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

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Limitations,  Limitations,  Aggregation, and  Aggregation, and  Constraints Constraints Lecture XIX Lecture XIX
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Limitations to Flexible  Limitations to Flexible  Functional Forms Functional Forms 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. Chambers demonstrates the limitations of the  functional forms based on limitations in  imposing separability.
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These arguments are similar to arguments  related to imposing separability on various  demand systems (i.e. the AIDS models). 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.
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( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 0 0 0 2 2 0 0 0 2 0 1 1 2 1 ! x x x x i i i i x x f x f x f x f x x x x x x x f x x x i x = = = = = + - + - + = - L ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 [ ] 0 0 0 2 2 0 0 0 2 3 3 * * 0 3 1 2 1 for some , 6 x x x x x x f x f x f x f x x x x x x x f x x x x x x x = = = = + - + - + -
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Focusing on the “residual term” 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. ( 29 ( 29 ( 29 [ ] 0 3 3 * * 0 3 1 for some , 6 x x f x x x x x x x x ε = = -
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Further, if we bring this concept together with our typical  notions of sampling theory, this approximation error may  confound the estimation of parameters. Finally, there is a problem with the estimation of a 
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Lecture 19-2005 - Limitations, Limitations, Aggregation,and...

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