Lecture19-2011 - Basic Functions Aggregation Issues...

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Unformatted text preview: Basic Functions Aggregation Issues Imposing Restrictions Limits, Aggregation, and Constraints: Lecture XIX Charles B. Moss1 1 University of Florida November 1, 2011 Charles B. Moss Limits, Aggregation, and Constraints: Lecture XIX 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 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 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 ) = f ( x0 ) + ( x − x0 ) + ∂ x ￿x =x 0 ￿ 1 ∂ 2 f (x ) ￿ ￿ ( x − x0 ) + · · · 2 ∂ x 2 ￿x =x 0 ￿ ∞ ￿ 1 ∂ i f (x ) ￿ ￿ = ￿ i ! ∂x i ￿ i =1 x =x 0 (1) ( x − x0 ) i Truncating the series to the third moment ￿ ∂ f (x ) ￿ ￿ f ( x ) = f ( x0 ) + ( x − x0 ) + ∂ x ￿x =x 0 ￿ ￿ 1 ∂ 2 f (x ) ￿ 1 ∂ 3 f (x ) ￿ ￿ ￿ ( x − x0 ) + for some x ∗ ∈ [x , x0 ] . 2 ∂ x 2 ￿x =x0 Charles B. Moss 6 Limits, 3 ￿x =x0 and Constraints: Lecture XIX ∂ x Aggregation, Basic Functions Aggregation Issues Imposing Restrictions Error in Approximation vs. Residual Point of Estimation Focusing on the ”residual term” ￿ 1 ∂ 3 f (x ) ￿ ￿ ￿ (x ) = (x − x ∗ )3 for some x ∗ ∈ [x , x0 ] 6 ∂ x 3 ￿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. Further, if we bring this concept together with our typical notions of sampling theory, this approximation error may confound the estimation of parameters. Charles B. Moss Limits, Aggregation, and Constraints: Lecture XIX Basic Functions Aggregation Issues Imposing Restrictions Error in Approximation vs. Residual Point of Estimation Point of Estimation Finally, there is a problem with the estimation of a functional form and the point of approximation. If one estimates the quadratic cost function, we parameterize the system based on approximations from the arithmetic average. If the Translog is used, the approximation is from the samples geometric average. This raises problems from two perspectives. First, the sample average may not adequately represent a relevant production point. Second, this point of approximation plays into outlier problems. Charles B. Moss Limits, Aggregation, and Constraints: Lecture XIX Basic Functions Aggregation Issues Imposing Restrictions Parsimony in Econometric Models Aggregation Issues Again, issues of aggregation can be addressed at several different levels. One level of aggregation involves the use of a single cost function to depict decisions of numerous farmers. One assumption is that farmers all face similar production functions. Second assuming that all farmers face the same production function, but possess heterogeneous unobserved inputs such as human capital. Charles B. Moss Limits, Aggregation, and Constraints: Lecture XIX Basic Functions Aggregation Issues Imposing Restrictions Parsimony in Econometric Models An extension to the heterogeneity issue can be found if we parameterize an aggregate cost function. Capalbo and Denny (AJAE, 1986) examine the impact of changes in technology on U.S. agricultural production using a cost function approach. 1 c (w , y ) = α0 + α￿ w + w ￿ Aw + β ￿ y + 2 (4) 1￿ y By + w ￿ Γy + θt 2 In this formulation, θ can be used to estimate the impact of changes in technology through time. However, to estimate this model we must assume that there exists an aggregate cost function. In other words, we could assume that agriculture in the United States is controlled by a single entity that minimizes cost. Alternatively, we could assume that the minimizing behavior of each individual is the same as an aggregate minimization. Charles B. Moss Limits, Aggregation, and Constraints: Lecture XIX Basic Functions Aggregation Issues Imposing Restrictions Parsimony in Econometric Models Parsimony in Econometric Models As mentioned in earlier lectures, a key element in the estimation of cost functions is parsimony. In general, the number of parameters in a quadratic system is (n + m + 1) (n + m) + (n + m ) 2 where n is the number of inputs, and m is the number of outputs. (5) For accounting purposes in farm level datasets and for degrees of freedom difficulties in when aggregate data is used, we often aggregate inputs and/or outputs. We may aggregate diesel, gasoline, and L.P. gas into a single fuel category. Adding to this we may aggregate fertilizer with fuel to form an agricultural chemical component. In each of these cases, we make fixed factor assumptions between the aggregated inputs. Charles