The Local Approximation View

To talk of ”approximation“ suggests that the concept of a flexible functional form is gene-rally understood in distinction to an underlying ”true“ data generation process whose functio-nal form and parameters are principally unknown. The researcher postulates a micro-theoretic model and a functional form, and then he evaluates the functional form’s parameters in a sta-tistical estimation procedure. About the relation between the supposed true function and the corresponding flexible estimation function three concurring hypotheses are possible:

1. The estimation function is a local approximation of the true function.

The approximation properties of flexible functional forms are only locally valid, i.e. re-stricted to a single point.¹²⁸ Thus, it can in no way be claimed that the assumed true function is approximated in its totality: value, gradients and Hessian of true and estimated function are equal merely at one single point – the point of approximation. Therefore, the estimated

126 SeeFUSS/MCFADDEN/MUNDLAK 1978: 232.

127 See CHAMBERS 1988: 170-171 for the single-output case.

128 See figure 2 below.

meters must be interpreted only locally. This considerably restricts the forecasting capabilities using these parameters if the variable values in the forecast period are relatively distant from the approximation point. In the immediate neighbourhood of the approximation point each flexible functional form provides theoretically consistent parameters if the true structure is theoretically consistent,¹²⁹ because the parameters of the approximating function and those of the true structure are identical at the approximation point, which is exemplified in figure 2 below where a known, globally consistent cost structure is locally approximated at two diffe-rent points P and Q. Since e.g. local concavity is a necessary condition for global concavity, i.e. concavity in the entire domain of the estimation function or all possible variable values, respectively, at least the necessary condition of the concavity hypothesis for the postulated true structure can be tested at the approximation point by means of the concavity of the esti-mation function – which, of course, must not be intrinsically concave.¹³⁰

But which point is the point of approximation, where is it located?¹³¹ Rather than approxi-mating a known algebraic structure at one point as depicted in figure 2, the estimation proce-dure fits the approximating function to a data sample over a more or less extended range, as figure 3 below elucidates. There is no way to infer from the approximation function to the location of the approximation point, or, more precisely, in a statistics context where the ap-proximated structure is intrinsically unknown the concept of an approximation point runs empty at all. Consequently, since there is no way to infer from an estimated function to the properties of the underlying true structure, if the point where the properties of the approxima-ting function are known is not the approximation point, the properties of the assumed true function remain unknown. From the rejection of the null hypothesis that the approximating function shows all properties of a well-behaved cost function, it can be followed that the exa-mined economic subjects are not cost minimizers – or that the test was not conducted at the approximation point. Hence, testing the null hypothesis is useless.

Commonly, the point of approximation is held to be located at some mean of variables o-ver all observations.¹³² However, this view emanates from erroneously interpreting the point of approximation and the point of expansion of e.g. a Taylor series as synonyms. All second order flexible functional forms can be interpreted as second order differential or Taylor series

129 See FUSS/MCFADDEN/MUNDLAK 1978: 233-236; CHAMBERS 1988: 177-180.

130 See LAU 1978: 418-420.

131 See MOREY 1986: 227 who raises this question.

132 See WHITE 1980: 150. For a recent example, see BARNETT/KIROVA/PASUPATHY 1996: 16.

approximations,¹³³ where the point of expansion is identical with the point of approximation of a known algebraic function. But the point of expansion, which is generally the point where all exogeneous variables assume unity value for a standard formulation of a flexible form, implies nothing about the approximation properties of the resulting specification with respect to a random sample, rather than to a known algebraic structure. Salvanes and Tjøtta prove the obvious: the estimated function is invariant with respect to the point of expansion.¹³⁴ This is, the estimation function, transformed to establish another point of expansion, yields the same graph because the estimated parameters exactly compensate the transformation, which is ob-vious because after all the same specification fits the same random sample. However, their sorrowful conclusion that second order flexible functional forms do not necessarily perform well at the point of approximation, because this point could be located outside the regular region, is groundless: there is no such point in an econometric estimation of a flexible functi-onal form.

