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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 nonzero 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 costfunction is not concave, then taking linear
combination in price space could further reduce cost.
Thus, a nonconcave 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 nonsingular 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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