Lecture19-2011 - Basic Functions Aggregation Issues...

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Basic Functions Aggregation Issues Imposing Restrictions Limits, Aggregation, and Constraints: Lecture XIX Charles B. Moss 1 1 University of Florida November 1, 2011 Charles B. Moss Limits, Aggregation, and Constraints: Lecture XIX
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Basic Functions Aggregation Issues Imposing Restrictions 1 Basic Functions Error in Approximation vs. Residual Point of Estimation 2 Aggregation Issues Parsimony in Econometric Models 3 Imposing Restrictions Homogeneity Symmetry Concavity Charles B. Moss Limits, Aggregation, and Constraints: Lecture XIX
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Basic Functions Aggregation Issues Imposing Restrictions Error in Approximation vs. Residual Point of Estimation Limitations to Flexible 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. 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. Charles B. Moss Limits, Aggregation, and Constraints: Lecture XIX
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Basic Functions Aggregation Issues Imposing Restrictions Error in Approximation vs. Residual Point of Estimation Error in Approximation vs. Residual Returning to the Taylor series expansion f ( x ) = f ( x 0 ) + f ( x ) x x = x 0 ( x x 0 ) + 1 2 2 f ( x ) x 2 x = x 0 ( x x 0 ) + · · · = i =1 1 i ! i f ( x ) x i x = x 0 ( x x 0 ) i (1) Truncating the series to the third moment f ( x ) = f ( x 0 ) + f ( x ) x x = x 0 ( x x 0 ) + 1 2 2 f ( x ) x 2 ( x x 0 ) + 1 6 3 f ( x ) x 3 x = x 0 for some x [ x , x 0 ] . (2) Charles B. Moss Limits, Aggregation, and Constraints: Lecture XIX
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Basic Functions Aggregation Issues Imposing Restrictions Error in Approximation vs. Residual Point of Estimation Focusing on the ”residual term” ( x ) = 1 6 3 f ( x ) x 3 x = x 0 ( x x ) 3 for some x [ x , x 0 ] (3) 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.
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