2. The estimation function and the true structure are assumed to be of the same functional form but show the desired properties merely locally.¹³⁵

This hypothesis accounts for the fact that most common flexible cost functions can either not be restricted to a well-behaved cost function without losing their flexiblility or cannot be restricted to regularity at all, as will be shown in the next chapter. All flexible functional forms can both be understood as a local approximation of an unknown structure or as a postu-lated functional form of the true structure. The latter is advantageous insofar as regularity can be tested at any data point, since the whole function is viewed as an approximation. Therefo-re, points of interest in the true structure can be examined by testing the respective points in the estimation function. But the uncertainty remains whether the estimation function and the-reby the true structure is still consistent with the properties of a well-behaved cost function if the data set does not accidentally equal a data set already examined. This problem could only be solved by systematically testing all possible data sets. This procedure, which is hardly ele-gant, is avoided with the assumption of another hypothesis:

133 See section 5.1.1 above. For a description of flexible functional forms as differential approximation or Taylor series approximation see CHAMBERS 1986: 162-164.

134 See SALVANES/TJØTTA 1995.

135 See MOREY 1986: 222-227.

3. The estimation function and the true structure are assumed to be of the same functional form and show the desired properties globally.¹³⁶

If one suceeds in finding a flexible functional form that can be restricted to global regulari-ty without losing its flexibliregulari-ty, i.e. a form which has the advantage that local properties are the necessary and sufficient condition for the respective global properties, another possibility ari-ses: only those functional forms allow one to infer from the estimation function to the true stucture and hence allow meaningful tests of significance, because otherwise the model lacks theoretical foundation.¹³⁷ Hence, a serious problem arises for the postulates of economic theo-ry if a properly specified flexible cost function which is globally well-behaved is not suppor-ted by the data. An unjustified rejection of the null hypothesis could be caused by missing, undersized, badly measured or ill-conditioned data, by a validation method that is inappropriate, or by misspecification. Misspecification includes e.g. omission of relevant vari-ables, a wrong functional form, or wrong modeling assumptions like separability and non-jointness hypotheses.¹³⁸ But, if one insists on the model being properly specified and the data being sufficient, it can be either followed that, claiming unlimited validity of neoclassical e-conomic theory, the ee-conomic subjects are not cost minimizers or profit maximizers,¹³⁹ i.e.

behave irrationally,¹⁴⁰ or that, allowing for multiple theories of economic behavior, they act rationally according to another rationality concept, i.e. neoclassical economic theory is not applicable in that case (or, taking the easy way out, they behave irrationally although there may exist other theories of economic rationality that are not considered).

The last approach of a flexible functional form does not labour under the illusion that the true structure of the data generating process can be locally approximated in analogy to the

136 See MOREY 1986: 219-222.

137 See sections 6.2.4 and 6.3 below.

138 Another possible misspecification could consist of the assumption that a second order flexible functional form is flexible enough really to provide an exhaustive characterization of technology [see section 5.1.4 be-low].

139 This conclusion not only assumes a well-behaved true profit or cost structure, it moreover presupposes a definite solution of profit or cost optimizing behavior. Thus, an underlying technology with affine pieces and the resulting multiple solutions must be excluded to allow a test of theory.

140 The identification of non-cost-minimizing or non-profit-maximizing behavior with irrationality [see MOREY

1986: 229] is of course only valid if the respective goal function is postulated. However, the argumentation remains the same when other optimization criteria are assumed. These generally imply other specific proper-ties of the respective behavioral function whose violation must be interpreted as irrationality too.

approximation of a known complex algebraic structure with the simpler Taylor series appro-ximation as suggested in figure 2 – in econometrically validated models the true structure is merely present through a random sample of observations, which is depicted in figure 3. Ra-ther, it promotes a concept of flexibility where the functional form – whose postulation is, in any case, an unavoidable component of economic modeling – just has to fit the data to the greatest possible extent, subject only to the regularity conditions following from economic theory and otherwise independently depicting all economically relevant aspects. The argu-ment that any flexible functional form can approximate any other flexible functional form and any arbitrary data generation process does not suspend the researcher from the issue of redu-cing the specification error to the greatest possible extent in selecting the most appropriate functional form for the entire data. In empirical applications and Monte Carlo simulations this has found its manifestation in the different ability of flexible functional forms to fit different technologies.¹⁴¹

141 See TERRELL 1995: 2 and the literature cited there.

c(w) local second order approximation at P

true cost function

P

local second order approximation at Q Q

w

(Figure 2)

c(w)

sample generated by the same true cost function second order flexible

estimation function

local second order approximation at Q

w

(Figure 3) (Figure 3)