 3 3 ** 0 3 1 for some , 6 x x x x x ε = 1
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AEB 6184 – Production Economics Lecture XIX Professor Charles Moss Fall 2005 a. As long as the third derivative of the function is non-zero at the point of approximation, we know that the Flexible Functional Form has a “specification” or “approximation” error. b. Further, if we bring this concept together with our typical notions of sampling theory, this approximation error may confound the estimation of parameters. C. Finally, there is a problem with the estimation of a functional form and the point of approximation. Implicitly, if one estimates the quadratic cost function, we parameterize the system based on approximations from the arithmetic average. Similarly, if the Translog is used, the approximation is from the sample’s geometric average.
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Lecture 19-2005 - Limitations, Aggregation, and Constraints...

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