B. Moss Limits, Aggregation, and Constraints: Lecture XIX Basic Functions Aggregation Issues Imposing Restrictions Parsimony in Econometric Models Given that these aggregation issues exist, what can be done? One alternative would be to give up on applied work altogether. Another alternative is to use the best data possible, but take a more Bayesian approach When do the results look right? Charles B. Moss Limits, Aggregation, and Constraints: Lecture XIX Basic Functions Aggregation Issues Imposing Restrictions Homogeneity Symmetry Concavity Given the development of the cost function, we are particularly interested in imposing three general conditions on the estimated parameters: Homogeneity, Symmetry, and Concavity. Charles B. Moss Limits, Aggregation, and Constraints: Lecture XIX Basic Functions Aggregation Issues Imposing Restrictions Homogeneity Symmetry Concavity Homogeneity The cost function is homogeneous of degree one in prices and the demand functions are homogeneous of degree zero in prices. The homogeneity restrictions are typically given by N ￿ αi = 1 i =1 N ￿ j =1 Aij = 0∀i (6) Back to the original definition, by Euler’ theorem, we know that λt c ( w , y ) = N ￿ ∂ c (w , y ) i =1 Charles B. Moss ∂ wi wi (7) Limits, Aggregation, and Constraints: Lecture XIX Basic Functions Aggregation Issues Imposing Restrictions Homogeneity Symmetry Concavity Continued With the cost function homogeneous of degree one t = 1 so 1= N ￿ ∂ c (w , y ) i =1 ∂ wi wi c (w , y ) (8) Or the factor elasticities of the cost function with respect to inputs prices sum to one. However, substituting the factor demands from the cost function back into this expression yields 1= N ￿ i =1 [ α i + A ·i w + Γ ￿ · y ] i Charles B. Moss wi c (w , y ) (9) Limits, Aggregation, and Constraints: Lecture XIX Basic Functions Aggregation Issues Imposing Restrictions Homogeneity Symmetry Concavity Continued Let’s simplify this structure by choosing both A and Γ to be singular so that Aw = 0 for all w and Γ￿ y = 0 for all y . In this case Equation 9 becomes 1= N N ￿ α i wi ￿ ⇒ c (w , y ) = α i wi c (w , y ) i =1 (10) i =1 To guarantee this restriction, we divide through by one of the prices (say wN ) yielding N −1 N −1 ￿ wi c (w , y ) c ( w , y ) ￿ wi = αi + αN → αN = − αi wN wN wN wN i =1 i =1 (11) For the Translog approximation, the well known αN = (ln (c (w , y )) − ln (wN )) − Charles B. Moss N −1 ￿ i =1 αi (ln (wi ) − ln (wN )) (12) Limits, Aggregation, and Constraints: Lecture XIX Basic Functions Aggregation Issues Imposing Restrictions Homogeneity Symmetry Concavity Symmetry Symmetry is a standard linear restriction. Aij = Aji ∀i , j Charles B. Moss (13) Limits, Aggregation, and Constraints: Lecture XIX Basic Functions Aggregation Issues Imposing Restrictions Homogeneity Symmetry Concavity Concavity As we have discussed concavity is a result of optimizing behavior. If the cost-function is not concave, then taking linear combination in price space could further reduce cost. Thus, a non-concave cost function is inconsistent with economic theory. Two problems: Imposing concavity and rejecting concavity. Charles B. Moss Limits, Aggregation, and Constraints: Lecture XIX Basic Functions Aggregation Issues Imposing Restrictions Homogeneity Symmetry Concavity Three Approaches to Imposing Concavity Two methods impose concavity with an equality The Lau Decomposition (Featherstone and Moss [1994]) ￿ x ￿ Ax = x ￿ P ￿ Px = (Px ) (Px ) ≥ 0 (14) This makes the estimation very nonlinear a11 P= 0 0 a11 ⇒ P ￿ P = a12 a13 0 a22 a23 Charles B. Moss a12 a22 0 a13 a23 a33 0 a11 0 0 0 0 a12 a22 0 a13 a23 a33 (15) Limits, Aggregation, and Constraints: Lecture XIX Basic Functions Aggregation Issues Imposing Restrictions Homogeneity Symmetry Concavity Continued Continuing the matrix operation a11 a13 a12 a13 + a22 a23 a13 a13 + a23 a23 + a33 a33 (16) The alternative approach is to use the fact that a positive definite symmetric matrix has all positive eigenvalues and a negative definite symmetric matrix has all negative eigenvalues. a11 a11 P ￿ P = a11 a12 a11 a13 a11 a12 a12 a12 + a22 a22 a12 a13 + a22 a23 Thus, we could simply constrain max [λi ] ≤ 0 ￿: λ = eigenvalue (A) (17) The non-singular method is by bootstrapping the estimation and retaining the estimates that obey Equation 17. Charles B. Moss Limits, Aggregation, and Constraints: Lecture XIX ...
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